Elektrodynamik

ElektrodynamikW. Glöckle23. Januar 20031

Inhaltsverzeichnis1 Elektrostatik 41.1 Das Coulombgesetz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Das elektrische Feld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Der Kraftflußsatz und der differentielle Zusammenhang zwischen elektrischemFeld und der Ladungsdichte . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Elektrostatisches Potential und Poissongleichung . . . . . . . . . . . . . . . . . 91.5 Elektrisches Potential und Potential an Grenzflächen . . . . . . . . . . . . . . . 121.6 Eindeutigkeitstheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.7 Elektrostatische potentielle Energie mit Anwendung auf Anordnung von Leitern . 171.8 Multipolentwicklung einer Ladungsverteilung und Energie einer Ladungsverteilungim äußeren Feld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.9 Zweidimensionale Probleme . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.10 Formale Lösung des Randwertproblems mit Hilfe der Greensfunktion . . . . . . 281.11 Die Greensfunktion am Beispiel der leitenden Kugel und einer Punktladung . . 311.12 Dielektrikum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371.13 Randwertproblem für Dielektrika (Methode der Spiegelladungen) . . . . . . . . 451.14 Lösung der Laplacegleichung durch Reihenentwicklung . . . . . . . . . . . . . . 491.14.1 Potential einer Punktladung . . . . . . . . . . . . . . . . . . . . . . . . 511.14.2 Dielektrische Kugel im homogenen Feld . . . . . . . . . . . . . . . . . 521.14.3 Multipolentwicklung einer Ladungsverteilung . . . . . . . . . . . . . . . 541.15 Energiedichte im Dielektrikum . . . . . . . . . . . . . . . . . . . . . . . . . . 552 Magnetfelder stationärer Ströme 572.1 Die magnetische Induktion B und Kräfte zwischen Strömen . . . . . . . . . . . . 572.2 Einfache Anwendungen des Biot - Savart’ schen Gesetzes . . . . . . . . . . . . 622.3 Differentielle Formulierung der Magnetostatik und Ampère’sches Gesetz . . . . 652.4 Vektorpotential und magnetisches skalares Potential . . . . . . . . . . . . . . . . 662.5 Das Magnetfeld einer entfernten Stromschleife . . . . . . . . . . . . . . . . . . 672.6 Potentielle Energie eines magnetischen Dipols im Magnetfeld . . . . . . . . . . 692.7 Magnetische Eigenschaften von Materie . . . . . . . . . . . . . . . . . . . . . . 702.8 Die magnetische Feldstärke H . . . . . . . . . . . . . . . . . . . . . . . . . . . 722.9 Beispiele zur Feldberechnung . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732.9.1 Die gleichförmig magnetisierte Kugel . . . . . . . . . . . . . . . . . . . 732.9.2 Magnetische Abschirmung . . . . . . . . . . . . . . . . . . . . . . . . . 773 Zeitabhängige Felder und die Maxwell-Gleichungen 813.1 Faraday’sches Induktionsgesetz (1831) . . . . . . . . . . . . . . . . . . . . . . . 813.2 Die Energie im magnetischen Feld . . . . . . . . . . . . . . . . . . . . . . . . . 843.3 Der Maxwell’sche Verschiebungsstrom und die Maxwell’schen Gleichungen . . . 863.4 Vektorpotential und skalares Potential, Eichtransformationen . . . . . . . . . . . 892

3.5 Energie und Impuls im elektromagnetischem Feld; Erhaltungssätze von Energieund Impuls für Systeme von Ladungen und elektromagnetischem Feld. . . . . . 924 Strahlung einer lokalisierten oszillierenden Quelle 984.1 Greensfunktionen für zeitabhängige Wellengleichung . . . . . . . . . . . . . . . 984.2 Felder einer oszillierenden Quelle . . . . . . . . . . . . . . . . . . . . . . . . . 1004.3 Dipolstrahlung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035 Freie elektromagnetische Wellen, optische Phänomene, Hohlleiter und Resonatoren1075.1 Ebene Wellen in nichtleitenden Medien . . . . . . . . . . . . . . . . . . . . . . 1075.2 Lineare und zirkulare Polarisation . . . . . . . . . . . . . . . . . . . . . . . . . 1095.3 Reflexion und Beugung elektromagnetischer Wellen an ebener Grenzfläche zwischenzwei Dielektrikas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105.4 Wellen in Leitern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1155.5 Hohlleiter und Resonatoren . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165.5.1 Resonator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216 Relativistische Formulierung 1236.1 Einführung in die spezielle Relativitätstheorie . . . . . . . . . . . . . . . . . . . 1236.2 Vierertensoren und Kovarianz physikalischer Gesetze . . . . . . . . . . . . . . . 1316.3 Die Kovarianz der Maxwellgleichungen . . . . . . . . . . . . . . . . . . . . . . 1366.4 Lorentztransformation des elektromagnetischen Feldes und Anwendung auf dasFeld eines geladenen Teilchens . . . . . . . . . . . . . . . . . . . . . . . . . . . 1406.5 Die Erhaltungssätze für Energie und Impuls . . . . . . . . . . . . . . . . . . . 1456.6 Relativistische Teilchenkinematik und Teilchendynamik . . . . . . . . . . . . . 1486.7 Relativistische Elektrodynamik materieller Körper . . . . . . . . . . . . . . . . 1526.8 Abstrahlung einer relativistisch bewegten Punktladung . . . . . . . . . . . . . . 1603

1 Elektrostatik1.1 Das Coulombgesetz¥¢¡ ¢¡©¨ §¦ ¥§¦£¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤¤ ¤¤¤¤¤¤¤¤¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤£§¦ ¢¡¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤Die Kraft auf die Ladung¥¡infolge der Ladung¥¦ist ¥¦¥¢¡§ ¡¨ ¦ ¡¨ ¦ und die Kraft¥§¦auf¥¢¡infolge¨ istist ¡und .. Die Kraft wirkt in der Verbindungslinie der LadungenGleiche Ladungen wirken abstoßend. Ungleiche Ladungen wirken anziehend.Die Konstante k hängt vom Einheitensystem ab. Für die Wahl wird die Einheitsladungdurch das Gesetz definiert, nachdem Kraft- und Längeneinheiten schon festgelegt sind. Nähereszu Einheiten später.Kräfte von verschiedenen Ladungen addieren sich vektoriell:% % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%% ¥! " ¥ " #¨ " $¨ " £ % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% %%% %%%%% % % % % % % % % % % % % % % % % % % %%% % %%%% % % % % % % % % % % % % % % % % % % % % % % % % % % %%% % % % % %% % % % % % % % % % % % % % % % % % % % %% % %%%% % % % %%%% %% % % % % % % % % % % % % % % % % % % %% % % % %%%%%%%%%%%%%% % % % % % %%% %% %%% % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % %%% %%%% % ¥ % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % £% % % % % %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Dies legt folgende Begriffsbildung nahe : Ladungen¥"Die erzeugen ein Kraftfeld im Raum, dasauf Ladung¥die wirkt.& ¥ 'mit '(& " ¥ " #¨ " $¨ " 4£% % % % % % % % % % % % % % % £¥§¦¥ ¥¢¡

mißt. Später bei zeitlich veränderlichen Vorgängenwerden wir sehen, daß' losgelöst von den Ladungen für sich existiert.Meßvorschrift für' :' ¨ " ¨ ¨ " "1.2 Das elektrische Feld'ist zunächst nur eine Hilfsgröße, da man nur '(¡ £¢¥¤¦¨§©¥Der Limes besagt, daß Probeladung¥die die das Feld erzeugende Ladungen und deren Anordnungnicht stören soll. Im makroskopischen bleibt¥auch im Limes noch makroskopisch,¥d.h.umfaßt immer noch sehr viele Elementarladungen. Im mikroskopischen stößt man¥mit imLimes auf die Elementarladung. Die makroskopische Vorschrift aus 1.1 hat sich bisher stets alsgültig erwiesen.Die Erweiterung der Vorschrift auf kontinuierliche Ladungsverteilungen wird beschrieben durchdie Ladungsdichte : ¨ ¨ Der Aufpunkt kann im Bereich der Ladungsverteilung liegen; dann wird die ¡Singularitätdurch das Volumenelement behoben.Als Spezialfall sind die Punktladungen enthalten : " ¥" " ¥ " ' Wir untersuchen nun die geometrischen Zusammenhänge zwischen' und den erzeugenden Ladungen.1.3 Der Kraftflußsatz und der differentielle Zusammenhang zwischen elektrischemFeld und der LadungsdichteDem elektrischen Feld zur Punktladung¥entspricht der Ausdruck:' ¥ ¥¡ Einheitsvektor Sei eine beliebige geschlossene um¥Oberfläche ,5

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % %% %% %%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % %%%%%%%%% %%%%% %%% % % % % % %%% % % %% % %%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % %%%% %% %%% %% %% %% %%% %% %%%%%%%%%%%%%%% %% %%% %%%% %%% ¥ ' £%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % %% % % % %% %%% %%%%% ¢¤£ ¦¥¢ ¦¥dann ist das gerichtete Oberflächenelement .ist dabei die Normale auf (lokal) und die Größe des Oberflächenelements.Zu bestimmen sei jetzt:§' £ Dies entspricht dem Fluß des elektrischen Feldes durch .¨% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%% %%% %%% %%% %%% %%% %%% %%% %%% %%% %%% %%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % %% %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % %% % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%¦¥©%%%%%%%%%%%%% %%%% %% %% % ¥ % %% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%© ¡ist die Projektion des Flächenelements auf die Fläche . Diese ist ¦¥¢£ , wobei das Raumwinkelelement ist.©¦¥Somit ¦¥§' £ ¥ ¡ © ¢ £ ¡ ¥ ¡ ¡ ¡© ¥Natürlich kann' auch von mehreren Ladungen innerhalb von S herrühren. D.h.,¥ist die in eingeschlossene Gesamtladung. Die Wahl von ist willkürlich, solange¥eingeschlossen ist.Dies ist der Kraftflußsatz:§' £ $! ¥ 6¢

Die Gesamtladung kann sich auch aus einer kontinuierlichen Ladungsverteilung aufsummieren :Dabei liegt ¢ innerhalb von .Wenn keine Ladung umschließt, ist§' £ ¡§' £ Dieser fundamentale Satz der Elektrostatik hängt ab von ¡ '' ¤£undSuperposition der FelderBei einfachen Geometrien läßt sich das' -Feld leicht gewinnen.Kugelförmige Ladungsverteilung:Bei einer kugelförmigen Ladungsverteilung muß das Feld aus Symmetriegründen radial gerichtetsein.Dabei ist¥¦¥%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % %% % %% %% %% %%%% % % % % % % % % % % % % % %% % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%£ ' £ ¡ ¥¦¥ ' ¥§¥¡die Ladung innerhalb einer Kugel vom Radius .Dies hat die Form, als wäre die Ladung im Zentrum der Kugel vereint. Ladungen außerhalb derKugel vom Radius tragen nicht dazu bei.Folge: Das E-Feld innerhalb eines ladungsfreien kugelförmigen Raumes innerhalbeiner kugelförmigen Ladungsverteilung ist Null.7

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Auswertung von' ¨ ¨ gewesen.Langer Draht mit linearer Ladungsdichte :¡%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%¢ ¡ £ ' £ ¡ ' ¢Unendlich ausgedehnte Schicht mit Oberflächenladungsdichte £ :%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 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§% ' £¡¡ £¡div' ! ¨ ¨ )(&Man beachte den verschiedenen Abfall des' -Feldes.Dies ist folgendermaßen zu verstehen:Beim langen Draht kommen die Beiträge von der -Umgebung: ¡ 'Bei der Schicht kommen die Beiträge von¡der -Umgebung:Nun zum differentiellen Zusammenhang zwischen Ladung und Feld:div' ¢ ! ¡¥© ¥ ¢¢ ¡£¢¥¤§¦©¨' £ ¢Da ¢beliebig ist, gilt:Wir zeigen nun, daß die Bestimmung der 3 Komponenten des Vektors'auf die Bestimmungeines skalaren Feldes zurückgeführt werden kann.1.4 Elektrostatisches Potential und PoissongleichungDas elektrische Feld' läßt sich aus der skalaren Größe gewinnen:' ¨ ¨ ' ¨ " ! Was ist nun die Bedeutung von ?¨ Probeladung¥Eine werde im Felddes Weges zu leisten:'von ¢$#nach gebracht; dabei ist folgende Arbeit längs&'¥ ' £ ¨9

¥ ¥ )(& (&&'& 'grad ¥ # ¨ # ist die potentielle Energie von¥in Bezug auf Beachte: %ergab sich als wegunabhängig: ¢£ ¢¤£ £ ¢ ¢ . # £% % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % %%% %% %% %% % % %% % %% % % % %% % % %% % %% %% %% %%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%¡ ¦%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%¢¤£' £ ¨ ¢¦¥Umformung mit Stokes’schem Integralsatz:Da der Weg beliebig ist, folgt:' £ ¨§ ' £ £§ ¢¦¥' £ ' £ £ § ' £ '( £rotD.h. das elektrostatische Feld' ist konservativ.10 ' £ rot¡ ¡

Beispiel: Das Potential ¨ § £¢¡ ¡¨ ¡ ©¡ ¡ ¨ #" " " !©" ¡ ¡ §¤£¡©¤¦ ' £rotfolgt natürlich ebenfalls aus rot grad & £ .Wie ist festgelegt ?'(! div ¨div grad ¨ Dies ist die Poissongleichung.Spezialfall: Im ladungsfreien Raum wird sie zu der Laplace-Gleichung: ¡ ¡ £Das Grundproblem der Elektrostatik ist die Lösung der Poissongleichung unter gegebenen Randbedingungen. für eine räumlich begrenzte Ladungsverteilung Genügt dies der Poissongleichung? ¨ ¡ ¡div grad ¡ div¡ ¨div ¡¥¤¦¡ ¨¢£¨§ ¨ ¡¨ Dies ist offensichtlich gültig füran ¡¦&& etwas Endliches ergibt. Offensichtlich¨ £ . Da der Poissongleichung genügen sollte, muß ¡kann das Ergebnis keine Funktion im gewöhnlichen Sinne sein. Zunächst der Nachweis, daßder Poissongleichung genügt: derart singulär sein, daß das Integral über ¡ ¡ ¡ £Wegen des Volumenelements ergibt eine verschwindend kleine Umgebung von keinenBeitrag zum ersten Integral. Dieser verschwindend kleine Raumbereich kann ausgespart werdenund der Gauß’sche Satz ergibt für eine beliebig große Oberfläche :£ ¨ ¡ © ¨ ¡ £ © ! ¡©11 ¨

genügt also der Poissongleichung. Das gesuchte singuläre Verhalten muß demnachsein.¡ ¨ ¨ ¨ Die rechte Seite läßt sich als Ladungsverteilung einer Punktladung, lokalisiert bei , auffassen,und die ganze Gleichung als Poissongleichung für das Potential dieser speziellen Ladungsvertei- resultiert daraus durch Superpositi-lung. Das Potential für beliebige Ladungsverteilungenon: ¨ ¨ (Linearität der Poissongleichung)1.5 Elektrisches Potential und Potential an GrenzflächenEs sei eine Ladung in einer Schicht an der Oberfläche eines Körpers angesammelt. Diese Situa-. tion wird gekennzeichnet durch eine £ OberflächenladungsdichteDas zugehöriges Potential lautet dann: ¦¥ £¨ Offensichtlich ist stetig beim Durchgang durch die Schicht. Wie verhält sich die Feldstärke?Wir verwenden nun den Kraftflußsatz%% % %% % %% % %% % %% % %% % %% % %% %% % %% % %% % %% % %% % %% % %% % %%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% £§' ¦ ' £ Für eine infinitesimal kleine Box mit verschwindender Höhe gilt:¢¢£ ' ¡ ¨ ' ¦ ¦¥ ' ¡©¨ ' ¦ £12 ¢ ¢ ££' ¡ ¢¦¥

% % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%% ¤ £¥ % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % £ %£D.h. die Normalkomponente von' erleidet einen Sprung, der proportional zu £ ist.Spezialfall des Leiters: Das Innere eines Leiters muß feldfrei sein, da sonst die frei verschiebbarenLadungen das Feld abbauen würden (Trennung von positiven und negativen Ladungen imLeiter). hat nur Normalkomponente mit' 'Eine Dipolschicht bewirkt, daß dort das Potential unstetig ist.Elektrischer Dipol:%%%%%%%%%%%%%%%%%%%%%%%%¨ ¥ £%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ¨ ¥¢ ¨ ¥ £ §©¨ ¨ Dies wird nun idealisiert durch einen mathematischen Dipol:¨ ¥ ¡¨ Man nennt ¤ elektrisches Dipolmoment ¢ £ ¥ ¢ £ "so daߥ ¥¤fest. Wir denken uns nun eine Schicht belegt mit elektrischen Dipolen, z.B. 2 Schichten mit entge- ¨ 13

gengesetzten Oberflächenladungsdichten und verschwindendem Abstand:% %% %% % %% % %% % %% % %% % %% %% % %% % %% % %% % %% % %% % %% % %% % %% %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % %%% % %% %%% % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % %%% %% % % % % % %% % %% % % % % % %% % %% % %% % % % % % % % %% % %% % %% % % % % % % % %%£% %% % ¢ d%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%£ ¨ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ¦¥ ¢£ ¦¥ ¨ ¨ £ ¦¥ ¨ ¨ ¢ ¦¥ £ £ ¦¥¥§¦ &¨©¨ & £ £ £ ¢ Dipoldichte¡£¢ ¤ Nun betrachten wir eine Annäherung von beiden Seiten © ¢ ¨ ¨ ¢¤£% % % % % % % % % %% % % % % %% % % % % %% % % % % % %% % % % % % % %% % % % % % % %% % % % % % % %% % % % % % % %% % % % % % %% % % % % %% % % % % %% %% % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%% %% % % % % % % % % % % % % % % % % % % % % % %% %% % % % % % % % % % % % % % % % % % %¨ ¢£% % % %%© %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% © ¨¢ 14 ¨ ¢ ¨ £¨

¢ £ ¢ £ ¢ © © ¨ ¡ © ¨ ¨ ¢#¨ © ¨ © ¨ ¨¢¢$¨ ¨¦¥ © ¨ © ¨ ¢ ¦¥ ¢ ¨¢ ¦¥ Zum ersten Integral trägt nur die -Umgebung von £und ¢ £die Klammer verschwindet. Für einen genügend kleinen Abstand von © steht senkrecht auf © ; außerdem sind dann auch die Beträge in den Klammern gleich unddas erste Integral verschwindet. © ¨ ¢ © bei, da für einen festen Abstand © ¨ Im£zweiten Integral projiziert der Limes wieder auf -Werte in der -Umgebung von . Für -Werte außerhalb ist der Integralbeitrag regulär und die -Multiplikation läßt diesen ©Beitrag ¢ verschwinden.Nun seiangesehen werden© ¨ ; in unmittelbarer Umgebung von © kann die Fläche, in der ¡ © £ ¢ ! also: £¢¥¤§© ¤© ! © 1.6 Eindeutigkeitstheorem © ©© ¡ ¢¡ £¢¡ ¦¥ ¢ ¨ ¡ © ¢ ¨ ¡ ¦ ¢¡ © ¨¢ ! %%%% %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% %% %%% %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%% % % % % % % % % % % % % % %%%%%%%% %% %%%%%%%%%% %% %%%%%%% % %% %% % %%% % % %% %% %%%%%%%%%% %% %%%%%%%%%% %% %% %%%% %% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% %%% % % %% % % % % % % %% % % % % % % % % % % % % %%% %%%%% % % % % % % % % % % % % % % % % % %%% % %%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%% %%%%%%%%% %%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % %% %% %% %% %%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%% %%%%%%%% %% %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% %%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% %% %% % %% %%%%%%%%%%%%% %% %% %% %% %% %% %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%% %% %% %% %%%%%%%%%% %% %% %% %%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%% %% %% %%%%%%%%%% %% %% %% %%%%%%%%%%%% %%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%% © liegt, als ebenDie allgemeine Situation der Elektrostatik ist, daß eine Anordnung von Leitern vorliegt, diegeladen oder nicht geladen sind. Deren Oberfläche zusammengenommen umschließt entwedervollständig ein Volumen ¢ % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%15

oder das Unendliche ergänzt % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% %%% %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%% Auf seien vorgegeben entweder die Potentiale (Dirichlet Randbedingung) oder deren Normalableitung, gleichbedeutend mit der Oberflächenladungsdichte (Neumann’sche Randbedin-¡gung).Dann 'ist eindeutig festgelegt; genauer gilt ist eindeutig für die Dirichlet’sche Randbedingung,ist eindeutig bis auf eine Konstante für die Neumann’sche Randbedingung.Beweis: In ¢ begrenzt durch giltSeien und zwei Lösungen, die die Poissongleichung und die Randbedingung erfüllen,dann gilt für U :Somit gilt:¡£¢ £ in ¢ £ ¢ auf für die Dirichlet’sche ¢Randbedingung¢¤£§ ¡ ¢ ¢auf für die Neumann’sche Randbedingung £ £¤ ¢ ¡¢ ¤£ in ¢ ¢const in ¢£für beide Randbedingungen¢ ¡¥¢§¦ ¢©¨¥ ¡© ¢ ¡ ¢ ¢ ££¢ ¢const, woraus' Da auf ist, folgt überall in für die Dirichlet’sche Randbedingung; demnachgibt es ein eindeutiges . Im Fall der Neumann’schen Randbedingung bleibtjedoch .¡Anmerkung: Die Vorgabe von und auf ist i.a. nicht verträglich, da beide für sich dieLösung festlegen und damit den jeweilig anderen Wert. Jedoch kann auf Teilen von der Wertund auf anderen vorgegeben sein.¦ ' ¡folgt und damit auch die Eindeutigkeit von '¡Einfache Folgerung: Der ladungsfreie Innenraum einer beliebig geformten leitenden Oberflächeist feldfrei. 16

Beweis: Auf ist das Potential konstant. Eine Lösung innerhalb von ist demnach diese Konstante. Wegen der Eindeutigkeit ist dies Lösung.1.7 Elektrostatische potentielle Energie mit Anwendung auf Anordnungvon LeiternZu bestimmen ist die potentielle Energie für eine Anordnung Ladungen¥ ¦§" ¥¢¡"¡ ¢ £ §¥vonWir beginnen mit 2 Ladungen.%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%% ¥§¦ % % % % % % % % % % % % % % % % % % % % % % % £% % % % % % % % % % % % % % % % % % % % % % % ¥¢¡ £ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ¦Gegen die Coulombkraft zu leistende Arbeit: ¨ %¥ ¡ !¥ &¥ & ¡ £ ¨ ¥¢¡ !Füer 3 Ladungen verhält es sich folgendermaßen:allgemein&¥ ¨ £ ¥ ¡ ¥§¦ %¡ ¨ ¦ ¥ ¦¡§ % "¤¦¥!¥ &' ¦ £ ¡ £ ¡ ¥¢¡ ¦¢"¥¥ " ¨ ¥¥ ¡ ¦¢¥¦ ¡ ¨ ¦ ¥¢¡¥¦ ¡©¨ ¦ ¥ ¥§¦ ¥¢¡¨ ¦ ¥ ¥¢¡ ¨ ¡ Dies ist die elektrostatische potentielle Energie als Funktion der Ladungen und Abstände. ¢ "§¥"¥¥ " ¨ ¥Nach dem Übertragen auf eine kontinuierliche Ladungsverteilung verhält es sich folgendermaßen: %¢ ¨ ¢¥ Dies läßt sich identisch umschreiben und führt zu einer neuen physikalischen Interpretation. Wireleminieren nun zugunsten von : ¡ ¨ 17.

% %% ¨ Feld der Ladung¥" "¡ "§' "¥ "¥ ¥¢ " ¥ "¥' "£¥ ' ¡¡'¥¥¨ ¥¥¥¤¦ ' "£'¥ ¡ Die potentielle Energie findet sich somit wieder in einer Energiedichte des Feldes: ' ¡ Während dieses stets positiv ist, könnte die diskrete Summe oben auch negativ sein. Der Grundliegt darin, daß beim Übergang zur kontinuierlichen Ladungsverteilung auch Selbstenergiebeiträgeder einzelnen Ladungen mitgezählt %wurden:Es '(£¢" ' ""sei.' "' ¡ " "¡ '¤ ¡£¢¤¦¥" ¥ §' ¡ " Die Selbstenergie der Ladungen¥"ist unabhängig von der relativen Lage der einzelnen Ladungen;sie hängt nur von der Existenz der Ladungen selber ab.%"¦ ' " ¢£% % % ¥" ¨ " ' " ( keine Oberflächenterme ) % % " " ¥ " " ¢Neben der Selbstenergie ergibt sich demnach der obige Ausdruck, der von der relativen Lage derLadungen abhängt.Anwendung auf Leiter:18

%¢¢¥" ¥ %¥¢¥¥"¥ ¢£"¥ " ¥¥¦ ¢ ¦ ¦ ¢ ¦¡¨ ¡ ¢ ¡ ¦ ¢ ¡ ¡¨ & ¦!¨ ¡ ¢ ¦ ¦ ¢ ¡ ¡ ¨ ¢ ¦¡ ¨ ¢ ¡ ¦% %¦¥ £ ¥¥£"¥¢"£ ¡¥Auf Leitern ist das Potential konstant und es liegt, wenn überhaupt, nur Oberflächenladungsdichtevor. Daraus ergeben sich vereinfachte Verhältnisse. £ ¦¥ wobei ¡ über die verschiedenen Leiter läuft. Wir wollen nun ¥ zugunsten der ¥oder die ¥zugunsten der¥eleminieren.Da die Poissongleichung linear ist, müssen die Potentiale linear von den Ladungen abhängen."¥ ¥¢Dabei hängen die"¥¢nur von der Geometrie ab. Dies läßt sich auch umkehren:" £"¥Kapazitätskoeffizienten¥Daraus folgt dann:"¥"¢¤¥¤¥Spezialfall: Zwei Leiter mit entgegengesetzten Ladungen nennt man Kapazität.Es gilt:¢ "¢ "Somit besteht ein linearer Zusammenhang zwischen den Ladungen auf einem der Leiter und derPotentialdifferenz zwischen den Leitern: £ " £KapazitätWie sieht die gespeicherte Energie aus? ¢ ¦ ¦ ¢ ¡ ¡©¨ ¢ ¦¡©¨ ¢ ¡ ¦ ¡ ¢¡ £ ¢Zwischen den Leitern wirken Kräfte. Diese lassen sich bei Kenntnis der gespeicherten Energiefür gegebene Anordnungen der Leiter bestimmen. Wenn der ¡ Leiter ein Stück ¥verschobenwird, dann leistet das System die mechanische Arbeit19

§%¥Kraftdichte (vom Leiter wegzeigend) ¡ ¥ ¢ %% %%% % %% % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % %% % %% %% %%% %%%%%%%¦ ¨ £ ¦¥ £ ¦ ¨£§¦¥ ¥ ¤¥auf Kosten der elektrostatischen Energieetc.¨ % % ¥ ¥ ¡£¢¥ ¨ Beispiel: Ein Leiter trage eine Oberflächenladungsdichte £ . Welche Kraft wirkt lokal?Energiedichte an der Oberfläche: ' ¡ ¡£ ¡ ¢ ¡£ Verschiebung eines Oberflächenelements um ergibt eine Verringerung des angrenzendenVolumens: ¡ % ¨ ¢ £¡¡ ¥ £ ¡ ¨ £ ¡ Anmerkung:£¡Lösungen der Laplacegleichung haben folgende Eigenschaft: Der Mittelwert von über eine Kugeloberfläche ist gleich dem Wert von ¡ im Zentrum der Kugel.£ Nachweis durch physikalische Argumentation:Sei durch eine Ladung¥generiert, die sich außerhalb einer gedachten Kugel befindet:%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ¥Es werde die Ladung¥ gleichförmig über die Oberfläche verteilt. Die Arbeit hierzu ist: ¦¥ &¦¥ ¥ Dieselbe Arbeit wäre wenn¥ aufzubringen, bereits vorhanden wäre und q aus dem herangebrachtwürde. Die verteilte Ladung¥ wirkt aber, als wenn sie im Ursprung konzentriert wäre.£ ¥ £ ¨§ ¥ ¤¥ ¨§ ¥ ¤¥ ©¥ ArbeitFolgerung: Man kann kein elektrostatisches Feld erzeugen, das eine Ladung im stabilen Gleichgewichthält. Denn sei die Ladung £ , dann müßte in der Umgebung des Gleichgewichtspunktesgrößer sein. Dies ist nicht möglich, da ¥ .20

1.8 Multipolentwicklung einer Ladungsverteilung und Energie einer Ladungsverteilungim äußeren FeldEine räumlich begrenzte Ladungsverteilung ist eine Quelle für ein elektrostatisches Potentialoder ein elektrisches FeldIn sehr großer Entfernung von erscheint die Ladungsverteilung als Punktladung und dementsprechendergibt sich ein einfacher Ausdruck für . Wie ändert sich dies bei näherem ”Zusehen”?Dies kann systematisch durch die Multipolentwicklung beantwortet werden: % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % %% % % %% % % % % % % %%%% %% %%%%%%%%%%%% %%%% %%% % %%% %%% %% % % % % % % % % % % %% %% %%%%% %%%% %% %% %%%%% %%%%%% %%%%%%%%%%%% %%% % %%% %%% %%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%% % % %%% %%%%%% % %%% % %%% % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%% %% %% %% %%%% %% %% %% %%%%%% %% %% %% %% %%%%%%%%%%%%%%%% % % % % % % % % % % % % % %%%%%%%%%%%%%% %% %%%% %% %% %% %%%% %% %%%% %% %% %% %%%%%%%%%% %% %% %%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%% %% % %% % %% %% % %% % %% % %% % %% %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % %% % % %%% % % %%% % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ¨ ¨ ¨¥ ¥ ¥ ¨ " ¢¡ ""¥¢¡ "¤¥ "" ¥ " " ¥ ¥ £ £ " ¥ £ £¤Für große Abstände ist der erste Term dominant, der nur die Gesamtladung enthält:In nächster Ordnung, ¢Dann in ¢etc.¦ ¤£¦ ¤¦¤¥Quadrupolmomente: ¦, tritt das elektrische Dipolmoment auf:¥ ¤ ¥ "¥ "21

¡ "¢¥ ¡ "¥¢ "¥"¥ "¥ ¥Dieser Tensor ist spurlos" ""¤£ "¥¥§ " £ ¨ "¥¥¤¥ "¡ ¨ ¡ ¦ Der Quadrupolmomententensor läßt sich stark vereinfachen.¥ ist symmetrisch und reell. Also läßt es sich diagonalisieren. Geometrisch bedeutet dies, daß"man ein Koordinatensystem finden kann, so daß "¥"Somit ist eine beliebige Ladungsverteilung durch drei Quadrupolmomente bezüglich der sogenanntenHauptachsen gekennzeichnet. Dies läßt sich sogar noch weiter reduzieren:Betrachten wir das Quadrupolpotential:"Dazu läßt sich Null addieren: ¥£ ¡ "für £ Wir wählen nun eine Konstante, die bei dieser Null geeignet ist:¤¦¤"¨§"¢& "¥¥Quadrupoltensor¤¦¤¥ "" § ¨ "¥¦¤¥"¥¨¡ ¦ ¤ § "¥Somit sind nach Diagonalisierung nur zwei Diagonalelemente unabhängig. Im Fall einer axialsymmetrischenLadungsverteilung genügt sogar nur eine Zahl, um das Quadrupolpotential festzulegen: ¤ § (2 transversale Komponenten sind gleich) Nun zur Energie dieser Momente im äußeren Feld22

¢" £¤ £¡ "" ¥¢ "¥ " ¤£'¥"¥¨'¥ ' £ ¡ "¥¢" £"¥¥" £ £ £¡ "¥ ¦ £ ¢ £'¥" £ £ £'¥¦" £ ¢'¥" £ £ ¢ ¥ §" wobei £ ¤£gesetzt sei.Für ein vorgegebenes Potential hat die Ladung¥die potentielle Energie:Dies bedeutet verallgemeinert auf eine Ladungsverteilung: um das Zentrum von (Da die Quellen von von verschieden sind, fehlt der Faktor ¦ ¡ ).Wir entwickeln nun : £ ""¥ £ £Für das äußere Feld gilt div. Wir können nun addieren: £ ¨ ' £ ¨ £¢"¥ '(¤£ ¢¤¥ £ !¨ ' £ ¨ ¢¤ §Somit ¤ §"¥¨ £ ¨ ¨ ¤¥ ¥ £ £ ¨ ' £ ¨ ¤¥Die Energiebeiträge sind also:¡Gesamtladung¡Dipolmoment¡QuadrupolmomentPotential im Zentrumelektrisches FeldFeldgradienten etc.1.9 Zweidimensionale ProblemeMathematische VorbemerkungSei ¥ eine analytische Funktion der komplexen Variablen £¢ ¢.23

Sei ¥ ¢ ¢ ¡ ¢ , Real- und Imaginärteil. Dann erfüllen und ¡ die Laplacegleichung.Nachweis: und ¡ erfüllen dort, wo ¥ analytisch ist, die Cauchy- Riemann’schen Bedingungen ¡¡ ¨ ¢ ¡ ¨ ¡ ¡ ¡¢ ¡¡ ¡¢ ¨ ¢ ¡ ¡ ¡ £Die Linien konstanten ’s und ’s stehen senkrecht aufeinander:Nachweis: ¡ £ ¢¡¢ ¨ ¡ ¡ ¡ ¡¡ ¢¨ % % % % % % % % % % % % % % % % % %¢ %%%% % % % % % % %%% % % % % % % % %%%% %%%%% %%%%% % % % %%%% % % % % % %% % % % % % %% %%%%%%%% %%%%% %% % %%% %% %% %% % % % % % % % %% % %% % % % % % % % %% % %% % % % %%% %% % % % % % %%% %% % % % % % %%% %%%% %%%%%% %%% %%%% %%%% %%%%% % % %%% % %% %% % %%% % %% %% % %%% % % % % % % % % % % %%% %% % % % % % % % %%% % % % % % % %%% %% %%%%%%%%%%%%% % % %% %% %%%%%% % % % %%%%% % % %%%%% %%% %%% %%% %%%% % % % % % % % % %% %% % % %%%%%%%%% %%%%%%%¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤%% % % % % % % % % % % % % % % % % % % % % %% % % % % %% % % % % %%%%%%%%%%%%% %%%%%% % %%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%% %%%%% %%%%%%%%%%% % % % % % % % % % % % % % %% % %% % % %%% % % % %% % % % % % %%%%%%%%%% %%% %%%%%% %%%%% %%% %%%%% % % % % % % % % % %%%% % % %% % %%% %%% %% %% % % % % %% % %% %% %%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% %% %% %%%% %%%%%%%%% %%% % %% % %% %% %% %%% %% %% % % % %% % % % % % % % % % % % % % %%%% %% % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%% %% % % % % % % % %%% % % %% % % % % %% %% %% % % %% %%%% % %%% %% % % % % % % %% % %% %% %% % %% % % %%%%%%% % %% % %% % %%% % % %% %%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ¢ ¢¢ £¡ ¡¤£ ¢ ¦¥ ¡¤£ ¢ §¥Somit kann man const auffassen als Linien konstanten Potentials und const als Feldlinienoder umgekehrt.Ferner gilt für eine analytische Funktion, daß die Ableitung an der Stelle unabhängig von derRichtung ist, mit der der Grenzübergang durchgeführt wird. Demnach: ¡"""""¦¥ """""" """©¨ "© ¢¡¡" " " """ """ 24"¢ ¡¢ ¨¢"""""¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤ '

Somit ergeben sich aus der Kenntnis von f die Äquipotential- und Feldlinien, sowie die Feldstärke.Einfaches Beispiel: Gleichförmiges Feld¢%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤ 'Äquipotentiallinien ¡¡ ' ¢Wie sieht die zugehörige analytische Funktion aus?Naheliegend wäre: ¥ ¢ ' ¡ ' £¢ ¢ & ¢ ¡Feldlinien ' constÄquipotentiallinien ¡ ' ¢ const ' """""""""" ' Allgemeine Problematik: Es liegt eine bestimmte Geometrie der Leiter vor, d.h es gibt Linienkonstanten Potentials.%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤ ¦ ¡ ¦¢%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Gesucht sind Feldlinien- und Potentialverlauf.25

Sei ¡ ¡ ¦eine Abbildung der komplexen ¦-Ebene auf die komplexe ¡-Ebene, in der dieGeometrie einfach ist¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤¤ ¤ ¤¤ ¤ ¤ ¤¤¢¡ % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%und die Lösung als ¥ ¡ angebbar ist.Dann ist ¥ ¦ die Lösung des ursprünglichen Problems.Beispiel:Wie sieht der Feldverlauf und der Potentialverlauf aus?¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤ ¤Die Lösung ist bekannt für:% %% % %% % %% % %% % %% % %% % %% % %% % %% % %% %% % %% % %% % %% % %% % % % % %% % %% % %% %% % %% % %% % %% % %% % %% % %% %% % %% % %% % %% % %% % %% %¦ % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%% %%% %% % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ¡¤£ ¢ ¦¥% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%% % % % % % % % % % % % % % % % % %¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤%% % %% % %% %% % %% % %% % %% % %% % %% % %% % %% % %% %% % %% % %% % %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ¡ £ ¢ ¦¥% %% % %% %% % %% % %% % %% % %% % %% % %% % %% % %% % %% % %% % %% % %% %%% ¡ £ ¢ ¦¥Die obere Halbebene¡läßt sich auf den Winkelbereich "¢¥¤§¦ ¦ ¡£¢% %% % %% % %% % %% % %% %% % %% % %% % %% % %% % %% % %% % %% % %% % %% %%%¢¥¤ ¡ ¢26%% % %% % %% % %% % %% % %% % %% % %% % %% % %% % %% %% % %% % %% % %% % %%%¨ ¥%% % %% % %% % %% %% % %% % %% % %% % %% % %% % %% %% % %% % %% % %% % %% %%% %% % %% % %% % %% % %% % %% % %% %% % %% % %% % %% % %% % %% % %% % %% %%%¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤' ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¡ % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%in der ¦-Ebene abbilden:¢¢¥¤

Ferner ist die Lösung in ¡-Ebene: ¡ ¡ ¡¥ ¦ ¡ ¡ ¡ ¦¤ ¢¢ ¡ ¦ ¤ ¢¢ ¦ £¢ ¢ ¦ ¥ ¦ Feldlinien¢ ¡ ¦ ¤ ¢ ¢ ¡ ¡ ¦ ¤ ¢¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤ ¤ ¡%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%""""¢ ¢ ¦ Äquipotentiallinien ¡ £ ¢ ¦¥¡ ¡¤£ ¢ ¦¥¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¦¥ ¦ ¡ ¦¥ ¦" "" ¦ ¤ ¢¢"" "Ein kleiner Winkel bedeutet: Geringe Feldstärke in der Ecke.¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤¦ % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % %% % % % %% % % % % %%% % % % % % % % % % % %%% % % % % % % % % %% %% %%% %%%% %%%%% %% %% %%% % % % % %%% %% %% %%%%%%%% % %% % %% %%%% %%%%%%% %%%% %%% %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% %% %% % % % % % % % %%% % % %% % % % % % % % % % % % % % % %%% % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ¦const ¦ ¤ ¢¢¦ ' größer % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % %%%%%%%%%%%%%%Feldstärke wächst an der Ecke über alle Grenzen27 ¡¤£ ¢ §¥% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%¡ ¡¤£ ¢ §¥¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤

" ¨¨¡ ¨ ¡ ¨ " " " ¨ """ ¨ £1.10 Formale Lösung des Randwertproblems mit Hilfe der GreensfunktionWir betrachten ein Volumen berandet durch eine Oberfläche . Darin sei eine Ladungsvertei-¢gegeben. Es liegen Dirichlet’sche oder Neumann’sche Randbedingung vor.lung Also:vorgebenoder vorgeben auf (Dirichlet)auf (Neumann) ¡ ¨ Der einfachste Fall ist uns bereits bekannt: im Endlichen, S im Unendlichen und auf zu Null angesetzt:Dies läßt sich in folgender Form schreiben: ¡ " ¨ Man nennt¨ eine Greensfunktion. Dieses G ist symmetrisch: Und es gilt auch: ¡ ¨ Ferner: " " £ ¤£ oder " " " Dies ist das einfachste Beispiel einer Greensfunktion für eine Dirichlet’sche Randbedingung.¨£¢ §Nun allgemeiner. Die Greensfunktion für ein gegebenes elektrostatisches Problem ist definiertals Lösung von¡ " ¨ 28

¡ § ¤ £ ¡¨¨ und ¢" " §" " " ¦ ¨¨für ¥¤ £ ¢" ¢ ¦¥ bei geeigneten Randbedingungen. Die allgemeine Lösungsform ist zunächst " " " wobei ¨ £¨Offensichtlich dient zur Erfüllung der gewünschten Randbedingungen.¡ " Wir benutzen nun zur Konstruktion einer Lösung und verwenden ein Green’sches Theorem: ¡ £¢ ¨ ¢ ¡¤¡ ¦ ¤ ¢ ¨ ¢ Dabei sind und ¢ beliebige, nicht singuläre Funktionen in ¢ .Beweis: £ ¢ ¢ £ ¤ ¢ ¡ ¢ ¦ Nach Vertauschen vonund ¢ und Subtraktion folgt das Theorem.Nun wenden wir das Theorem auf " an: ¨ ¡ ¡ ¦ ¨ " £ ¨ " ¨ ¢Wir stellen nun Forderungen aneinzubauen:, um eine Dirichlet’sche oder Neumann’sche Randbedingung Dirichlet: " £ ¥Dann gilt:29

¡in ¢div grad ¡¨ " §Die einfachste mögliche Randbedingung auf" wobei ¡§ ¦¥ §§ ¨ " § §¥ " £¡" ¢ ¦¥ ¦¥ ¦¥ " ¨ ¥ Damit ist das Problem, formuliert für , auf ein etwas einfacheres für ¥ zurückgeführt worden.Bei Kenntnis von ¥ ist die formale Lösung gefunden.2 Beispiele:¤£auf ¥ " ¤£¥ " § auf ¨ ¦¥ Nun zur Neumann’schen Randbedingung: vorgegeben £auf wäre wünschenswert, auf entsprechend der allgemeinen Formel oben. Dies ist jedoch nicht verträglich mit:¡ " ¨ Hieraus folgt nämlich:¦¥ ¨ ist folglich¢¡# der Wert der gesamten Oberfläche ist. ¨ Somit £¡ " ©¥ § ¡ " ¢¡ " ¢¡$ 30

Ein Problem dieses Typs ist, daß aus einem endlichen Teil und £ besteht:£¡ £und § £ ¥Im Neumann’schen und Dirichlet’schen Fall erfüllen die Greensfunktionen somit einfachereRandbedingungen als das Potential, nämlich £ ¥ auf auf "die nur von der geometrischen Anordnung, jedoch nicht von den speziellen Randwerten des Potentialsabhängen.¢¡ ¨ Trotzdem sind nur bei einfachsten Geometrien Lösungen fürangebbar.Bei der Konstruktion der Dirichlet’schen Greensfunktion hilft deren symmetrische Eigenschaft:¥ " Nachweis: Wir wenden das Green’sche Theorem aufan. Daraus ergibt sich: ¤ ¤ ¥ ¥ ¥ ¥ " " " " ¥ " ¢ ¥ " ¢ ¢ ¢ ¥ " ¢ ¡ ¢¥ ¡ ¡ ¥ ¢¢ ¨ ¥ " " ¨ ¥ ¢ ¨ ¥ ¢" " ¨ ¢ ¨ ¥ " ¡ ¢¥ ¡ ¡ ¥ ¢ " ¢ ¨ " " ¢ ¦¢ ¦ ¢ ¡ £ ¨ ¢ ¢ £ Im Neumann’schen Fall kann man die Symmetrie zusätzlich fordern, sie ist nicht automatischvorhanden.1.11 Die Greensfunktion am Beispiel der leitenden Kugel und einer Punktladung%%%%%% %%% %% %% %% % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% %% % %%% %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %¥ ¤%¡31

¡ ¤ ¨¢%%% %%% %% % %% %% % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % %% %% %% %% %%% %%%%%% %%%%%%%" " ¥"""¦ ¡ ¨ ¡ ¡¢¦¦¦¥ ¢¨ £¦¤¢ ¤ ¦£¨£¨ ¦ ¨¢ ¦ ¨¨¨¥"auch""""( ¦ ¦ ¨¨% % % % % % % % % % % % % % % % % % % % % % % ¢¥ ¤ £ ¨¢£¨ £ ¤ ¨ ¦£¨& ((¦ £§¨¨¨¨¢ ¦ £ ¨¢ ¥ ¢ ¥¨ ¨ ¨¢¨¨ £besteht aus einer Kugeloberfläche und .ist konstant auf der Kugel und 0 im Unendlichen.Es liegt also der Dirichlet’scher Fall £ vor.Gesucht wird mit ¥ ¤£ auf und¡ ¨ .¥ ¥ " " Intuitiver Zugang: Die Ladungen sind auf dem Leiter frei verschiebbar. Demnach wird die Kugeldurch die äußere Ladung polarisiert werden:+++ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+¥++---+- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ¥ - - ¨ Wir versuchen nun, ob diese Ladungstrennung (Polarisation) durch eine gedachte Ladung¥ simuliertwerden kann. Aus Symmetriegründen muߥ auf der Verbindungsgeraden vom Ursprungzu¥liegen.Ansatz:Wobei ¢¡ ¡¥ " . Wegen der Symmetrie muß mit der Forderung¨ ¨ erfüllt sein.¨ ¢oder ¨¢ ¥ ¢ ¤£¨¡ ¨ ¡ ¢ ¦£¨ ¥ ¤ ¦ £¦¦¨ ¢£¨¦ ¥¦§¨¦ ¨¦¦¨ ¢(©¨ ¦ £¨¦¢ ¤¨ ¦£¨¨ ¦£¨¢ ¨ ¦ ¤£¤ ¤ ¢¦¢ ££¨£¨ £( ¥¥ & " ¦ & ¨ &32¦¨

¥""""" " ""¨¡""¥¡ ¡ " " """" "" ¥ ¢ ¤¡ ¡ ¡ ¡£¢ ¨ ¢¡¡¨¥ " " ¨¡""¥¥ ¡¡¢¡ ¡ ¨ ¢ ¦ £¢¡ ¢ ¥¨ ¢¨" © § ¡¡¡¡© " " "© " " "¡ © " " ¥"¡© " " "" """§¡¨§¡¡ ¢ ¥ ¡¡" ¥¨¥ ¨ ¢¡ ¦ ¦¥¡" " "" ¡Dieser Ausdruck verschwindet für ¡und ist symmetrisch:"In dieser Form sieht man auch, daß der 2.Teil, wie gefordert, auch die homogene Poissongleichungerfüllt: ¡ ¡ ¡ ¥ ¨¢ ¥ £Wir benötigen ferner die Normalenableitung:¨ ¤ ¥ ¥ ¡ ¡ ¨ ¢ ¨ ¨ ¢ ¢ ¨ ¢ ¢ ¥ ¨ ¢ ¤ ¥ ¨¥ ¨ ¢¨ ¢ ¦ £¢¡¨ ¢ ¨ ¢ ¡ £¢¡ ¨ ¢ ¡ £¢¡ ¡ ¡¡ ¨ ¡ ¡ ¡¡¡ ¨ ¡¡# ¡ ¡¡ ¨ ¢ ¡ £¢¡So läßt sich nun der allgemeine Ausdruck auswerten: ¦¥ ¨ ¨ ¥ § § "¨ © ¨ ¥ ¡ ¥ ¨¢ ¥ © ¡ ¨ ¡¨ ¢ ¡ £¢¡¡© ¡ ¡¡ ¥ ¡ ¡¡¨ ¢ ¡¥ £¢¡¨ ¢ ¡ £¢¡ ¢ ¡ ¡¡¦¦ ¡ ¡ ¡¡ ¢¨ ¢¡ ¥ ¦ ¢¡"§¦¢¢ ¡ ¡¢¡ ¨ ¡ ¡ ¨ ¡ ¢ ¨ ¡¨ ¥¨ © ¨ ¥ ¡¥ ¥ ¨¢ ¥33

¨ ¥ ¡¡ ¨ ¢¨ ¢ £¢¡¨£¢ ¥ ¡ ¡ © ¨ ¢ © ¡ ¡ © ¨ ¢ © £¢¡ ¨ ¤¢ ¨ ¢ ©¡ ¢ ¡ ¥ ¨¢ ¨ ¢ © ©¨ ¢¡ ©¡¨ ¢¡ ©¥¨ ¡ ¢¥ ¥¥¥ ¥¢¥ ¦ £¢¡" """"" ¦ £¢¡ ¤¤£¦ ¡ ©¡ © ¡ ¨ ¡¤"¡¡ ¢¤""""""" ¢¢¤¡ ¥ ¡ ¨" "¥ ¥¢¥ ¦ ¦ ¢¡ ¤¤£¦ ¦ £¢¡ ¤¤£¦ """"""" ¢Betrachten wir nun zwei Fälle:a) Kugel geerdet (freier Zufluß und Abfluß von Ladungen zur Erde), § £b) Kugel wird auf dem Potential § £gehaltenFall a) © ¨ ¥ ¡© " © " ¥Wie ist die induzierte Ladung auf der Kugeloberfläche verteilt? ¨ ¥ ¨¢ ¥ £ ' ¨ ¨ ¥ £ ¦ ¢¡ ¨¡ ©¡ ¢¢¡ ¥ ¥¨ ¢ © ¨ ¥ ¢¥ ¡ ©¨¥¨ ¢ ©¥¢¥ ¨ ¥ £¢¡ ¡ © ¡¡¥¨ ¢¡ © ¡ © ¡¡ £¢¡ ¨ ¥ ¡© ¨¢ ¥ ¥¥¤ ¢ ¥ ¥34

Demnach ist die induzierte Ladung hin zur Punktladung konzentriert.¢ (¡©¨ £¥¤§¦0321¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%¦¡%%(%%%% ¡%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %%%%%% %%% % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % %% %%( % ¢%¦¡% %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0Wie groß ist die gesamte induzierte Ladung?£©¡¡¨ ¥ ¡¡ ¡© ¦ ¦ ¥¢ ¨ © ¨¨ ¥ ¢¡© © Dies muß so sein nach dem Kraftflußsatz. ¨ ¥¡¡¡¤ ¢¡¨ ¨¡¢ ¢ ¥ ¤¥ ¥¨ ¡ ¢¡¤ ¢ ¥ ¢ ¨¡ ¥ "¤ " ¥ ¡ ¡ © ¥ ¦ £¢¡¢ ¤ ¡¡ ¡ © ¥¥¨ ¡ ¢© ¡¥¥¦ ¦ ¢¡"¦¥" """¦ ""¤¢£¨¢¥ ¨ ¤ ¢ ¦¥Welche Kraft spürt die Ladung¥infolge der induzierten Ladung auf der Leiteroberfläche? DasOberflächenelement spürt: ¢ 35£¡ ¦¥¦ ¤¦¤

%% % %¦¥% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % %%% % % % % % %% % % % % % % % % % % % % % % % % %%%%% %%% %%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % %%% % % %% % % % % %%%%%%%%%%%%%%%%%% % %% % % % % % % % % %%% % % % % %% % %% %% % %% %% %% %% %%% %%%% %%% %%%%%%% %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Aus Symmetriegründen ist nur die Projektion auf die Verbindungsgerade wirksam: ¢ ¢ ¡¢£¡ ¢ ¡¡¡ ¥¡ ¡¡¦£¦ ¡©¥ ¥ ¥ ¡¡ ¡¡© ¤ ¨¢ ¥ ¥¥¡ ¢ ¥ ¤¥ ¦ ¡¥¨ ¡ ¢©¦ ¡Diese Kraft wirkt auf den Leiter. Die auf¥Kraft ist entgegengesetzt. Es ist gerade die Coulombkraftund der Ladung¥zwischen¥" ¤ " ¥ fiktiven :Fall b) ¥ £ ¥ "¤© ¨¢ ¥ ¥ ¥¤ " ¥ ©¦ ¡ ¥¤ ¤¨¢ ¥ ¥¥ ¢ ¦¡ ©¤ ¨¢ ¥Sei nun die Kugel auf Potential § £Potential: ¡¥¦ ¥¥¡ ¨ ¥ ¡¡¡¡ ©¦§Dem entspricht die zusätzliche Oberflächenladungsdichte§ ¨ £¥¥ £¥ ¦ ¤¨¢ ¥ ¥¥ gehalten. Dies führt auf den folgenden Zusatzterm zum """ Sie ist gleichverteilt über die Oberfläche und summiert sich auf zur Gesamtladung¡§¡ §Was ist nun die Gesamtkraft auf Punktladung¥die ? Sie rührt her von der gesamten Oberflächenladungsdichte.Der £ § konstante -Anteil verhält sich so, als § ob im Ursprung wäre§" ¢ § £ ¥ ¡ ¦¥ ¥ " ¤ " ¥ © 36 ¥§ ¨¢ ¥¤ ¦©¡¥ ©¡¦ ¡

¥¥ ¥¤¤ ¤ ¤¤ ¤ ¤¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤¤ ¤ ¤%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%§¥ ¨¡¡¢¤¨¢ ¥ ¥¥ ¥¥¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤¥ ¡¡ © ©¦ § ¡¤£¦¥¡Wir finden für eine gewöhnliche Coulombkraft zwischen den Ladungen¥zwei § und .Dagegen wird für © ¢ ¡die zusätzliche attraktive Kraft infolge der induzierten immerwichtiger: ©¨¥ ¦¦ ¦1¤£¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¦%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ¨ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%¦Man versteht nun klassisch, warum eine Überschußladung die Oberfläche nicht verläßt: Die Ablösungwürde zu einer induzierten Ladung führen, die eine rücktreibende Kraft ausüben würde. © ¡1.12 DielektrikumWir betrachteten bisher isolierte Ladungen oder Leiter, auf denen Ladungen frei verschiebbarsind. Nun nehmen wir Nichtleiter hinzu, Gase, Flüssigkeiten oder Festkörper, in denen die Ladungenin den Atomen oder Molekülen gebunden sind, aber im elektrischen Feld sich etwasräumlich aus ihrer Gleichgewichtslage verschieben. Die Materie wird polarisiert. Die einzelnenAtome oder Moleküle bekommen elektrische Dipolmomente oder bereits vorhandene permanenteelektrische Dipolmomente werden im elektrischen Feld teilweise ausgerichtet. Dies führt dannebenfalls zu einer Polarisierung.Wie äußern sich diese Phänomene makroskopisch?Diese einzelnen atomaren Dipole ergeben in einem kleinen makroskopischen Bereich (Volumenelement),der jedoch mikroskopisch sehr viele Atome und Moleküle enthält, ein resultierendesDipolmoment. Dieses resultierende Dipolmoment kann von Bereich zu Bereich variieren undläßt sich durch eine makroskopische, ortsabhängige Dipoldichte beschreiben.37

Wir bestimmen nun das Feld herrührend von dieser Dipoldichte.Potential eines elektrischen Dipols: £ ¤Die Dipoldichte beschreibt das makroskopische Dipolmoment pro Volumeneinheit. DasPotential generiert durch ein Stück polarisierter Materie besitzt den folgenden Ausdruck:¢ ¢ ¨ ¨ ¨ ¨ Dieser Ausdruck läßt sich umformen mit der resultierenden neuen Interpretation: § &§£¢ ¢¨ ¤¦ ¨ ¨ ¢¨ ¨ ¢ ¨ £¡¨¦¥ ¢£ ¨ ¢ ¨ ¨ ¢ ¤¦ ¨ Die Polarisation äußert sich somit in einer Oberflächenladungsdichteund einer Raumladungsdichte¢&£¡&¢ £¢div ¢ ¨¢Dies ist anschaulich klar. Bei einem homogenen annulieren sich Ladungen benachbarter Dipole,so daß die Nettoladung in einem bestimmten abgegrenzten Volumen (£ Null ist ).¤£Eininhomogenes dagegen führt auf ein unvollständiges (¤ £Auslöschen ):¢%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% %%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%% %%% %%% %% %% %% % %% %% % % %% % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % %% % %% %% %% %% %% %%% %%% %%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%% % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % %+ + - %%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %-%%%%%Die Dipole an der Oberfläche haben keine Partner zur Auslöschung. Es bleibt also stets eineOberflächenladungsdichte zurück.Man kann nun zwischen zwei Quellen für das elektrische Feld unterscheiden:38

¡¢£¢ ¤' ¢ $¨ ' ¢' Frei verschiebbare Ladungen; das sind Einzelladungen oder Ladungen in Leitern und werdenmit bezeichnet Gebundene Ladungen in Nichtleitern. Diese werden mit ¤ bezeichnet.Somit:div ¢mit ¨div ¢ '(Dies legt nahe, den elektrischen Verschiebungsvektor durch ' ¢ ! div&div einzuführen.¢Der Verschiebungsvektor rührt demnach allein von den verschiebaren Ladungen her. ' &¢Wie gewinnt man ? Er wird erst durch ein angelegtes Feld induziert. Bei üblichen Feldstärkenist ein linearer Zusammenhang ausreichend:wird elektrische Suszeptibilität genannt. In Kristallen istin verschiedenen Richtungen verschieden polarisierbar:¡¢ ¡i.a. ein Tensor, d.h. das Medium ist¡ ¢ ¤ ' ¤Damit haben' und ¢ nicht mehr die gleiche Richtung.Wir wollen hier jedoch den isotropen Fall weiter betrachten: ' ¡ ' ¡ ' & Man nennt & ¡die Dielektrizitätskonstante.Der obere lineare Zusammenhang schreibt sich dann als39

Das Dielektrikum hat eine Berandung und grenzt an ein zweites oder mehrere. Wie sehen dieRandbedingungen an Grenzflächen zwischen verschiedenen Dielektrika aus?¢%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ¦¡%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%§£ ¥ "¥ verschiebbare Ladung¢ ¡ ¨ ¦ £ " £ verschiebare OberflächenladungsdichteFür lineare isotrope Media gilt dann:¢¤£ §¡ ' ¡©¨ ¦ ' ¦ £Auch in Dielektrika gilt rot '(¤£, da sonst auf geschlossenem Weg Energie durch Herumführeneiner Ladung gewonnen werden könnte. Demnach gilt ferner:' ¢ ¤¡ ' ¢ ¤¦Elementares Beispiel:'%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%' ¤ ¦¤ 'Dielektrikum'%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Das Feld im Dielektrikum ist um den Faktor reduziert. Demnach auch die Spannung ' ¤ £ ' £ 40

gegenüber dem freien Raum.Verhalten einer Kapazität mit Dielektrikum%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%¦¤ ' ' %%%%%%%%%%%%%%%%%%%%%%%%%%Zwischen den Platten herrsche die Spannung .Ohne Medium gilt: £ £ ' £ ' £ Mit Medium gilt: £ £ ' £ ' £ ¡£¢ ¤¡ £Also wird die Ladung auf den Platten um den Faktor erhöht, um die gleiche Spannung aufrechtzuerhalten.Entsprechend wird die Kapazität um erhöht.Schließlich kommen wir zu einer einfachen klassischen Betrachtung zum Zusammenhang zwischenmakroskopischer Suszeptibiltät ¡ und mikroskopischer Polarisierbarkeit .Welches Feld spürt ein Atom oder Molekül in einem Dielektrikum? Wir denken uns eine kugelförmigeUmgebung des Atoms, in der eine mikroskopische Betrachtungsweise angestellt wirdund das Äußere der Kugel beschrieben durch makroskopische Größen.41

'£ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Atomgedachte Kugelz¢%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Das Atom spürt die folgenden Felder:Das mittlere Feld 'Das Feld erzeugt durch die Oberflächenladungsdichte an der KugeloberflächeDas Feld von den Dipolen der Atome und Moleküle innerhalb der KugelDie als homogen angenommene Polarisierung ¢erzeugt eine Oberflächenladungsdichte:£¡ ¢ £¢ ¢ ¢ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%z¢¢¢----+++%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Die generierte Feldkomponente in z-Richtung am Ort des Atoms sieht folgendermaßen aus:' £¡¦¥ ¡ ¡¤© ¢ ¡ ¡ ¡ ¢ £ ¢ ¡ ¡ ¢ ¢ £¢§ § ¢ Somit:' § ¢42

£ £ ' ¤ £ ¦ ¨ ¨£ ' ¡ ¢ ¢ §¤ ¤ § £§© ¨§ ¤£ ¡ ¤¡ ¢ ¤ ¤£ ©' ¤ £ ' ¡ ¢' ¢ §¥¥' Nun zum Feld der einzelnen Dipole: Eine Summation über alle Dipole innerhalb der Kugel ergibt: ¨¦ $ ¨ Nun seien die Dipole isotrop innerhalb der Kugel verteilt: ¡ & ¡ ¢ ¡ ¡ ¡§¢ & ¢ £ Das resultierende Feld im Kugelmittelpunkt ist demnach Null. Dies ist auch richtig für die Dipoleauf den Gitterpositionen eines kubischen Kristalls (siehe Jackson für weitere Information). ¦ $ £ ¦¡ ¦Somit ist das effektive Feld am herausgegriffenen Atom:Dies polarisiert nun das Atom.' ¡ ¢ Dabei istdie atomare Polarisierbarkeit.Die Dipoldichte ist£ ¤ " £ Anzahl der Atome pro Volumeneinheit¢¤£ ¢ £ £ ¨ § $¨ oder ' ¨§ ¢§ $¨ £ ¢ § £ § £©$¨43

¢¥©¤ £ ¤© ¢ ¨ ¢ £ ¤©¤¡ ¦£¢ ¥¥¤§¦©¨¥ ¢"¤¢¡wobei&¤©' ¤ ¦ ¨¦ ¦ ¤£ ¤© £ ¤© § § ¢ § #¨' ¢ § Dabei ist die Massendichte, das Molekulargewicht und £ © die Avogardro’sche Zahl.Dies ist die Clausius-Mossotti Gleichung und gibt den Zusammenhang zwischen der makroskopischenGröße ¥ und der mikroskopischen Größe an.Es mögen nun zusätzlich permanente Dipole vorliegen und im Feld teilweise ausgerichtet werden- entgegen der Temperaturbewegung, die einer Ausrichtung entgegenwirkt.Die Energie eines Dipols im elektrischen Feld ist:Der Boltzmannfaktor gibt die Wahrscheinlichkeit an, bei der TemperaturEnergie zu ¦ ¢finden:¤© £den Dipol mit der'( ¨ ¤©' Daraus resultiert eine Dipoldichte: ¦ ¦©¨¥ ¢ ©¢ £¦©¨¥ ¢ ©¦ ¦ ¦¨¥ ¢ ©¨Wir betrachten nun :£ ¤© ©¤Dabei ist'( ' ¢ das lokale Feld.& ¡ £ ' mit ¡ ¤ ¡ ©¡ ¤§ © £Daraus folgt, daß temperaturabhängig ist.44

1.13 Randwertproblem für Dielektrika (Methode der Spiegelladungen)Einfachste Beispiele zur Methode der Spiegelladungen:Leiter%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%-q% % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%qRandbedingung Randbedingung ¤£ läßt sich offensichtlich erreichen durch fiktive Spiegelladung¨ ¥: ¥ ¨ © ¨ ¥ © Leiter%%%%%%%% %%% %%% % %% %% %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % %%%%%%%%%%% % % % % %% % %%% %% %% %%% %% %%% %%%%%%% %%%%%%% %%%% %%% %%% % % %%%%%%%% %%%%% % % % %%%%% % % %£ %%%%%% % % %%%%%%%% % % % % % %% % %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%% %%%% % % %% % %% % %%%% % %% %% % %%%% %% % % %%%%% % %%% % % % %%% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% % % %%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% £% % % % % % % %% % % % %%%q % % %%% %% % % %%%% %% % %% %%% %%%%% %%%% %% % % %%% %%% %%% % % % %%% % % %%%%% %%% % % % % % % %% % %%% % %q´% %%%%%%% %%% % %% % %% % %%%% % %% %% % %% % %% % % % %%%%% % %%% % % % %%% % %%% %%%%% % % % % % % % %% % % % % %%% % % % %% %% % % % %% % %%%%% %% % % % % % % % % %% %%%% % % % %% % % % %%%Die Spiegelladung¥ fiktive ermöglicht die Realisierung,halten.45¤£auf der Kugeloberfläche konstant zu

Nun zum Dielektrikum:¡%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%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zq¦£div ¦ div '( £div §¡ div '( £ ¡ £rot '(¤£überallDa keine verschiebbare Oberflächenladungsdichte vorhanden sein soll, gilt: ¦ ' """" ©¢¡ §¡ ' """" © £Ferner ist ' und ' ¡ stetig an der Grenzfläche.Wir versuchen nun durch Einführung einer Spiegelladung eine Lösung zu finden:£¥¤¦¦ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%ZP§£©¨qdd’q’¦46

¦¢ ¦¥¡ ¦¥ ¦¢¦ ¡¡ ¡¡ ¨ ¡¡ ¡£¡¡ ¡ Lösung der Poisson-Gleichung% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %¦%¢ §¡ ¡§ ¡d”q” ¡¥Lösung der Laplace-Gleichung¡ ¨ ¡Lassen sich mit diesen beiden Ansätzen die Randbedingungen erfüllen?¦ ¡ ¦ ¡ """""""""""""""""""" ©©©© ¦ ¡ ¡ £¢¡ ¨ ¥ ¡ ¡ £¢¡ ¥ ¥ §¡ ¡ ¡ £¢¡ ¦ ¨ ¥ ¡ ¡ £¢¡ ¥ ¡ ¡ £¢¡ §¡ ¨ ¡ ¡ £¢¡ ¦¥ ¡ ¡ £¢¡ ¥ ¥ ¡ ¡ ¡ £¢¡ ¨ ¥ ¥ ¡ £¢¡ §¡¢¡ ¡ ¥ ¥ ¡ £¢¡¡ ¡ £¢¡Dies muß identisch für alle erfüllt sein. Dies ist nur möglich falls .¤ £ ¥ ¥ ¦ ¦¥ ¥ ¥#¨¤ ¥ ¥ 47£ ¡ £¢¡¤ ¤ £¨¥ ¡¤ ¥¥¢ ¤ £ ¤ ¤ £¨¥ ¥¥ ¤ ¥ ¥¢

Wie sieht der Feldlinienverlauf aus?¤¢¡ ¨%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%¤¢£ ¨%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Welche Polarisationsladungsdichten liegen vor?¢ ¨ div ¢¢ ¡ 'div ' Daraus folgt: £mit der Ausnahme der Position der Ladung¥ . An der Trennfläche jedochgibt es Oberflächenladungsdichten, die sich wegen ¦ §¡ nicht aufheben:Sei nun ¢der Einheitsvektor von Dielektrikum 1 nach Dielektrikum 2.£¡ £ ¢ ¦ £¢ ¦ ¨ ' ¦ £¢ ¨ ¦ ¨ ¨ ¦""""" © ¦!¨ ¦ ¡ ¡ £¢¡¢ §¡ ¦ ¡ ¥ ¥¢ ¦ ¨ ¦§¡ ¦ §¡ ¡ ¡ £¢¡£¡ ¥ ¢ ¡ £¢ §¡¨ ¡""""" © §¡ ¨ §¡ ¡ ¡ £¢¡¢ §¡ ¦ ¡ ¥ ¥¢ §¡ ¨ ¦ §¡ ¡ ¡ £¢¡48

¡ ¡ ¤Orthonormalität:£¡£ ¨£¡¡ ¡© ¥¢ ¥ ¦ ¨ §¡ ¢ ¡ ¡ ¢§ £¨§¡£ £ "¢¡¡ £¥ ¥¨Daraus kann die Nettoladungsdichte berechnet werden:Somit ergibt sich für ¦ §¡eine Nettooberflächenladungsdichte von gleichem Vorzeichen wie¥und für ¦ ¡ §¡ eine Nettooberflächenladungsdichte von entgegengesetztem Vorzeichen wie¥.1.14 Lösung der Laplacegleichung durch Reihenentwicklung ¦ ¦ ¡ ¡ £¢¡§¡ ¢ ¢ ¢¡ ¡ ¢ ¡ ¢ ¡ ¢ ¡ £Dies ist separierbar: ¡ ¢ ¡ ¡ ¢ ¢ Beide Anteile müssen für sich konstant sein:¡ ¡ ¨£¢ ¤¢ ¡ ¤£Die Kugelfunktionen sollen eindeutig und endlich sein bezüglich ¢ ¢ ¡ ¢ ¡ ¥¢¦¢ ££¨§©§und regulär. Es ergeben sich (ohne Beweis) die sogenannten Kugelfunktionen:¥ mit¢ £ "" ¢"¡ £ £ ¨¢©"¨¢ "¡¤¡©¥ ¤ ¨¥¨ ¨ ¢ ¡¡ ¢ ¤49&¡©

wobei die Konstanten % ¥¢% ¨#¥ ¨ ¡ ¥ ¢¡ ¦¢ ¢ ¢ ¨ ¢¢ ¢¤ ¥ ¤ ¨$¥ ¡ ¦ ¥ ¥¥ ¤ % und ¥ ¥¥£¥¥ ¢¨ ¦ ¨Die unendliche Menge der Kugelfunktionen ist vollständig, d.h. es sei % eine beliebigeFunktion. Es gilt also: ¥ mit¤¦¤¡ ¡¡©¥ ©¥ % £Es gilt:¥ ¢ ¢ ¢ ©¨© ¦¢ ¨© ¥ ¦"¥¥Dabei sind ¢zugeordnete Legendre Polynome:¦¢ ¥¥¥ ¢ §" ¤£ £ " ¢Legendre Polynom: ¢Die erfüllen die Orthogonalitätsrelation¢¢ ¡ ¨ ¨¦ und sind vollständig auf dem Intervall:Es gilt:Nun zur Radialgleichung:¨ § §¥ ¡ ¡ ¨¢ ¦¢ ¡¡ £Man findet leicht zwei linear unabhängige Lösungen:¡Somit ergibt sich die allgemeinste Lösung der Laplacegleichung¥ ¦¥ ¥über die Randbedingungen festzulegen sind.Wir betrachten nun drei Beispiele:50

1.14.1 Potential einer Punktladung% % % % % % % % % % % % % % % % % % % % % % %%% % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%% ¨ % ¨ ¡ ¨ ¢ Da keine -Abhängigkeit vorhanden ist, genügt die Entwicklung nach den ¢ ’s: ! Für ¡ ist regulär. Das bedeutet ¥ ¤£Spezialfall: Da wobei© % ; es gilt ¢ ¤£ % ¤ ¡ ¦¥ .% ¨ ¨ ¦ ¢ ¨ ¦ ! ¢ ¨ ¦ ¢ symmetrisch in und ist, muß allgemein geltenDiese Entwicklung wird oft gebraucht. ¤¤¨ ¦ ¢ ¤ ¢ " ¤¢¡¤£ " 51© % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%

1.14.2 Dielektrische Kugel im homogenen Feld'©% ra% % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % %% % % %% %%% % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%% % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% %% %% %% %% % %%%% % %%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Die Kugel wird polarisiert werden und das Feld verändern.Da keine verschiebbaren Ladungen vorliegen, gilt im Innen- und Außenraum:Da eine axiale Symmetrie vorliegt gilt:" ¤ ¡Die Randbedingung im Unendlichen lautet:Alle anderen ¥ ’s¤£¤ ¨ '©¡ .für ¢ % ¡ ¤£ ¢ für§¡¥ £ ¦ ¨¢ für£ ¡ ¨ '© ¥ ¦ ¨ '©Es liegt ein Sprung der Normalkomponenten von'an der Kugeloberfläche vor:% ¢ ¡¨ "" ""¥ ¢"" ¨ ¤ ¡" " ¦ ¢ ¥ ¦ ¢ ¦¢ ¨ % ¦ ¥ ¦ ¨¢ £ ¦ ¢ % ¨ ¦¢ £ ¡¡¡¦ für ¨"" ¥ ¢"¤¢ £¡¡¢ ¨Weiterhin liegt eine Stetigkeit der Tangentialkomponente von ' an der Kugeloberfläche vor:¡ ¨"""" ¥ ¢ " " ¨ ¡¤ £ " ¨ ¡¤ ""¨ ¡" " ¥ ¢const"52¡¤ ¢""" ¥ ¢ ""% % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%'©

%% ¥ ¦ % ¦ ¥ ¦ £ ¦ ¡ ¦ ¨ '© %" ¨¡¨£ ¦¡¦ £ ¥ ¦ ¨¢ £ ¦¡¡¥ ¦ ¡§ '© ¨¢' "¢%%%%% %%% %%% %%% %% %% %% %% %% % %% %% % %% % % %% % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % %% % % % %% % %% % %% % %% %% %% %% %% %% %%% %%% %%%% %%%%% %%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % %§§§In der Differenz müssen die Komponenten £für ¢einzeln verschwinden:Zusammen mit der oberen Bedingung folgt:Weiterhin gilt: ¡ £ £für " ¢ £¦ ¢ ¦ £¡¢ ¢'© ¢ ¥ ¦¢ ¨ ¨ #¨ Folglich: ¢'© ¨$¨¢ '©Und somit herrscht innerhalb der Kugel ein konstantes elektrisches Feld:¢ '©' " §in -RichtungOffensichtlich ist ' " ¡ '© . Diese Schwächung rührt von der Polarisierbarkeit her:¢ '©¢ ¡Sie erzeugt eine Oberflächenladungsdichte¢ #¨ '©£¡¢ § ¢'©$¨ -------++%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%++++++53

'¤ ¤ £ ¢¡¡ ¡Also liegt eine Überlagerung des homogenen Feldes' © und des Feldes eines Dipols vor.¢Das letztere resultiert als Volumenintegral über die konstante Polarisierung ¤§ ¡ ¥ ¢ ¨ %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % %'©% % % % % % % % % % % % % % % % % % % % % % % ¢ ¥¥ ¥ ¢ ¢&¥ ¢ ¢ ¢ ¢ ¨ ¦ ¥ ¨ ¦ ¥ ¥¤% % % % % % % % % % % % % % % % % % % % % % % ¥ ¥ welche dem äußeren Feld entgegenwirkt.¡¤ ¨ '© ¢'©$¨ mit¤ #¨ '© ¨ :'© ¢ § $¨ § ¡1.14.3 Multipolentwicklung einer Ladungsverteilung ¨ ¦¢ falls ¨ ¡ Diese Wahl ist angemessen, das Potential außerhalb der Ladungsverteilung zu untersuchen.%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Nun gilt ein Additionstheorem der Kugelfunktionen:%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ¨ ¥ ¨ ¦ ¥ Eingesetzt ergibt dies:¥ ¢ ¢54

¤£©¨© ¤ ¢ ¦ ¦ ¨¢ ¥ ¥ ¥ ¦ ¦ ¡ ¥£ ¤ §¡ ¥ ¦ ¥ ¡ ¥ ¨¡Dabei ist das Multipolmoment von definiert als:Niedrigste Ordnungen:¤ ¤ ¥ Dies ist das Monopolfeld© ¢ § ¦ ©¢ § £¢ ¢ ¦ ¦ ¢ § ¨ ¢ ¢ ¥¦ ¦ ¥ ¥ § £ ¥ ¦ ¡ ¦ ¥ wobei das Dipolmoment ist. Somit finden wir das Dipolfeld: ¤¢ ¢ Nach Einsetzen der Kugelfunktionen läßt sich der Zusammenhang mit den kartesischen Komponentendes Quadrupolfeldes erkennen.1.15 Energiedichte im Dielektrikum Das polarisierbare Medium wird zusätzlich zum freien Raum als Träger der Energie auftreten.Es sei nun bei gegebener Ladungsverteilung und gegebenem Potential zusätzlich hinzugebracht.Dies bewirkt eine zusätzliche potentielle Energie. ¤ £ ¦ 55

§ ¤ Bei einfachster Verknüpfung zwischen im Dielektrikum. Vor-DieEnergiedichteausgesetzt esgilt '.© ¥¦ ¨ £ ¢ ¤ ¡©©¥und ' ¡ © ¢£ ¥ £ £gilt'dann: £ ' ¦ ¤ Das Oberflächenintegral ist¤£, da im £ die Felder genügend schnell abklingen sollen.' ', ' ¡ ¢ ¤¦' £ ' £' ¡im Vakuum wird ersetzt durch 56

2 Magnetfelder stationärer StrömeDie Beschreibung und Erklärung magnetischer Phänomene war historisch gesehen schwierigerals das Gebiet der Elektrostatik, da man keine magnetischen Ladungen (magnetische Monopole)fand. Dies gilt bis heute. Die einfachste Einheit ist der magnetische Dipol (Magnet), der eineAuslenkung infolge eines Drehmomentes erfährt, hervorgerufen durch ein Kraftfeld,das manheute magnetische Induktion nennt. Die entscheidende Einsicht war, als man den Zusammenhangzwischen bewegten Ladungen, Strömen, und der magnetischen Induktion erkannte:1819 Oersted:1820 Biot-Savart1820-1825 Ampèrestromdurchflossene Drähte erzeugen Ablenkung von magnetischenDipolen.quantitativer Zusammenhang zwischen Strömen, magnetischerInduktion und den Kräften zwischen den Strömen.2.1 Die magnetische Induktion B und Kräfte zwischen StrömenMan findet zwischen 2 Leiterschleifen ( geschlossenen Stromkreisen ) folgende magnetischeWechselwirkung ( Kraft ):¡ ¦ ¡¡¡§¦§¡ Dies entspricht folgender Anordnung: ¦ ¦ ¡ ¦!¨ ¡ ¦ ¨ ¡ ¡¡% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%% %%%%% %% % %%%% %%%%%%% %%%%% %%%%%% %%% %%%%% %% %%% %%% % % %% % %% %%%%% %%%%%%%% %%%%%%%%%%%%%%%%%% %%% %%%% % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % %% % %%%% %% %% % % % %% % % % % % % % % % % % % %% % % % % % % % % %%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% %%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%% %%%%¦ %%% ¦!¨ ¡ % %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % ¡ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ¡ ¦ ¡ Wie sind Ströme I definiert ? Seien N Ladungsträger pro Volumen und der Ladung q vorhanden, mit einer¢mittleren Geschwindigkeit . Dann passiert in einem Zeitintervall eine Ladung ein vorgegebenes gerichtetes Flächenelement:¥ ¢ £ ¥%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% %%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%% %%% %% %% %% %%%% %% %% %% %%%% %% %% %% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % %%% %% %% %%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%¢57¢ ¦¥ &

¢ ¦¥ ¥Somit ist die transportierte Ladung pro Zeiteinheit :&¥ ¥ £ ¢ £ wobei ¡& ¥ £ ¢eine Stromdichte darstellt. Also: ¥ £ ¥ £ ¢ £¢ ¦¥£ ¢£ ¡Über den Querschnitt des ladungsführenden Trägers aufintegriert, ergibt sich der Strom: §¡£ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% %% %% %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%% %% % % % % % % % % % % % % % % % % % % % % % % % % % % % %%% % % % %% % % % % % % % % % % % % % %% % % % % % % % % % % % %% % %%%% % % % % % %%%%%% % % % % % % % %%% % % % % % % % % % % % % % % % % % % % %% % % % % % %% % %% % %%%% % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % %%%%%%% %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % %%%% % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Schon hier sei die Bilanzgleichung für die Ladungserhaltung angegeben. Sei eine geschlosseneOberfläche, dann muss im Fall eines positiven austretenden Nettostroms durch die Ladung indem von umschlossenen Volumen abnehmen :¢¡£ §Dies muss demnach auch lokal gelten: ¨ ¡¥ ¢ ¢£¡ ¡ £¥Diese Beziehung nennt man Kontinuitätsgleichung. Bei stationären ¥¤ £Verhältnissen ist¢¡ und £somit ( Es tritt soviel an Strom ein wie aus ). Die Einheit von ist über die Einheit der¦Ladung festgelegt.Der ¥ Faktor im obigen Kraftgesetz ( = Lichtgeschwindigkeit ), wird im Rahmen der Relativitätstheorienatürlich erklärt werden. Der obige Ausdruck für ¡ ist scheinbar nicht symmetrischunter Vertauschung von Leiter § 1 2, was nicht akzeptabel wäre. Dem ist aber nicht so. Wir¦verwenden % ££ %und erhalten dann :%¡ ¦ ¡ ¥¡¡¡§¦§¡¡¡¢¡ £ £ ¥(¨ ¥ ££¡§ ¦ £ ¦¡ ¨ ¦¡§ ¦ £ ¡ 58 ¦¡ ¢£

Nun gilt :folglich : §¦ £ £ ¦¡ ¦ §¦ ¨ ¦ ¡ ¦¡§ ¦¡ £ ¦¡ ¡§¦§¡ ¦ £ ¡ ¦¡ ¡Dieser Ausdruck hat nun die erforderliche Symmetrie und ändert sein Vorzeichen bei Vertauschungund . vonGehen wir nun zurück zur ersten Form. Sie lässt sich zerlegen als:¢mit ¦¡¥ ¡§ ¦ ¡ ¡¡ ¦ §¡§ ¡ ¡¥$¡§ ¦ ¦ ¨ ¡ ¦ ¨ ¡ ist die magnetische Induktion hervorgerufen durch die Stromschleife 2.¥$¡Dieses lässt sich nun verallgemeinern für die Stromdichten : ¡¥ ¡ ¡¡ ¡ ¡¥ ¨ ¨ Der Ausdruck¥für ist die Verallgemeinerung des Biot - Savart’schen Gesetzes, welches indifferentieller Form angegeben wurde: ¡Im Ausdruck für die Kraft erkennen wir die elementare Lorentzkraft ¥¡ ¡ ¨ ¨ ¥ ¢ ¡¥¡Sie ist die Kraft, die auf eine bewegte Ladung wirkt. Sei nämlich eine stromdurchflossene Leiterschleifeim Magnetfeld¥gegeben:%% % % %% % % % % %% % % % % %%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % %% % % % % %% % % % % %%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % %% % % % % %% % % % %%% % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%59¥ ¦

Da ¢¢¡ ¡ % ¡£ ¡ § ¡% ¢ §¡¡§ ¡ ¢¡¥ ¥¥¡ ¥¥Kraftelement¤£ % Ladung pro Volumen£Volumen % ¥& ¡ £ ¥ und Kraft auf die Leiterschleife¢ ¥ ¡ ¡§ ¡Sei nun¥ein konstanter Vektor über den Bereich der Schleife, dann folgt§ ¡Die gesamte Schleife spürt demnach keine Kraft. Sie spürt aber ein Drehmoment, das auf dasmagnetische Dipolmoment der Schleife wirkt: ¡¥ ¤£ £ ¨ ¥ £ Wir betrachten jetzt nur die x-Komponente :§¤£ ¥ ¢ ¥ ¡ ¥!¢ ¨ ¥ ¢ £¢ ¦¥Nun ist$¡ £¡§ ¢ £¢ £somit ist£ § ¢ ¥ ¡ ¥ %Einführung des Vektors :¨§©¨Welche Bedeutung haben diese Komponenten? ¢ ¨ ¢ ¨ £¢ ¨ ¢ 60

§ ¢ ¢% % % % % % % % % %%%%%%%%%%%%%%%%%%% % % % % % % %%%%¦¢ ¢% %%% %%% % %%% % ¡§ ¢% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% %%% % % % % % % % % % % % % %% % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%¡%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%§ ¢ ##¢¢£ ¦¢ ¢ £¢ £ ¡§ ¢ £¢£ ¡§ ¢ ¨ ¦¢ ¢ £¢schraffierte Fläche¢¢#Projektion von Weg C auf die¢- Ebene % % % % % % % % % %%%%%%%%%%%%%%%%%%% % % % % % % %%% ¡ ¢ % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%% % % % % % % % %%%%% % % % % % % % % %% % % % % % % % %%%%%%% % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ¦ ¢ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ¡ ¢ ¦ £ ¢ ¦ !¨ ¢ ¡ schraffierte Fläche¨¢ ¡ £ ¢Folglich schraffierte Fläche = Fläche aufgespannt von Projektion von auf die % £ ¡ % ¨¨-EbeneEntsprechend sind , die Flächen aufgespannt von den Projektionen von auf die undEbenen.Somit61

und£ ¡ ¥! %¡ ¨ ¥ ¡¡ ¥ Man definiert das magnetische Dipolmoment der stromdurchflossenen Leiterschleife wie folgt:&£ ¡oder allgemeiner für beliebige Stromdichte %¡% ¢ §¢ ¡ ¡ Die Stromschleife verhält sich demnach als magnetisches Dipolmoment, welches in konstantemMagnetfeld ein Drehmoment spürt£ ¡¡¥¡§ ¡ ¢2.2 Einfache Anwendungen des Biot - Savart’ schen GesetzesIm allgemeinen lässt sich das Integral für¥bei komplizierten Stromverteilungen nicht analytischauswerten. Wir betrachten nun einige symmetrische Anwendungen, wo dies möglich ist.Langer gerader Draht:¥ ¥ % ©%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% % % %%%% % % % % % % % % %% % % % % % % % % % % % %% % % % % % % %% % % % % % % % % % % % %% %% % % % % % % % % % %%%% % % % % % % % % % % % % % % %% % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % %%% % % % % % % % % % %% %%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ¨ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% % %%%%%%%%%%%%%%%% %%%%%%%% % % % % % % %%%%% % %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% %¡¡! ¢ ¢ ¡! ! © ¨ ¨ ¨ 62 ¥zeigt in ¨Richtung

¡ ¢ ¨ ¥ ¢¡ © ¢¢ ¡ ¡¨¥© ¡¨¥ ¢ ¡ ¨ ¢ Wegen der Rotationsinvarianz um die - Achse sind die ¥Zentrum.Kreisstrom: %% %% % %% % %% % %% % %% % %% % %% %% % %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % %%% % % %%%%%% %%% % % % % % % % % % %% % %%% % % %%% % % % %% %%% % %%% %%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% %¡% % %% % % % % %%%% % % % % % % % % % % % % % % % %% % % % %% % % %%% % %%% % % % % %% %%% % % %%%% %%% % % % % % % % % % % % % % % %%% % % % %%%% % % % % % % % %%% % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % %% % %% % %% % %% %% % %% % %% % %% % %% % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% %% %%% % % % % % % % % % % % % % % % % % % % % % % % %% % ¢ ¥ ¨ ¨ ¤¦ ¡¨ ¡¡¡§¦ ¨ ¢ ¡ ¢ -Linien Kreise mit der -Achse als ist nur auf der -Achse einfach¥zu berechnen¨¡ ¡ ¨ ¨ ¦¦ ¡ ¨ ¡ ¢ ¢ ¡ ¨¡ ¡ 63

¡ ¨ ¡ ¨ ¦ ¨ ¢ ¦¡ ¨ ¡ ¨ ¦ ¨ ¡ ¨ ¦¡ ¨ ¡ ¢ ¦ ¨ ¡ ¡¦ ¨ ¢ ¡ ¦¡ ¨ ¢ ¦ ¨ ¡ ¡¦ ¨ ¡ ¡¦ ¨ ¦ ¨ ¡ ¦¡ ¨ ¡ ¢ Aus Symmetriegründen kann nach der Integration nur die Komponente in -Richtung bestehen¥ ¡ ¦ ¨¡ ¡ ¢ ¡ ¡ ¡ £¥Anwendung: Helmholtz -Spule%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%¡¡¡ ¢Durch geignete Wahl von ¡ und kann das Feld in derUmgebung von ¢sehr homogen gemacht werden.Axialfeld für Spule%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ¡%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%©%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%¢£¢¤¢¥¢¤¢¤¢¥¢¢¢¢¢¢¢¢£¢¤¢¥¢¤¢¤¢¥¢¢¢¢¢¢¢%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ¡%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%¨ ¦ ¡£ Windungen auf Länge ¡¥!§ © ¡¦©£¡ ¡ ¡ ¢ ¡ ¡ ¨ © ¡ £¥ ¨ © ¡ ¡ 64

¢ ¡ ©¥ £¢ ¡¢ ¡£¡¢¡££¡ ¡£¡¥ ¢ ¢ £¥ ¢ ¢ £ ¨ ¡ ¨ ¢¡¢ ¨ ¢£ £ ¡¡¡¡¡ ¡ ¡¡ ¢©©¡ ¡¡¡¡©¤¨ ¡©¡¡¢© ¡© ¡¡¡ ¦ ¤¤££¦£££¥ § © ¢ ¡¡ ¡ ¡ £ ¥Sei ¡ ¨¡und © nicht in der Nähe der Enden:¡ ¢ ¡ ¢ ¦ ¢ ¡ ¡ ¨ ¡ ¨ ¦ ¦ ¥ ¨ ¥ ¢ ¥ ¢ ¦ ¡ ¨ ¡¥ § ¡ ¨ © ¡ ¡ ¡¡ ¤¡£¢Korrekturterme zum gewohnten Ausdruck©¨ 2.3 Differentielle Formulierung der Magnetostatik und Ampère’sches GesetzBerechnen ¢¡und ¢der magnetischen Induktion¥: ¨ £ ¡ ¡ ¡ ¨ ¡ ¡ ¨ 65

¨ ¡ £ Es gilt¢£¡ demnach¡£ ¤ ¦ ¡ ¨ £¡ ¥dieses¥-Feld hat keine Quellen. Quellen wären magnetische Monopole.¥ ¨ ¡¡Verwenden Stoke’schen Satz:§¡¨¡ ¡¡¨¡ ¡ ¡ ¡£ ¡ ¡ ¨ ¨ ¤ ¦ £ ¤ £¡ ¥ £ rot§¡ ¤ ¡ ¡£¦¨¤ ¡£¢ © für Magnetostatik¥ ¡¥ £ ¢ ist dabei der Gesamtstrom durch SDies ist das Ampère’sche Gesetz.Bei einfachen Geometrien kann man damit daslanger Draht¥%% % %% % %% % %% %% % %% % %% % %% % %% % %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%¥¤¡© ¡ ¡¡§¡ ¨ ¦ ¡£ ¨ § -Feld berechnen:¥ 66 ¢¥¡¡ ¨ ¨ ¡£¢ ¤¨ ¨

Im Fall, dass ¡ohne ¢Durch geeignete Wahl von ¡%%¢¡ % ¢¡ % ¢ ¡% ¡ % ¨ ¡¢¡ % ¡£ ¡ ¤£ ¡ %%2.4 Vektorpotential und magnetisches skalares PotentialKann man das¥-Feld auch durch ein Hilfspotential ausdrücken ?Im Gegensatz zu ¢ £gilt nun: ¦ ist, kann man nicht wie 'bei verfahren. Aber es gilt¥ £ ¢¡, somit £ & £da .Man kann nun das sogenannte Vektorpotential folgendermaßen abändern¢¡¥ ¢% "¥zu ändern.um ´einfache´ Gleichungen für ist also durch¥nicht eindeutig bestimmt. Diese Freiheit lässt sich benutzen,aufzustellen: ¡% ¨ ¡ kann man stets erreichen, dass¡ £ % ¤£ und demnach gilt%d.h. jede Komponente von erfüllt eine Poissongleichung. Eine Lösung ist¡¡ Dieser Ausdruck ist einfacher auszuwerten als der¥von Biot-Savart für ; jedoch muss manhinterher noch die Rotation bilden. Die Bedeutung von liegt vor allem in der Behandlung desStrahlungsfeldes (siehe später)¤£Im stromfreien ¡ Bereich, , gilt: ¨ demnach wie in der Elektrostatik¥ ¨ ¢ ¥ 67

% ¡% %¡" ¤ £¡ ¤ ¤ ¡ " ¤ " ¤¢ ¤ ¡¤¡¡¢ ¤ ¤¨ " ¡ ¤ £ ¢¨ ¤¤ ¤¨ ¤¤¨ ¤¡¤¡©¦Da weiterhin stetsGleichung¥ £ ¢£¡, gehorcht das magnetische skalare Potential¢ ¥der Laplace-¡ ¢ ¥ £Hauptanwendung in der Behandlung magnetischer Materialien (siehe später)2.5 Das Magnetfeld einer entfernten StromschleifeWie erscheint eine entfernte Stromschleife ?¡ ¨ " ¡ £ ¢ ¡Der 1. Term verschwindet in der Magnetostatik:¡ ¡" " ¨ ¢¡ ¡£¢ ¤Folglich £¡Umformung des Integrals¡"""¡ ¡ "£ £ " ¡ " ferner ¡folglich führender Term ¢ "¡""¡""" Betrachten Komponente" "¡ ¦ ¨ ¦ ¡ " ¡§ ¡ ¡ ¦ ¨ ¦ ¡ ¡ ¡ ¦ ¨ ¦ ¡ "68

¡% ¢¥ ¡© §¨ ¡ ¡¡ ¤¡¡ ¡ ¨ ¡§ ¡ ¡ ¤¡¢¡ ¦¡ £ ¡ Somit:¡ ¦ ¢ £¡in dem wir das magnetische Moment ¢ ¡ ¡ ¦ ¢ ¡der Stromverteilung wiedererkennen.Der führende asymptotische Ausdruck für das Vektorpotential ist demnach:% ¡ ¨ ¡ Daraus folgt die magnetische Induktion¨ £ ¡ ¡ £ ¨ £ ¨ § £© Diese Form ist dieselbe wie bei einem elektrischem Dipol, daher der Name magnetischer Dipol.Diese Form werden wir verwenden zur Beschreibung magnetisierter Materie, da die atomarenKreisströme makroskopisch gesehen weit weg sind.2.6 Potentielle Energie eines magnetischen Dipols im MagnetfeldWir entwickeln das äußere Magnetfeld¥ um das Zentrum der Stromverteilung: ¥ ¡ ¡ £ ¥¡¡ ¡¡¡ ¥ £ £ £ £ 69

¡¡¡ ¡¡ ¨ ¢ ¡ ¡ ¡¡¡ ¡¤ £ ¨ ¡ ¡ ¤ ¡ ¦ ¡ ¡ ¨ ¡ wegen "¥¢¥ £¥ ¡¥ £©Es folgt die potentielle Energie eines magnetischen Dipols im Feld¥¦¡¤ ¨ ¤¡"DerMagnetostatik.Ferner gilt:- Operator differentiert nur die¥- Komponenten. Das erste Integral verschwindet in der £¥ ¨ £ Beachte,Somit:¡ £ wirkt nur auf¥.¡ ¢ ¤ ¡für das äußere Feld¥ ¥ £¥ ¢ £¥ ¨ ¢ ¡ £¥ ¨ ¡Nun gilt folgende Identität £¥ ¡ ¨ ¢¥¡Nachweis:¢¥¡¨¥ £ Somit¡¡ ¨ ¥¡ ¡£¢ ¤ ¥¨ £¥ wegen rot¥ £¨ ¥ £2.7 Magnetische Eigenschaften von MaterieNeben den wahren Strömen, getragen von freien Elektronen oder Ionen, gibt es Ströme getragenvon gebundenen Elektronen. Diese ´Kreisströme´ erscheinen makroskopisch, d.h. weit entfernt,70

¢ ¥ ¢¢% ¢ £¨ £ ¨ £¢ £¨ £ ¨ £¥ & ¡¢¨ ¡¥ ¢¥ "" ""¨ ¡als magnetische Dipole; darüber hinaus tragen Elektronen selber ein magnetisches Dipolmoment.Makroskopisch gesehen resultiert aus der Summe dieser mikroskopischen Magnete in jedemmakroskopisch kleinem Volumen ein makroskopisches Dipolmoment ¥Dabei isteine Dipoldichte, die von Ort zu Ort variieren kann. Diese Magnetisierung derMaterie erzeugt ein magnetisches Induktionsfeld¥. Verwenden das Vektorpotential¥ %¥ ¨ Das Resultat der Magnetisierung lässt sich auch äquivalent aus Magnetisierungsströmen abgeleitetsehen. Es gilt¡¨ ¨ ¡demnach¢£¢¤¦¢¢£ % ¥Das zweite Integral lässt sich in ein Oberflächenintegral umwandeln: ¨ ¤¦¨¡ ¡(¦(¦¨ ¡£¢¥¤ ¨ ¢ ¢ ¨ ¨ ¨ " "" ¨ " ""¨¡£¢¥¤¡¥¡¨ ¥¡ ¨ ¨ ¨ ¡ ¨ ¨¥¡¥ £¡ Somit¨ ¡¥ Der Vergleich mit der Form¨ % ¡zeigt, dass ¨ die Rolle einer Stromdichte spielt. Man nennt sie Magnetisierungsstromdichte.Dies lässt sich veranschaulichen:¡71

Sei Dipoldichte durch kleine Stromkreise hervorgerufen:Wegen %¡ gilt dann¢%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % %%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % %% % % % % % %%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%¥ § ¥ § %%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ¡ ¦ ¡ ¥ ¡ £¢ ¡ ¢ ¥ £¢ £¢ der resultierende Nettostrom in y-Richtung ist dannoder die Stromdichte¥Dies ist Spezialfall ¡ ¨ ¡ ¢von .Hinsichtlich der Berechnung von oder ¡¥¥ ¨ ¡¢¡& ¦ ¨ ¡ ¨ ¡ ¡ ¡ ¨ ¡ £¢ ¥ ¥ können wir uns das Innere der magnetisierten Materie durch¥beschrieben denken.2.8 Die magnetische Feldstärke H¡ und den Magneti-Die magnetische Induktionsierungsströmen ¥:Es liegt nun nahe, analog zuherrührt:¡¥rührt von 2 Quellen her, den wahren Strömen ¢¡ ¦ ¤im Dielektrikum, ein neues Feld ¢¡ ¨ ¡72¢ ¡einzuführen, das nur von ¡

¢ £ ¢ ¦ £ ¡ ¡¨¢ ¦ £ ¡ ¤ ¡¨ ¦ ¤ ¦¡¥¡¤ & ¥(¨ ¢ ¤ ¦¦oder in Integralform§ £ ¤Wie ¥hängt mit den Magnetfeldern zusammen? Es ist üblich im Fall linearer Medien anzusetzen:¥ ¡ ¥ ¤ " ¡ ¥ ¤ ¡¢¡ ¤£ ¢ ¥£§¦©¨ £ ¢ ¢¥ £¢ ¥Mikroskopische Grundlagen von benötigt mehr atomare Kenntnisse (siehe FestkörperphysikVorlesung ¡ ¥ ).£paramagnetisch; vorhandene magnetische Momente werden ausgerichtet; ist¡temperaturabhängig.¥ £diamagnetisch; Momente werden induziert. Paramagnetische und diamagnetische¡Suszeptibiltäten sind klein.¥ferromagnetisch, ist groß. Ferromagnetische Substanzen können spontan magnetisiert sein¡( Ferromagnete ) ohne äußeres Feld.Wir erhalten¡¥ ! ¥ ¤ ¡ ¥ ¤ ¤ PermeabilitätSchließlich Randbedingung zwischen 2 linearen Medien:¡ ¢£¡¡ ¦ £¦ £ ¦ ¦ § £ £ ¤ ¦ ¢ ¨ ¤ ¡ ¢ ¢ ¤ ¡ ©¡ © wahre Oberflächenstromdichte in Grenzschicht.73

2.9 Beispiele zur Feldberechnung2.9.1 Die gleichförmig magnetisierte Kugel¥ % ¡¡¥ % ¡Benutzen die Kugelfunktionen£¦¡%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % %%% %%% % % % % % %%%%%% % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%% % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % %% % % % % % % %% % %% %% %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% %% %% %% % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % %% % % % % % % % % % % % % % % % % %¡ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% % % % % % %%¢ ¥¥ ¡ ¨ ¡¥ ¡¡ © ¥¡¡¦ ¥ ¦ ¨ ¥¡ ¡¥ ¡ ¦¦ ¢ ¡ ¨ ¡¢ ¨ ¡¦© ¡ ¥ ¢ ¢¦ ¨ ¦¦ ¦ ¦ ¨ ¦¦ ¨£ ¦ ¦ ¨ ¦¡¥¥¢ § ¢ ¢§ ¢§ ¢% ¡¤ ¦ ¦ ¦ ¦ £¢¡© ¥ 74¦ ¦ ¡ ¦ ¢ ¢ ¢ ¦ ¨ ¢ ¨ ¦ ¢ ¢ ¤¦ ¥ ¨¢ ¦"¨ ¢ ¢ ¢ ¢ ¢ ¢ ¤¦ ¥ ¨¡ ¥ ¦ ¡© ¥ ¢ §¢¨ ¤ ¦ ¦ ¨£ ¦ ¦ ¦¥ ¢

Das Kugelinnere:¤Das Kugeläußere:¤ ,¡¡ ,¦¡% § ¥ ¦ ¡ ¨¦ ¢ ¥ £ ¥ ¥ ¨ " ""%""§¢¦""¦ ¡ ¨§¢ ¢§¢ ¢ © ¨§ £ ¦¨¦ ¢ ¢ ¥ ¢ ¦¡ ¦¡¡% """"""" ¨ ¥ §§¢¢¢ © ¢ § ¥¤¡£ ¡¡ %§ ¥¡Das gesamte magnetische Dipolmoment der Kugel ist § ¥¡ Das Kugelinnere¡ % %¥ $ £¥ ¡ £homogenes Feld.Das Kugeläußere¥!§ ¥ ¥" § ¥ ¥ # ¨ ¨ ¨ © ©¥ ¡ ¨ ¢ ¨§ ¢ © ©¢ © ¨ ¢ ¢¥ ¢¥¤¦¥ § © oder¢¥¤¦¥ ¥ § £75

Dies ist das Feld eines Dipols%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%¥Wie sieht das Magnetfeld ¤aus?¤ " ¥ " #¨ ¥ ¨ § ¥¤ ¢¤¦¥ ¥ ¢¥¤§¦¥ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%¤Es müssen Quellen für ¤an der Oberfläche vorhanden sein:¢¡ ¢£¡ ¡¡ ¢ ¤©¨ ¢£¡¢¡ ¡¨ ¡ ¨ £ ¡ £¢ ¡ ¡ 76

¨ ¡¨ ¡ ¢ ¡ ¡ ¡Somit¢£¡ ¢£¡ ¤und ¥ ¤¤¥" ¥ §§ § ¢ ¡¡¤" Speziell für sehr großes strebt " ¢§ ¥ © ¥¥ ©§¨ ¡ ¨ ¡ ¥ ©¥ © ¤Nun gilt ¨ ¨ ¨ ¨ ¥ ¨ ¥ In der Tat, auf der Oberfläche und winkelabhängig liegen Quellen, die die Unstetigkeit vonerklären.Nun zu einer permeablen Kugel im äußeren Feld. Permeabel heißt, dass die Magnetisierung erstdurch das äußere Feld hervorgerufen wird.Nehmen zunächst an, dass die Magnetisierung bereitsvorhanden ist.Wegen der Linearität der Gleichung ¥kann man eine gleichförmige¨ ¨ magnetische Induktion ¥ © © überlagern. Daraus folgt¥" § ¥ ¨ ¤¥ ©¥ und¤" ¨ § ¥ ¥ © ¨ § ¥ ¥" $¨ ¥©Nun berücksichtigen wir,¥dass aus der Anwendung des äußeren Feldes resultiert.Verwendendazu den linearen Zusammenhang¥ © ¨ ¥ © §¥ §¥ ¨ ¢wasexplizitFerner gilt dann¥als Folge von ¥ © zeigt.¨ ¥" ¥ © ¢77

Anmerkung: Bei Ferromagneten gilt nicht der lineare Zusammenhang ¥ ¤, sondern eineHysterese Kurve ¥ ¥ ¤ Elemenieren zunächst aus die Magnetisierung ¥ :¥ " ¢ ¤ " § ¥ ©Mittels des nichtlinearen Zusammenhangs¥ " ¥ " ¤ " gewinnt man dann ¥ " und ¤ " als Funktion von ¥ © und daraus dann ¥ .2.9.2 Magnetische AbschirmungLassen sich Magnetfelder abschirmen?%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%¥ ©¥ © %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ¡ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%permeables Medium¢¤ ¤£ ¢¡ ¤ ¢¡ ¥ £Dies gilt innerhalb der 3 Bereiche.¤ ¨ ¢ ¥ ¡£¢ ¥ £Ansätze: ¡ ¢ ¥ ¨ ¥ © ! ©¨ ¦ ¢ ¡ ¡ ¡ ¡ ¢ ¥ ! ©£¢ ¥¤ ¨ ¦¢ ¡ ¡ ¢ ¥ ! © ¢ Randbedingungen an ¡und :¤¢ , ¥ ¥ stetig¢ ¥ ¨ ¢ ¥ 78

¦ ¨¦ ¢ § ¢¢¥ ¢ ¡ ¢ ¥ ¡¦ ¢ ¨¦ ¤¡¡¨ ¨ ¥¤ ¦ ¢ ¦ ¨¢ ¤ ¦¡¦¡ ¥ ¢ ¡ ¢ ¥¥ ¢ ¡ ¦ ¢ ¡¨¨ ¢¡¨£ ¨ ¡§ ¨ ¡#£ ¨ ¡ ¥ ©# ¥ © ¢ ¥ ¨ Hieraus folgt:¥ ¢an der Stelle .Da die Legendre Polynome orthogonal sind, müssen die Koeffizienten der ¡ ’s gleich sein:¢¢ ¥ ¨ ¨¢ ¥ £¢ ¥ ¨ ¨¢ ¥ unabhängig von . Dies gilt auch für ¡ ¢ ¡¥¤ ¦¡ ¢ ¦ ¤ ¦ ¡ ¢ ¢ ¦ ¡¥ © ¦ ¨¦ ¡Nachdem in radialer Richtung differentiert wurde, gilt wieder das gleiche Argument wegen derOrthogonalität der ¢ ’s:¨ ¤¢ ¨ ¡ ¢ ¢ ¦ ¨ ¤ ¢¡ ¢ ¨ ¡ ¦ ¢ ¢ ¡ ¦ ¨ ¤ ¡ ¨ ¥ © ¨¢ ¦Es folgt fürund¢ ¢ ¤ £ ¢ ¦ ¨¢ ¤ ¦ ¢ ¦ £ ¢ ¢ ¨ ¢ ¢ ¨ ¢ ¢£ ¢ ¢ ¢ ¨ ¢ ¢£79

¦¡ ¦¤ Im Hinblick auf die gesuchte Abschirmung wählen wir ¨ ¦ ¢¨ § ¡ ¢ ¡ ¨ ¢ ¢£# £ ¡ ¨ ¡¢¦ ¢Damit lassen sich die Magnetfelder angeben: :¥ © ¥ © ¡ ¨ ¢ ¢£# £ ¡ ¥ © ¨ ¡ ¢¨ ¡¥ ¢¥¤ ¨ ¨ ¥ ©¢ ¨ ¡¦ ¢ ¨§ ¢ ¨ ¡¦ ¢ ¨§ ¡¤ ¢ ¨ ¡Dies ist Überlagerung des homogenen¥ © -Feldes und eines magnetischen Dipolfeldes.¥ § ¢ ¨ § ¥ © ¨ ¡ ¥ " ¨ ¡¢ ¨¡£¨§ ¢ ¥ ©¨ ¡ £ ¨ ¡ ¥ ©¨ ¡¥ © #¨¡ ¡ §¢ ¡ ¥ © § ¡¢ ¨ ¡¥ ©© ¥© ¥¥ © ¡ §Das Feld im Hohlraum ist demnach um Faktor unterdrückt. Da es Materialien mit ¢¡£¤£gibt, ist Abschirmung möglich. Wegen des Faktors¦verlaufen die Feldlinien hauptsächlich in der Schale.¦% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%% % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%% % % % % % % % % % % % %%%%%%%%%%%% % % % %%%%% % % % % % % % % % % % % % % % % % % % % % %% % % % % %%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%% %% %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%% % % %%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%% % % % % % % % % % % % % % % % % % % % %% %% % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%% % % % % %%% % % % % % % % %% %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % %%%% % % %% %%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % %% %% % % % %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % %% % %%%% %% % % % %% % % % % % % % % % % % % % % % %% % % % % % % % % % % % % %%% %% % % % % % % % % % % %% % % %%% %%% % % %%% %%% % % % % % % % % %% % %%% %% %% %% %%% %% %%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%80# £ ¢£, der groß gewählt werden kann,£ ¨

3 Zeitabhängige Felder und die Maxwell-GleichungenDie statischen Probleme behandelten elektrische und magnetische Felder als getrennte Phänomene.Die einzige Verbindung waren die Ladungen, die in Ruhe Quellen für elektrische und inBewegung auch Quelle für magnetische Felder sind. Ein enger Zusammenhang zwischen' und¥wird sich bei zeitlich veränderlichen Vorgängen ergeben. Mit den Einsichten der speziellenRelativitätstheorie werden wir sogar erkennen, daß oder ¥dominant in Erscheinung tritt, ist nur eine Frage des Bezugsystems.'und 3.1 Faraday’sches Induktionsgesetz (1831)Beobachtungen von Faraday¥%% % % % % %% % % % % %% % % % % %%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % %% % % % % %% % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % %% % % % %% % % % % %%% % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %%%%% % % % % % % % % % % % % % % % % % % % % % % % % %%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % %% % % % % %% % %% % %%% %% %%% %% %% %% %% %% % % % %%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % %% % % % % % % % % % % %%%%%%%%%%%%%% %% %%%%%% %%%% % % % % %%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% %% %% %% %%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % %% % %% %% %% %%% %%% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % %% % % % % % %% % % % % % %%%%% % % % % % % % %% % %%% % % % % % % % % % % % %% % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% geschlossene Drahtschleife 1% % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % %% % %% %%% % %%%%%%% %%% % %% %%%%%%%%%%%%%%%%%%%%% % % % %%% % % %%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% % % % % % %% %%%%% % % % % %% % % %%%%% %%%% % % %%%%% %% % % % %% %%%%% % %%%% %%%%% % % %% %% % % %%%% %%% % % % % %%% %%%% %% %%%%%%%%% %%% %%%% % %%% % % % % %% % % % % %%%% % %%%% % %% % % %% % % %% % % %% %% % %% % % %%¥von gleicher Natur sind, und ob £ %Das Magnetfeld¥sei durch eine zweite entfernte Stromschleife hervorgerufen. Es fließt einStrom in der Drahtschleife 1, falls1. der Strom in der zweiten Stromschleife an- oder abgeschaltet wird2. die zweite Stromschleife in bezug auf die Drahtschleife 1 bewegt wird3. falls¥durch zusätzliche Permanentmagnete verändert wird.Der Stromfluß in der Drahtschleife 1 muß durch eine sogenannte elektromotorische Kraft& § ¢' £ ¢ £ hervorgerufen werden. Nur so kann der Strom längs £Wir führen den magnetischen Fluß ein:Dabei ist eine beliebige Fläche, die von £ &§zeitlich, und das Faraday’sche Induktionsge-Durch die drei genannten Veränderungen variiertsetz lautet:¥ £ aufrecht erhalten werden.umrandet wird. Die Fläche ist beliebig, da div¥ £ 81'.

¨§ ' ¢ ¨ ¥§¥ £ ¨ ¥dabei ist Konstante £die , und das Wegintegral wird im mathematisch positiven Sinn durchlaufen.Das Minuszeichen drückt die Lenz’sche Regel aus, die besagt, daß der induzierte Stromin einer solchen Richtung verläuft, daß er durch sein Magnetfeld der Änderung des angelegtenMagnetfeldes entgegenwirkt.Dies sei nun verifiziert:Der induzierte Strom generiert das Magnetfeld¥ , wobei gilt:oder¥ ¡rot¥ £ ¢ § ¥ bildet mit eine Rechtsschraube. Es zeige nun ¥¡%% %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % %% % %% % %% % %%% %% %% % %% % %% % %% % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % %%Drahtschleife%1%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% %%%%%% %%%%%%%%% %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% %%% %% %%% %% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%nach oben und nehme zu. Wegen des Minuszeichensläuft der Strom im mathematisch negativen Sinn, und¥¥ wirkt¥entgegen.Im Gauß’schen Maßsystem ist ¡ .Das Induktionsgesetz lautet also:¢§' £ ¢ ¨ ¡% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %¥ §%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%% % %% % %% %% % %% % %% % %% % %% % %% % %% %Zunächst wollen wir die Dimension von untersuchen:¡ ' £ ¢£¢ Ladung% £ ¥¥ £ £Länge¡¥ £ ¦¥ ¢ ¥¤§¦Länge Zeit¤§¦ Zeit LadungZeit£Länge Länge Zeit¡¡ ¤§¦ ZeitLänge£§Geschwindigkeit82LängeLänge £ ¡ "Länge¡

Warum ist gerade ¦ ? Für einen speziellen Fall können wir dieses Gesetz auf bereits Bekannteszurückführen. Im Fall 2. wird die von konstantem Strom durchflossene zweite Schleife gegenüberder ruhenden 1. Schleife bewegt. Es erscheint plausibel, daß dieselben Phänomene auftretensollten, falls die erste Schleife gegenüber der zweiten nun ruhenden bewegt wird. Wenn dieRelativbewegung der Schleifen gleichförmig erfolgt, ist dies nur ein Übergang von einem Inertialsystemzu einem anderen, und die physikalischen Phänomene, hier , dürfen sich nicht ändern(siehe Kapitel über spezielle Relativitätstheorie).Wir bewegen also nun die Schleife 1 in einem vorgegebenen Feld¥:% % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % %% %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % ¢%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%¥% % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Die Ladungen in der Schleife werden bewegt und spüren die Lorentzkraft ¡¥ ¡¥ ¢Daraus resultiert die elektromotorische Kraft längs der Schleife¡§ ¢¡ £ ¢ ¥Dies läßt sich in der Form des Induktionsgesetzes schreiben:% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % %%% % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%% %% %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%% %%%% %% % % %%% % % % %%%%%%%%%%% %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % %% % % % % % % % % % % % % % %%%%% %% % % % % % %% % % %%% %%% %% %% %%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % %%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % %% %% %% %% % %% %% %% %% %% %% %% %% %% %% %% %% %% %% %%%%% %%% %% %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%¢ %% %% %% % %% %% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%% %%%%%%% %%%%%%% %%%%%% %%¢ ist das Flächenelement für die Mantelfläche zwischen den zwei Lagen des Stromkreises.Es weist nachaußen.¡¢ ¥ ¥ § ¢ £ ¢ ¥ § ¢¡83¢ ¥ ¢ ¥ £¥¡

¢¡§¥¥¥¥ £ ¥ MantelflächeWegen div¥ £ Gebildes Null:ist der gesamte Fluß durch die Oberfläche des von den 2 Lagen aufgespanntenMantelfläche ¨ £Hier ist zu den beiden Zeiten bezüglich entgegengesetzter Normalen berechnet. Wir wollenden Fluß auf die gleiche Normalenrichtung beziehen und vergleichen:Mantelfläche ¨ ¨Mantelfläche ¨ ¨Somit ¨ Mantelfläche ¥ ¥ ¨ ¨ ¡¡Dies hat die Form des Induktionsgesetzes mit ¡ .Das Faraday’sche Gesetz ist jedoch wesentlich allgemeiner: £auch, falls Leiter ruht, und das-Feld sich zeitlich ändert! Die ruhenden Ladungen erfahren im Magnetfeld keine Lorentzkraft!¥Es tritt demnach ein neues Phänomen auf. Ein zeitlich veränderlicher Fluß erzeugt ein nichtwirbelfreies elektrisches Feld :' § ¢ ¨ ¤£ £' £ Dies könnte für beliebige geschlossene Kurven gültig sein, unabhängig vom Vorhandenseineiner Drahtschleife. Dann würde mit dem Stoke’schen Satz folgen:£ ' ¨ ¡ rotDiese Verallgemeinerung ist in der Tat Realität. Ein zeitlich veränderliches¥-Feld induziert einelektrisches Feld mit rot ' £. Diese Beziehung ist die Grundlage des Phänomens der Wellenausbreitungvon ¥im Vakuum.'und Auf dieser Gleichung beruht auch unmittelbar das Funktionieren des Betatrons.3.2 Die Energie im magnetischen FeldBeim Aufbau der Ströme und den dazu verknüpften Magnetfeldern treten Induktionsvorgängeauf, gegenüber denen die Kräfte zum Stromaufbau Arbeit verrichten müssen, um den Strom aufrechtzuerhalten.84

¡§ ¡§Wir eliminieren nun zugunsten von ¤ somit ¡¤ £ ¡ % £¡ £¡ £ % ¢¥¡% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%£ ¨ ¥ ¤ ¥ £¡ % ¢ £ ¦ %Die induzierte elektromotorische Kraft leistet an dem Stromfluß folgende Arbeit pro sec:££.Aufgrund des Induktionsgesetzes ist dies£ ¥ ¨ ¡%%%%%%%%%%%%%%%%%%%%%äußere QuelleDem entgegengesetzt ist die Arbeitsrate der äußeren Quelle,¥ . Also: ¥¡Wir wollen die rechte Seite in Feldgrößen ausdrücken:¡ ¥ £ §¡¡¤ % £ ¤ ¡ Nun gilt die Identität£ £ £ £ ¡ ¡¤ ¤ £ % Bei lokalisierter Feldverteilung verschwindet der zweite Term und ¤ £¤ £Für ein lineares Medium gilt: 85

¥¢ ¥¢ ¢ % ¤£% £¤ ¥¤¡% £ % ¦ SomitDie gespeicherte Energiedichte im Magnetfeld ¤ £¥ ist bemerkenswert analog zum Fall der gespeicherten Energiedichte im elektrischen Feld¤ £' £ Im elektrostatischen Feld läßt sich die gespeicherte Energie auch schreiben als:¢ Die analoge Form im magnetischen Feld gewinnt man ebenfalls ¤ £¥ ¤ £ % rot£ rot¤ ¨ ¢ ¡ ¡3.3 Der Maxwell’sche Verschiebungsstrom und die Maxwell’schen GleichungenWir haben bisher folgende Gesetze kennengelernt: £ ¡¤ Coulombgesetz¡Ampere’sches Gesetzkeine magnetischen Monopole'( ¨ ¡¡ ¥ Faraday’sches GesetzDas Ampere’sche Gesetz ist unter stationären Bedingungen abgeleitet worden. Bleibt es gültigbei zeitlich veränderlichen Vorgängen?¥ £ £Nein, denn das Gesetz ist mathematisch nicht konsistent:div ¤ & £ ¡ rot div 86

¢ £ £ ¤ ¡ ¡rot' £¥ ¤ ¥ rotrot' ¨ ¡ '¢¥¥¢' ¥¢¥' Wegen der Kontinuitätsgleichung ist div bei zeitlich veränderlicher Ladungsdichte. Entwedergilt ein neues Gesetz oder man kann mit Hilfe der Kontinuitätsgleichung eine Erweiterung£ versuchen: Dies läßt sich schreiben als:¥ ££ ¥ £ ¥ ¤¦ ¢£Die rechte Seite ergibt eine Möglichkeit für eine erweiterte verallgemeinerte divergenzfreie Strom-dichte: ¢Man nennt¥ den Maxwell’schen Verschiebungsstrom.Maxwell postulierte folgende Erweiterung des Ampere’schen Gesetzes: ¢£¥ ¤¦Der Maxwell’sche Verschiebungsstrom enthält 2 Anteile: ' ¥ ¥Die zeitliche Änderung der Polarisationsdichte hat eine Ladungsverschiebung und damit eineStromdichte (¥ hat die Dimension einer Stromdichte!) zur Folge. Dieser zweite Teil sollte injedem Fall in Erscheinung treten. Im Vakuum jedoch bleibt nur ¥ und dort gilt dann:¡ Diese Gleichung ist in bemerkenswerter Symmetrie zu:Wir werden sehen, daß dieser¡ TermMöglichkeit gibt, Licht aller Frequenzen als elektromagnetische Wellen zu verstehen.¥ die freie Wellenausbreitung ermöglicht, und uns die87

£ ' ¥ ¡¤¥¥¥¥¥' ¥¤Kurz gesagt: Diese Max’wellsche Erweiterung hat sich als richtig erwiesen.Wir erhalten somit die endgültige Form der Maxwellgleichungen: £ ¥ £ £¤ ¡ ¡ ¡' ¨ ¡¡ Dies gilt im Dielektrikum und in magnetisierbarer Materie. Zu ergänzen ist, wie sich Materiegegenüber¥verhält, d.h. die Abhängigkeit der Polarisierung, der Magnetisierung und'und der Stromdichte von ¥. In den einfachsten Fällen gilt:'und und es gilt das Ohmsche Gesetz:¥ £LeitfähigkeitDarüberhinaus gilt die Kontinuitätsgleichung, die jedoch über die divergenzfreie Stromdichtebereits eingebaut ist. £Im Vakuum bei Vorhandensein von Ladungen und Strömen gilt:' "' ¥ £ £¡ ¡ ' ¨ ¡¡ Die geladenen Teilchen selber erfahren natürlich Kräfte im elektromagnetischen Feld. In klassischerBeschreibung gilt:¥ £ ' ¥ ¢¡Schließlich gilt im reinen Vakuum:¡ ¥¥88

% ' rot' ¡¥%£¢¥%¡¡¥ %' ¥ ¨ ¦ ¥¥' ¢£&¤div' div ¥%¤¦ ¥%' ¥£¥ £ £' £¡ ¡ ' ¨ 3.4 Vektorpotential und skalares Potential, EichtransformationenSowohl für praktische Lösungsverfahren der Maxwellgleichungen, wie auch für die formale Weiterentwicklungder Theorie, nämlich die kovariante und quantisierte Form der Elektrodynamik,ist die Ersetzung von und wesentlich.¥durch die Potentiale 'und Wegen ¥ £ £gilt¥ rot ¥ £ ¢rot¡ ¥ ¤¦ £¡ ¥ ¨grad ¡ oder'( ¨ Mit diesen beiden Ausdrücken 'für sind die beiden homogenen Maxwellgleichungenidentischerfüllt. und werden nun über die verbleibenden beiden inhomogenen Maxwellgleichungenfestgelegt. Wir betrachten nur den Fall von Ladungen und Strömen im Vakuum:¥und ¡ ¡rot¥ ¨div grad % ¡¨rot % rotgrad ¡ ¡Wir verwenden nun ¡ % ¡ £ % ¨ ¡ % 89

%Dabei bleiben ¥und % ¨ ¡ % ¡ £% £ % ¡ ¡¥ rot % ¨ % ¢ ¢ ¨ ¥% % ¨ ¡¥¥% ¡ & ¡dabei Eichinvarianz. Durch diese Umeichung in ¥ %gegeben, dann ist i. £a. ¡ £ ¡ £ % ¥ £ % ¥%¡ ¡und erhalten:% ¡$¡ ¨ ¥¡¡ ¡ £% ¨ Die vier Maxwellgleichungen von erster Ordnung in der Zeitableitung sind auf zwei gekoppelteGleichungen von zweiter Ordnung in der Zeitableitung umgeschrieben. Sie lassen sich sogarentkoppeln infolge der Freiheit in der Wahl der Potentiale:und ¥zu ändern:sind durch' und ¥nicht eindeutig festgelegt. Man kann und ändern, ohne' und' ¨ ¡ Freiheit :¡ ¥& die Invarianz vonerreichen, daßungeändert. Man nennt diese Transformation Eichtransformation, und%'und läßt sich'und ¥¤£%Nachweis: Seien undNach Umeichung gilt % ¡¡ ¤£ ¥ ¡ £ ¨ ¡ ¥¥ ¡¡ ¨ ¡¡ ¡ durch geeignete Wahl von ¡ .Man nennt¡ ¥ £Lorentzbedingung.Mit dieser Wahl von und erreicht man, daß entkoppelte Gleichungen gelten:90

¡ ¨ " ¥ ¡ ¡ ¡ ¨ ¥ ¡ ¨ ¥ ¡ ¨ ¥¤¡¨¤ ¨ ¨ ¥¨ ¡ ¨ ¡¨ " ¥ " ¥ ¡¨ ¥ " ¥ ¥ ¡ ¨ ¡¡ % ¡Selbst die Lorentz-Eichung %legteiner Funktion , die¡% ¨ und ¡ ¡¡noch nicht eindeutig fest. Die Umeichung vermöge¡ ¡ ¡erfüllt, ändert nichts an der Lorentzbedingung.¥ ¡ £¡¡ Neben der Lorentz-Eichung wird häufig auch die Coulomb-Eichung (oder transversale Eichung)verwendet:In diesem Fall gilt % £ £mit der Lösung ¡ ¨ " ¥ " ¥ " ¥ , daher der Name Coulombeichung. " ¥ das momentane Coulombpotential gemäß Also ist ¨ Ferner % ¡% ¨ ¡ ¡ ¡¡¡ ¡ aus-Die rechte Seite läßt sich über die Kontinuitätsgleichung allein in bestimmtem Teil vondrücken:¦ & Zerlegen¡& ¡ ¡mitrotdiv ¡ £Es gilt die Darstellung:¡ £¨ ¨ ¡ ¨ ¡ ¡ ¡¡ 91

¨ ¨ ¨ ¡ ¨ ¨ ¡ ¨ ¨ ¨ ¨ ¡ " ¥ ¨¨¨ ¡ £¨ ¡¡ ¨ ¡¨ ¡ ¡ ¨ ¡ ¡ ¥¡ " ¥ ¨ %¡ ¨ ¡ ¨ ¡¥Man benutzt diese Eichung manchmal im Vakuum, wo% ¨ ¥ rot % %%¥ " ¥ . Dann ist & £Nachweis: " ¥ ¡ ¨ " ¥ ¡¨ " ¥ ¡ ¤ ¡ " ¥ ¦ ¨ ¨Wiederholte partielle Integrationen ergibt: ¤ " ¥ ¦¨ ¤¡ " ¥ ¦ ¨ ¡ " ¥ ¨ ¡¨¨ Somit " ¥ und¡ % ¡¡ ¡ ¨ ¡ ¡¡¡ ¡ £¡ Die physikalischen Felder ergeben sich dann als¢ ¡ ¥ ¡ £¡¡ ¡ ' ¨ 3.5 Energie und Impuls im elektromagnetischem Feld; Erhaltungssätzevon Energie und Impuls für Systeme von Ladungen und elektromagnetischemFeld.Das elektromagnetische Feld verrichte Arbeit an den Ladungen. Dabei wird dem Feld Energieentzogen und in mechanische Energie umgewandelt.92

¡¨¡¨ £ ' ¤¡& ¥¡¡£ ' ¡¥ ¡£¢¡'¢£¤ £ ' ¨ ¡ rot¥¥ ¤£ ' ¡¢ ¤ ¨ rot¥¥¤ ' £ ¥ £ ¤ ¦ ¤ £ ¥¡§¡ ¤ £ ¥¥¤¦ ¥¥¤¦ ¥ Die pro Zeiteinheit durch' und ¥an einzelner Ladung geleistete Arbeit ist:Das Magnetfeld leistet keine Arbeit, da ¢ £ ' ¥¢¡Demnach ist die Arbeitsrate an einer Stromverteilung in einem festem Volumenelektromagnetischem Feld geleistet:¢¥ und vom¥ ' Sie tritt als mechanische Energie und letztlich als thermische Energie wieder in Erscheinung. Siewird dem Energieinhalt des elektromagnetischen Feldes entnommen.¡£Präzisieren wir diese naheliegende Aussage durch Verwendung der Maxwellgleichungen:¡£ ' ¥ ¤¦ Identität ' ¨ ' £rot ¤rotferner¤ ¦ Somit '¡£' ¨ ¡£ ¡Annahme linearer Medien:' £ ¢£ist die aus der Statik bekannte Energiedichte des elektromagnetischen Feldes.Somitoder¤ '¡£Wie ist das Oberflächenintegral zu interpretieren ?¡£' ¡ '¡ ¤ £ 93

haben dieselbe Dimension ( z.B. betrachte rot ¥ £und' ' ¥ ' ¤ ¥ ' ¥¥Poynting Vektor¡ '¡¤ 'und ¤ '( ¨ ¡ EnergieFläche £ Zeit¢ ¤¦ Energiedichte £ Geschwindigkeites liegt nahe als Energiestromdichte des elektromagnetischen Feldes zu interpretieren. Damitbeschreibt die obere Gleichung die Energiebilanz: die Abnahme der elektromagnetischen¢Energie links ist gleich der Arbeit an den Ladungen plus den Abflüssen von elektromagnetischerEnergie durch die Oberfläche.Dies läßt sich auch als differentielle Kontinuitätsgleichung schreiben:'Der Erhaltungssatz der Energie für das System geladener Teilchen plus elektromagnetischemFeld lautet:¡£ ¨ mit¥ ¥ ¡£ ' 'gilt ' ¡ ¤ ¡ £Die von der Elektrostatik her bekannte Energiedichte des elektromagnetichen Feldes erweist sichdemnach als allgemein gültig, auch bei zeitveränderlichen Feldern. ¥ ¨ § Nun eine Betrachtung zur Impulsbilanz für elektromagnetische Felder und freie Ladungen.Die Gesamtkraft auf eine Ladung¥ist: ¥ ¡Sie bewirkt nach dem Newton‘schem Gesetz eine Impulsänderung.¢¡ ¥ ¥ ' ¡ ¢¡¥ 94

¢ ¥ ¥ ¡ ¡¥ ¢ ¥ ¥ ¡¥ ¡ £¢ ¡ ¡ '' '¡ rot¥¡'¡'' ¡¡ ¥¡'¥ ¥ ¥div '¡¥ '¥ ¥ ' rot¡¥ £ ' ¨ ¥¥ ¥¨ £¥ ¡'¡¥ ¨ ¡ rot¥ ¤¦ ¥¥ ¤¦ ¨ ' ¨ ¥¡ rot¥ ¡ ¨rotVerallgemeinert auf alle Ladungen¡ ¡¥ ¡ ' Wir eliminieren nun und ¡ zugunsten der Felder:¥¨ div ¡ ¥ ¤¦¡¡ rot ' ¢¡ ¢£' £ '¡ ¥ ¢'div ¡¢£¨ 'div ¥¨ ¡¨Somit ¥¡ rot¥ ' £ ' ¨ Die rechte Seite läßt sich in ein Oberflächenintegral umwandeln. Dann liegt die Interpretationder linken Seite nahe:¡'Diese Größe hat die richtige Dimension: ¡ ¥ Impulsdichte des elektromagnetischen FeldesEnergiedichte durch GeschwindigkeitDemnach ist der Gesamtimpuls des elektromagnetischen Feldes im Volumen '¡¥ ¢) ¡95

¥ £¥¡ ¥ ¨ ¡¥¡ rotdiv¥ ¢ ¤¤¤ ¢ ¥ ¨ ¥ ¥ ¥£ ¥¤ ¥ ¤ ¥ ¢ ¨ ¥ ¥ ¢ ¨ £¡£ ¥£ ¢ ¥¡¥£¥¥ ¢ ¥¡ ¥ ¤ ¤ ¤ ¥ ¢ ¨ ¢ £¦ ¢ ¥¡¢¦ ¢ ¢ ¤ £§¤¢ £¦ ¢ ¥¡¦&¡ ¤¡¦ ¢ ¥¡§ £ £¦ ¢ ¥¡¢ ¤ ¢¥¡ ¤¥¡ ¢ ¦ ¢Nun zur Umwandlung der rechten Seite in ein Oberflächenintegral: ¥¨ Läßt sich dies als Divergenz darstellen ?¥ ¢¥ ¢ ¨ ¢ ¥ ¤ ¤ Zusammengefaßt zum Vektor ergibt sich:¢ ¢ ¥ ¥ ¢ £¦ ¢ ¨ ¢ ¢ £¦ £§¦ ¢¤ ¤ ¥ ¤ ¥ ¢ ¨ ¢¢ ¡£¢ ¤wird als Dyade bezeichnetDamit schreibt sich die rechte Seite alsmit dem Maxwell’schen Spannungstensor:¢ ¥ ¤ ¨ ¢ ' ¡¦¥¡ ¥ ' ¢ ' ¤ Schließlich lautet dann die Impulsbilanz:¥ ¢ ¥ ¢) § £ Offensichtlich muß die rechte Seite interpretiert werden als die vom elektromagnetischen Feldüber die Oberfläche auf das Volumen übertragene Gesamtkraft. Auf diese Art läßt sich in derTat die Kraft auf ein Stück Materie hervorgerufen durch ein elektromagnetisches Feld berechnen.96

Es sei Punktladung¥eine im homogenen elektrischen Feld' © :¢ ¤¦ ¢ ¢ ¤¦'¦ ¡ ¤ §£¦ ¢¢'© ¦ ¢ § £ ¦¤£ ¡ © ¦ ¢ § § ¥ © '©'£© ''© '©§¡ §¤ §¦ ¡'© ' ¡ §¦ '© '© ¨ ' ¡©£ §£ £¡'¦ £ '© ¢¦ ¢ ¦ ¢¦ § £§ ¦¦ ¢£¦ ¢Dies sei an einem einfachen Beispiel demonstriert:¦ 'Der Maxwellsche Spannungstensor lautet dann:'©' '¡ ' ¨ ' ¡¢ ¦ © '¦ £ '¦ ¨ ©'Das eigene Feld sollte nicht auf Ladung¥die wirken!Ist dies der Fall?'¦ ¨ ¦ ¢ £'¦ ¢ ¤ ¡§ &Wir betrachten nun eine gedachte Kugeloberfläche mit Radius um¥.§ ¡ © £ ¡ ¡ ¨ ¢¡ ©£ ¦ ¡ © ¤£ ¢ ¢Anteil von' © allein:£ © Schließlich der Interferenzanteil: '¦ © '¦ ¨ '¦ £ '© ' © '¦ ¨ '¦ £ '© ' ¥ ¡£¢ ¤ ¥ 97

¡ ¡ ¨ ¥ ¡ " ¥ ¡¡¡¡ ¢ " ! ¡!! !! £¢ " ¦" ¤& ¡!¤¨ & ¦ "¦ ¥¥ ¨ ¨ ¨ ¦ " ¤& ¨ ¤ &¨ ¦"¥ ¨ ¨¤ ¨ © ¦ "&¤¦"&¤¦ "&¤¨ ¦"¨ ¦"4 Strahlung einer lokalisierten oszillierenden Quelle4.1 Greensfunktionen für zeitabhängige Wellengleichung%Die Wellengleichungen für und sind vom Typ¡ ¡wobei f eine bekannte Quellenfunktion ist . Wie in der Elektrostatik suchen wir eine Greensfunktion,jetzt als Lösung von¡ ¨ ¥ " ¥ ¥ ¥ " ¥ ¨ ¨ ¡ ¨ Dann ist die spezielle Lösung für ¡¡ ¡verknüpft mit der rechten Seite¨ ¥ ¥ ¥ " ¥ ¥ " ¥ ¥ Fourierentwicklung für G in Bezug auf die ¥ Abhängigkeit :&"¥ ¤ ¥ " ¥ ! Dies eingesetzt ergibt :¡ ¢ " ¦" ¤&¡¡&"¥ ¨ ¤ ¨ ¥ ¡ ¨ Multiplikation mit ¦ "&¤¨ ¤ & ¦"¥ ¨ und Integration über und ergibt :¥ ¡¨ ¥ &¤ ¥ ¦"¦ & ¡ ¨ ¡Nun gilt£¢ ¦ "¦ ¥ ¢ Somit¨ ¤ &oder¥ ¨ ¨¡¡ ¢ " ¨ ¡¢ ¢ ¨ ¡ ¤ & " ¢ ¢¢ ¥¥ ¨ ¡ ¨ ¥¥& &¨ ¨ ¦ "¥ ¦ ¨ ¨¦ ! ¥ " ¥ ¥¡ ¡ ¨ ¥¥ ¢ ¢98

¡! !!! !!für ¥¦"% %% % %% % %% % %% %% % %% % %% % %% % %% %%% ¦"&¤¡ ¥¦ " ¦"&¤¦ "% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %¤¤¡Mit diesem Ansatz ist die Differentialgleichung erfüllt, aber das Integral ist wegen der Singularitätnoch nicht definiert . Welche Randbedingungen sind zu erfüllen ?Quelle an für ¥ ¥ produziert auslaufende Welle für ¥ ¥ .auslaufende Welle für ¥ ¥ £Wir betrachten das nicht definierte Integral in :¥ ¡¡ ¨ ¡¡¡ "£¢ ¥ ¨ ¥ und ersetzen es durch¥ ¡£¢¥¤§© ¤2 Pole in der komplexen - Ebene¡ ¨ ¡ £¢ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Im ¡ £¢ Re% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%¨ ¡ ¨ ¢ ¡ ¨ ¢ Für£lässt sich das Integral in der oberen Halbebene schließen und der Integrand ist Null.Für£lässt sich das Integral in der unteren Halbebene schließen und es ergeben sich nach¢dem¢Residuensatz Beiträge von den beiden Polen. Der Limes ¢ £garantiert, dass weiterhineine Lösung vorliegt.£ ¡ Sei . Dann erhalten wir ¨ ¥ " ¥ £¢¥¤¤¨§© ¡& ¤&¡¡ £¢ ¨ ¡¡ ¥ ¡¦" ¨ ¡ £¢ © ¦ "¡¡ ¢¤& ¨ ¢ ¢ ¨ ¡¡ ¡ ¡ ¦" ¤&¢Winkelintegration& ¢ ¡ ¢ ¤¨ ¢ ¡ 99

¢ ¡ ¡% " ¥ ¡¡©¢ !©¤ ¦" ¤ ¢ ¡ ¢ ¡ ¢ ¡¢ ¨¡ ¡ ¤ ¥ ¨ ¥ ! !! ¤ ¥¨ " ¥ ¥! ¨ ¥ ¨ &¡""¥ ¡ ¨ ¦¦¡¨¦ " ¤& ¤¨ ¦¤¦¡¢ ¡ ¢ ! ¢ ¡ ¢¦¡ ¢ ¡ ¡¨ ¡ ¢ ¢ ¡¡¡¥ " Diese Greensfunktion nennt man retardiert, da der Effekt ausgehend zur Zeit zu einer späterenZeit¥ ¥ ¨ ¨ an auftritt. Damit wird die Lösung¥ ¥ && ¨ " ¥ ¡ " ¥ ¥ ¥ ¨ " ""¨ ¡£¢ £¢ ¦¦¨ ¡4.2 Felder einer oszillierenden QuelleZerlegen allgemeine zeitabhängige Stromdichte nach Fourier Diese Fourierzerlegung überträgt sich auf das Vektorpotential¦"¥ ¡ " ¥ !¡¡ ¥ % ¡ % ¨ ¡¡¡¡ ¨ " ¥ ¨ ¥ ¥ ¡ ¡¨ ¥ " ¤ ¡£¢£ ¢ ¦ ¥¦¨ ¡¡¡ ¥ !¡ ¨ 100

! !¡¢ ¡ ¤! &¥ ¡ ¤ ¤' !%¥ ¡%¥ %¥ ¥ % ¥¡ ¢ ¢ ¥ ¡¨ ¡ ¥ ¡ ¥ ¥ ¨ ¦ ¥ &&¨ ¦" ¤¡ ¦ " ¥¡ ¡ ¦ "Somit können wir jede Fourierkomponente für sich studieren¦" ¤& &¨ Hieraus gewinnt man die Fourierkomponenten der physikalischen Felder¡ ¥ ¨ %¥ und aus der Maxwellgleichung lokal außerhalb der Quelle¥ ¥¡¡ ¥¢¡ ¨ ¨ ¢¢¥ £ ¢¡¥ £ Welche Längen sind im Spiel ?Dimension der Quelle dWellenlänge ¡ ¤Aufpunkt¡ ¢ ¢¡¥ £ ' ¥ Wir wollen annehmen, dass stetsNahzone oder statische Zone¨ gilt, und unterscheiden zwischen 3 räumlichen Gebieten:Zwischenzone oder Induktionszone ¡Fernzone oder Strahlungszone Nahzone ¨ ¢ :¦" ¤& &¨ ¡"Der damit einhergehende Zeitfaktor isther ¦ bekannten Felder.Erinnere an Multipolentwicklung¥ harmonische Schwingung der von der Statik ¨ ¡ ¥ 101

oder¡ ¨ ¢ ¨ ¡¥ ¢ ¥ ¥¢ ¨ &&¨ ¡¥¦" ¤ ¡ ¥ ¡¥ ¢ ¢ ¨ Deswegen¥ ist , also transversal und von der ¢ Ordnung¤"¨ ¢ ¥ ¨ ¥ ¨"¤ ¨ ¥¢%¨ ¢¢% ¥%¢¨¤ &¢¨¨ % ¡¨ ¢ ¡ ¢¤£¦¥ ¤¥ist ebenfalls transversal und ¦ ¡¢¥ ¢ ¡'¥¦ ¤¥¦Fernzone :Exponentialfunktion oszilliert stark.Entwickeln¨ ¨ £ ¨ ¤ ¡ ¡ ¨¢ £ ¡ ¨£ ¢ ¦" ¤ ¦ " ¤NennerSomit¨ ¥ ¦" ¤% ¡ ¢ ¦ " ¤¤ &"Dies ist eine auslaufende Kugelwelle, zusammen mit Zeitfaktorund ¦ richtungsabhängiger Amplitude.Wie sehen die physikalischen Felder aus ?Der einfacheren Notation wegen unterdrücken wir im folgenden den Index ., mit Abfallverhalten ¢¥ £ ¢ %¨ ¨% ¥¨ % ¢ ¢¢ ¨ ¢ %¥ ¨ ¢¢¡ ¥ %¢ ¢ ¢ % ¢ ' ¢¡'¨¢ ¥'¢102

¦¦ " ¤¥ !¨ Betrachten führenden Term für das Fernfeld für nach Fourierentwicklung ¥© ¡ ¥ ¡ ¢¡ ¦ £ £ ¨ ¦" ¥¢ ¨¥¥ ¢¡ ¦ ¥ ££ £ ¥ ¨ ¢ ¤ ¥ %¥ ¨ ¢ ¤ ¥¥¦¦¡ ' ¤ ¤ ¦¥ ¥¥¡durch jede Kugeloberfläche geht gleich viel Energiees findet Abstrahlung von Energie aus der Quelle statt.Das Integral über die Stromdichte lässt sich nochfür vereinfachen :¥ ¢Der Abfall ¢bedeutet, dass die Energiedichte ¥ ¤ &¢ £ ¢¨¥ ¦" ¤%¨ ¢ ¢ ¥ £ ¡Zwischenzone: es muss exakte Entwicklung von(siehe etwa Jackson ) ¨ benutzt werden.¡ ¦" ¤& &¨ Wegen fallen die höheren Terme schnell ab. 4.3 Dipolstrahlung%¥ ¦" ¤Nun ist¦ außerdem folgt aus ¢¡ ¦ ¤£¥Somit ¢ ¥ ¥ ¨ ¡ ¨ ¢ ¤ ¥ mit elektrischem DipolmomentHieraus resultiert das Vektorpotential¦" ¤Führender Term in der Nahzone103

¥und ¡¥¡ %¥ ¡¤ £ ¢ ' ¥ ¡¥¡¥£ ¤ ¥ ¡ ¥ ¡ ¨ ¢ ¤ ¥ ¤¥ !¨ ¤ ¥ ¨¢ ¤¥ ¥¥£ ¥¥' ¥ ¡¥¢¦" ¤¤ ¡ ¡ £¥¢¥ ¥¡ ¥ ¦" ¤Diese Form ist demnach überall gültig.Nun zu den'- Feldern : ¨ ¢ ¥ ¥ ¢£ ¥ ¡¦" ¤ ¡ ¡' ¥ ¢¦" ¤¥ ¨ ¢ ¤ §' ¥ ¡ ¡¡ Nahzone :der harmonischen Schwin-"dies ist elektrisches Dipolfeld mit noch anzubringendem Faktorgung.¦¥ ¨ ¤ ¥ ¤¥ ' ¥ §¥ ¢ ¥¡ ¤ ¥ ¡ Dies ist um ¢ Faktor kleiner' ¥ als . Also dominiert das elektrische Feld. ¥¢ £Für, d.h. der statische Limes, verschwindet das magnetische Induktionsfeld und das'-Feld derNahzone reicht bis nach .Strahlungszone :£¥ ¥ ¡ ¡¤ ¥ ¢ £ ¥transversal und orthogonal aufeinanderSchließen wir nun das Zeitverhalten mit ein und betrachten den Poynting Vektor' ¥ ¥ ¡¥Das ¥ -Verhalten führt wieder zu Abstrahlung.¦¡ 'wobei und¥die reellen Felder sein sollen'¡¥ ¥ ¥ ¡¥ ' ¥ ¥¥ ¥ ¦ "¦"¦"¦"¢ ' ¥¢ ¥ ¡¥¥ ¦ ¡"¥ ' 104

Zeitgemittelter Poynting Vektor.' ¥ ¡¥ ¥ ¥§ ¡ ¡ !¡"¥ ¦©¡ ¥ ' ¥ ¡¥ ¥ ' ¥ ¡¥ ¥ ¢zeitgemittelte abgestrahlte Energie pro Sekunde in das Raumwinkelelement , hervorgerufendurch das oszillierende elektrische Dipolmoment © ¨ ¢ ¡ © ¢ ¡©¡¡¡¡¡¡Winkelverteilung der Dipolstrahlung¥¡¢ ¡ © ¡¢ ¢ ¡ ¡' ¥ ¡¤ ¥ :¥ ¥ ' ¥ ¡¥ ¥ ¥ ¡¤ ¡ ¡ ¤ ¥ ¢ ¢ ¡ ¤ ¥ ¡ ¡ ¡ ¤ ¥ £ ¢ ¡ ¡ ¤ ¥ £ ¡ ¡ ¤ ¥ ¢ ¤ ¥ ¨ ¤ ¥ ¡ £ £¢ ¤ ¥ ¡ ¨ ¢ ¤ ¥¡ ¨ ¢ ¤ ¥ ¡£ ¢ ¤ ¥¡ ¨ ¡ ¢ ¤ ¥¡ ¢ ¡ Totale abgestrahlte Energie pro Sekunde%% % %% % %% % %% % %% % %% % %% %% % %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ¤ ¥ ¡ ¤ ¥¡ £%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% % %% %% % % %% % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % %% %% %% %% %%% %%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%¢ ©¤ ¥¢ ¡ ¢§ ¤ ¥¡©105

In analoger Weise lassen sich Multipole der nächsten Ordnung behandeln: magnetischer Dipol;elektrischer Quadrupol etc. mit typischen Strahlungscharakteristiken (siehe Jackson) .106

ot ' rot ¡ ¨ ' ¨ ¡¡ ¡¥ ¨ ¡ ¡¡¡¥'¡¤ ¥' ¡5 Freie elektromagnetische Wellen, optische Phänomene, Hohlleiterund Resonatoren5.1 Ebene Wellen in nichtleitenden MedienWir betrachten nun ein homogenes Medium ohne Quellen und zeigen, daß die Maxwellgleichungenfreie Wellenausbreitung zulassen. Wir haben dieses Phänomen bereits in der Abstrahlung ausoszillierenden Quellen kennengelernt. Nun betrachten wir die Wellen losgelößt von den Quellen. ' £ £ ' rot¥ £ £¥ ¨ ¡ rot¡ ¥¤£¥ £oder¡ ¥ rot¥ £Ebenso folgt¥ ¡¤£¡ ¥ ¡¤£¡ Jede Komponente erfüllt somit die Wellengleichnung.¥ ¡ £¢ ¡¤ mit¢ ¡ Einfachster Lösungstyp ebener Wellen: " ¥ ¦" ¤& &"¥ ¤Die Wellenlänge ist offensichtlich definiert durch¢ ¡ ¨ ¡ ¤£ ¢ ¢ ¤ ¢ ¨ ¡ ¨ und durch ¢ ¢ ¡¤ Frequenz ¢ Positionen konstanter Phase von sind offensichtlich Ebenen ; demnach schreiten die Wellenfronten(Ebenen konstanter Phase) in -Richtung fort, mit der Phasengeschwindigkeit:¢ ¢ 107

Wie sehen nun die Lösungsvektoren Dabei ¦sind , ist der Realteil zu nehmen.¥aus ?und'' " ¥ ¦ '© " ¥ ¥ §¡ ¥ © Einschränkung infolge der Maxwellgleichungen :'und ¦"&¤¦"&¤ "&"§¡Einheitsvektoren und ' © , ¥ © komplexe Amplituden. Von diesen Ausdrücken£¥ £ £'( £ ¦ £ ¤£ ¤£¡ £sind demnach transversal zur Ausbreitungsrichtung der ebenen Welle.¥Daraus folgt, daß' und ¢ ¢ ' rot¤ ¡¡Welche Energie wird dabei transportiert ?Poynting Vektor zeitlich gemittelt :¢¡ ¦¦ '© ¨¥ £¥ ¡¦ ¡©¡ '© ¤ '©¥¡ ¦% % %% % % % % %% % % % %% % % % %% % % % %% % %% % %% % % %% % %% % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%% %¡% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % %% % % % % % % % % % %% % % % % % % % %% % % % % % % % % % %% % % % % % %% % % % % % %% % % % % % % % %%% % % % % % % % % % % % % % % % %%%%%% %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % %%%%%%%%%%%%%%%%& ¥ ¥§¡ ¥ © ¢ £¥in Phase schwingen mit festem Amplitudenverhältnis.¥Re¡¡ '¡ ¤ ¡ '¡ '©¡ ¡¤ ¢¤ ©¡'Die Energiedichte weist demnach in Ausbreitungsrichtung der Wellenfront.108

¥ Re ¢ '©¡ " ¥ '© %% % % % % %% % % % % %% % % % %% % % % ¥ £ ¤ ¥ £¥ '% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%% % % % % % % % % % % % % % % % % % % % % % % % % % %%% % % % % % % % % %¢ '©¡ §¡ ' ¡ ¦"&¤ ¦ ¢§¡ ¦"&¤% % % % % % % % % %%Energiestromdichte in der Welle zeitlich gemittelt :' £ '£ ¢ '©¡¢ Sei die Geschwindigkeit des Energiestroms, dann folgt daraus: ¤ '©¡¤¢ ¥¢ ¢ ¥¤ '©¡ ¡¤¤ ¢ ¢ 5.2 Lineare und zirkulare PolarisationMan spricht von linear polarisierten Wellen, falls die elektrische Feldstärke in eine feste Richtungzeigt. Der allgemeinste Polarisationszustand ist eine Linearkombination aus zwei transversalenKomponenten:"" ' " ¦" ¤& &"¥ und ¥" ¤ ¡' "¢ " ¢' Somit heißt die allgemeine Lösung für ebene Wellen in Richtung&"¥ ¤" ¥ ¦ ' ¦ Falls ' ¦und ' ¡verschiedene Phasen haben, liegt eine elliptisch polarisierte Welle vor. Diesbedeutet, daß'für einen festen Ort und variable Zeit eine Ellipse durchläuft. Der Spezialfall' davon ist die zirkular polarisierte Welle :' ¡ ¢ ¢' ¦§"& "¥ ¤' ¦ '© reell ¦ ¢' % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%§¡%%%%%%%%%%%% 109

Für festes läuft 'im Kreis.% %% % % % % %% % % % % %% % % % % %% % % % % %% % % % % %% % % %%¢% % % % % % %%%%%% %% %% %% %% % %% % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % %'% % % % % % % % % % % % % % % % % % % % %% % % %% % %% %% %% %%% %%% %%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % '©'¡ '©' ¨ ¥ ¢ ¨ ¥ 5.3 Reflexion und Beugung elektromagnetischer Wellen an ebener Grenzflächezwischen zwei Dielektrikas %% % % % %% % % % % %% % % % %% % % % %% % % %%% % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % %i r’% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%einfallende Welle¢r gestreute Wellereflektierte WelleWir werden sehen, daß die aus der Optik bekannten Phänomene auftreten. Dieses rechtfertigt,zum Beispiel Licht als elektromagnetische Wellen aufzufassen. Alle diese optischen Phänomenefolgen aus den Maxwellgleichungen.Man unterscheidet kinematische EigenschaftenReflexionswinkelSnellius’sches Gesetz:und dynamische Eigenschaften:Einfallswinkel ¢ ¢¢ ¢ ¢ ¢ ¤ 110Brechungsindex

Intensität von reflektierter und gebeugter WellePhasenverschiebung und PolarisationszustandEinfallende WelleGebeugte Welle:Reflektierte Welle :Randbedingungen: daTangentialkomponenten von ¡ £'und ' ¥ ¤ '©¦"&¤&"¤¡ ' ©' ¡¥ ¥ ' " ¤¨ ¤ & "¥ ¨ & ¦ ' ©' ¤ ¡¥ ¦"&¤¨ ¤ &"¨' ¥ ¨ ¨ ' sind die Normalkomponenten¤vonstetig.und¥stetig und dieSchon aus dieser Bedingung folgt, daß die Frequenzen gleich sein müssen und ebenso die -Funktionen für ¦ auf der Grenzfläche.¢! Außerdem wegen desselben Mediums istund' ' ¨ ' ¤£ £ £ £ ¡Aus der oberen Grenzflächenbedingung folgt, daß die 3 Vektoren in einer Ebene liegen.¦ ¦¡¡¤ % %% % % % % %% % % % % % % % % % % % %%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % %% %%% % % % % % % %%%%%%%%%%%% % % % % %% % % % % % % % % % % % %% % % % % % % % % % % %%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Die Projektion ¦von auf der Grenzfläche liegt ¦in -Richtung.¦¦111

Für aus der Ebene folgt ¦¨ ¦ ¡ ¢ ¦ ¡ ¢oderDa ¡ £ ¦ ¡ ¦ ¦¢ , d.h. Einfallswinkel = AusfallswinkelNun zu den dynamischen Eigenschaften:Stetigkeit der Normalkomponenten von ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¤ ¤ ¢ ¢ Snellius’sches Gesetz ¤ ¡ '© und¥: Da ist dies gleich '© ¤ ' ¡¡'© Stetigkeit der Tangentialkomponenten von ¡'© Wir führen nun die Einheitsvektoren ein:£ ¨ © © £ ' © ¨ ¡' ¡' ¡¤': ' ' © ¡¨ '©und© ¨ %% % % % % %% % % % %% % % % % %% % % %%' © ¨ ¡¡¤ ¢ £ ¡ ©¡ ' © £% % % % % % %%% % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % %%% % % % % % % %¢¡%% %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%¢ ¢%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%¢ ¡¢112 © £ '¢ £¢ £ ¢ £ §¢ ' ©§ ¢ £ ¡

¡ ¡¡ ¢' '©¨¢ ' ' ©'£¡¡ ¢¢&¤&¢¡ ¢ ¢ ¨ ¨ ¢ ¢ ¨ ¢ #¨' ©¤' £ ' ¢ ¨ ' ' '£¤' ¢ ¨ ¢ ¢ ¢ ¢ ¡ ¤ ¢'¢ ¢' £¨££¨£¨£¨¢¢£¨£¢¢¤ ¢' £¡¨¤¡£¢ ' ¢ ¨ ¤¢' ¡¤ ' ¡¡ ¢ ¢££¤' £¤ ¢ ¢' ¡ ¢ ¢' ' ' ¨£¡ ¡¡££££ ¢In diesen lassen sich ausdrücken:¢¡ ¢ ¢¢¡ ¢ ¢¢ ¢Bedingung der Transversalität:¢ ' ¢'© £Damit lassen sich die Bedingungen (1)-(4) formulieren:¤£ ' ' ¨ ' £¨ ' ' ¡¨¡ '¢ ' £¨ '§' ¤£ ' ¡¨¡Vorbetrachtung zu (2): '¢ ' ¢ ' ¢¢ ¢ ¢' ¢' ¢¢ ¢ ¢' ¢ ¢' ¢¡ ¢' ¢ ¢ ¢' © ¨ ¢ ' ¡¢ ¢ ' ¢ ' ¢ ¢' £¨ £¤£ ¢ ' ¢ ¢ ¢' ¢' £¨ ¢' ¢ ¢' ¨ ¢' ¡¡ ¢ ¢' £ ¢ ' £ Die Vektoren spannen die Einfallsebene auf. Wir sehen aus (1)-(4), daß die E-KomponentenEinfallsebene durch den Prozeß an der Grenzfläche nicht in die Einfallsebene umkippen undund 113

Lineare Polarisation Einfallsebene: ' ¢'' © ¢& '' ¢¡ ¢ ¡¤' ¢ ¢ ¨¨¨"¢¡¤£¥ ¢¡¤£"¢¡¤£¥ ¢¡¤£"¢¡¤£¥ ¢¡¤£ ¢ ¢ ¢¨ ¢ ¢ ¢umgekehrt.Die allgemeinste elliptische Polarisation wird aus einer Linearkombination von linearer PolarisationEinfallsebene und linearer Polarisation in Einfallsebene gebildet. Nach (1)-(4) könnenbeide Fälle getrennt behandelt werden und die Ergebnisse linear überlagert werden.§ ' ' ¨ ' £¢ schon im Snellius’schen Gesetz enthalten ¢ ' ¨ ' ¢ ¢ ¢ ' ' ' ¨Lineare Polarisation in der Einfallsebene: ¢ ¨ ' ¢ ¢ £Transversalitätsbedingung:' ' ' ¢ ¢' ' ¢ ' ' ¢ ¢ ¨ ' ¢ £§ ' ¨ ' ¢ ¨ ' £ ' ' ¨ ' ¤£ Gleichungen (1) und (4) sind verträglich wegen dem Snellius’schen Gesetz.114

' '¢ ¢ ¢¢ ¢ ¢ ¢ ¢¨ ¢ ¢ ¢ ¨ ¢ ¢ ¢ ¢ ¢¢ ¢¡¨ ¢ ¢ ¢¡ Totale innere Reflexion: Für ¢ ¢ ist ¢ ¤¡ für ¢ ¢ ©! ¢¡£¡£ ' ¤ ¨¡ ¡'ist der Zusatzterm gegenüber dem Dielektrikum. ¡ ¡¥ ¡ ¨ '¤' £¥¨ ' ¦¡£ ¢¢ ¦ ¢¢Zwei interessante Phänomene:Brewster Winkel für Falls ¢ ¢ ¢ ! ¢ ¢und 'in Streuebene: ist, tritt keine reflektierte Welle auf. ¢ ¢ ¢ ! ¢ Unter diesem Einfallswinkel ¢¢kann man aus gemischter Polarisation rein linear polarisierteStrahlung herstellen. Der reflektierte Strahl hat nämlich nur Polarisation Einfallsebene.Für diesen Einfallswinkel dringt die gebeugte Welle nicht ins zweite Medium ein. ¢ © Für hat man eine exponentielle Schwächung der Welle im zweitem Medium (siehe Jackson).¢5.4 Wellen in LeiternDas elektromagnetische Feld kann wahre Ströme auslösen. ¤ £ ' ¡¥¤£'(¤£ ' £ ¡' ¨ ' ¡¥ £' ¨ ¡¡ ¡ 115¡¡

Ansatz:'( '©¦"¦ &¤&¥ ¨ ¤¡ ¡¡¡£¢ ¤ '( '© exponentiell gedämpfte WelleAußerdem ist wegen div' ¤£ '© Für einen guten Leiter ist¡ ¤ ¦¢¢¤ &¤ & £ ¢ ¦"¢ " . ££& £¢ ¢ ¥ ¨ ¢ ¤ ¢ Abfall der gedämpften¡¡¢ ¤ ¢ £Welle aufd.h. die Eindringtiefe ist besonders gering bei hohen Frequenzen.Aus'¡ ¤£rot ¥©¡ ¡ folgtSomit für guten Leiter¤¤'©¡ ©¤"""' © "£¢ ¢ ¢"¡"""was zeigt, dass das Magnetfeld dominiert.¦ nach ¡ '© ¡5.5 Hohlleiter und Resonatoren ¢ ¤ ¢ ¡£ ¢ ¤ ¢ ¡£¢komplexHohlleiter haben grosse Bedeutung zum Transport elektromagnetischer Strahlung. Wir wollenuns unter Hohlleiter einen Metallzylinder vorstellen und unter Resonator einen Hohlleiter mitgeschlossenen Enden.Gibt es im Hohlleiter Wellenausbreitung ?%% % % % %% % % % % %% % % % %% % % % %%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % %% %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %¢116% % % % % % % % % % %%%%%%%%%%%%%£

¡¦ ' '¡¦¦!¨ ¡¡' ¢ ¡ ¦¦ ¨¡¥' ¢ ¨ ¡ ¦ ¨¢ ¡£ ¡¦¦ ¨ ¦¦¡ ¢ ¢ ¡£¡ ¦ ¨ ¦ ¡ ¢ ¦¡ ¦ ¡' ¦" ¤¦ ¢ ¨ ¡ ¦ ¢ ¨ ¡ ¦ ¢ ¦¦ ¨ ¦¢¦¦¦" ¤"Das Innere des Hohlleiters sei mit einem Dielektrikum, gekennzeichnetdurch , gefüllt.Wir betrachten eine bestimmte Fourierkomponente¥ Maxwellgleichungen im Innern:¦ "¥ ¨ ¢ ¥ £ £ £ 'Wir interessieren uns nur für Lösungen vom Typ£' Wir zerlegen weiter:¥ " ¢ " ¦ " ¢ ¦" ¤¦Wir können und ¦durch und ausdrücken. " ¢ " ¢ " ¢ ' ¦¢ ¢ ¨ ' ¡ ' ¦¡ ¨¦' ¡ ¦¦¡ ¢ ¦" ¤ §entsprechend §¦ Daraus folgt¢ ¡ ¦¡ ¢ ¨¢ ¡¦ ¢ ¡¡ ¨¢ ¡ ¦¦ ¢ ¢ ¡¦ ¨¢ ¡ ¦¡ ¢ ¢¡ ¢¦ ¢ ¨¢ ¡117

¡¡¡ ¨¡¡ ¡ ¡ ¦ ¡¡ ¡ ¢ ££ ¦ ¥ ¨ ¡£ £ ¡£' ¦¦ ¦¡¦¤¤ ¢ ¢¦ ¢¢¦¦¦ ¢ ¨ ¡ ¦ ¡ ¡¦¦ ' §¦¦Dies läßt sich nun auflösen:oder¡¡¡ ¢ ¢ ¡ ¢ ¡ ¦¤¨ ¡ ¦¦©¨¢ ¡ ¨¢ ¡oder¡ ¡ ¨ ¡ ¦ ¡¡ ¡ ¢ ¡ ¨¢ ¡ ¡¦ ¥¥¤¥ ¨ ¡ ¢ §©¨ ¢¢ ¡¨ ¢ ¡ ¡ ¢ ¡ ¢ ¡ ¨ ¢ ¡¢ ¢ ¢ ¡ ¨¡¡ ¡ ¡ ¡ ¡¡ ¡ ¢ ¢ ¡¡ ¥¥¤¥ ¨ ¡ ¢ §¢ £¢ ¥¥¤¥ ¨ ¡ ¡ ¢ ¨ ¢ ¡¢ §Multiplikation mit ¦ " ¤ ¡¥! " ¢ " ¤ ¥¥¥ ¢Entsprechend findet man¤ ¥¥ ¨ ¢¥ §Somit sind in der Tat die transversalen Komponenten durch die longitudinalen ausgedrückt. ¥ ¨ ¡' " ¢ " Aus den oben angegebenen Maxwellgleichungen folgen die Wellengleichungen ¡ ¡¡ ¡¥ " ¢ " £'Für unsere spezielle Form folgt dann ¡ ¡ ¨ ¡ " ¢ £¡118

TM – Wellen: ¥ TE – Wellen: ' ¥! ¥ ¥¢ £ ' £Oberfläche ¥ £ Oberfläche Oberfläche £ ¤¥¤¡ ¦ ' £ ' ¤£§ ¤¡ ¡' ' ¡ ¤ ¡ ¡ ¤ ¡ ¡ ¤ ¡' und ¦ § " ¢ und § " ¢ sind als Lösung dieser Gleichung zu bestimmen.Wir nehmen nun Randbedingungen des idealen Leiters an (in Wirklichkeit muß auf endlicheEindringtiefe und Ohmsche Verluste korrigiert werden)Kein elektromagnetisches Feld im Leiter' ¢ £¢£¥! ¢£Oberfläche¥ ¢£¥ ¢Oberfläche £sind i.a. nicht ver-Diese ¤£beiden Oberfläche und Randwertprobleme'träglich es gibt zwei mit' £Lösungstypen oder ¥ ¤£¢¤£Oberflächeinnerhalb des Hohlleiters.¥ ¤£¡¡ ¨ ¡¡¢ ¥! ¡' ¤£ ¡ ¤¥¤¡ ¦¥ ¥!¤£¢§¤£ ¥ ¥!¢ ¦¥!Diese Eigenwertprobleme führen auf Eigenwerte ¤ ¡ £ " " ¢ " ¡ ¡ist dabei gegeben und fest(siehe Beispiel unten); ¨¡¡ ¨ ¤¡¡119

¡¤¡ ¤ ¥%% % %% % % %% % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%¡ ¡dann¡¢% %% % % % % % %% % % % % % %% % % % % % %% % % % %% ¡ ¡¤ ¤¤ ¡ ¨ ¡ ¥ " ¢ ¥ © ¡¡¡ ¡%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % %Sammelindex& £ " ¢ ¥% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %Wir definieren die Abschneidefrequenzen:Wir sehen, fürreell: Die Wellen können sich längs des Hohlleiters fortbewegen,imaginär: Die Wellen können sich nicht ausbreiten.istIm Bild:ist¦ ¡ % % %%%%%%%%%%%%%%%d.h. zu jedemgibt es nur endlich viele Moden.%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Betrachten wir nun ein konkretes Beispiel:Rechteckiger HohlleiterAls Beispiel betrachten wir die TE – Moden:%%%%%%%%%%%%%%%¢ ¡ ¤¡ ¤£¢ £an¤£ " ¡ ¤£ " ¢Separationsansatz ¡ ¢ ¢¤¡ ¥ ¡ ¡¡¢ £ £ " ¢ £ £ " " ¢ganz120

Abschneidefrequenzen:Für ¡ ¥ ¡ ¤¢¡¡ ¡¡ist die niederste Abschneidefrequenz für " ¢ £Somit sind die Wellen in dieser Mode¥ ¥ ©¥ ¨ ¢¡¡ £'¥ £ '¡ ¢¡© ¡¤ ¡¦ ¥ © ¡ ¢ ¦" ¤ ¡¢¡"" ¤ "¦¡ Von diesen Ausdrücken ist natürlich der Realteil zu nehmen.5.5.1 Resonator¥ © ¢ ¡Zylinder mit Endflächen im Abstand %%%% %%%% %%% %% %%% %% %% %% % %% %% % %% % % %% % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % %% % %% % %% %% %% % %%% %% %% %%% %%% %%%% %%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%% %%%%%%% %%%% %%%% %%% %% %% %%% %% %% % %% %% % %% % % %% % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % %% % %% % %% %% % %% %%% %% %% %%% %%% %%%% %%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%% % % % % % % % %%%%%%% %%%%%%%£stehende Wellen mit – Abhängigkeit ¢ ¡ ¢ oderBeispiel: TM – Mode mit ¥ Die Randbedingung' ¢ £¤£¦" ¤an der Oberfläche bedeutet:' ¤£ an £121¥ "und ¥ ¥

' ' an £und ¡ " ¢ ¢¨ ¢ ¤ " ¤ £ "" ¢"¡¡¡ ¡ ¤¥¤¡ ¦ ¡ " ¢ ¤£mit der Randbedingung' ¡ ¡ " ¢ Zylinderoberfläche £ ¡ ¡£ ¤ ¤ ¤ ¡ " ¢ ¡¡§Es gilt' ¤¡ £ ' § " ¢ " ¡ " ¢ ' Ferner genügt ¡ " ¢ als relevanter Teil von' der GleichungundDie Randbedingung führt zu ¤ ¡¤¡ für jeden Wert von ¤ ergeben sich Eigenfrequenzen¡¡ ¨ ¡¤¡ ¢ ¡¡ ¡Die transversalen Felder sind' ¨¤ ¤¡ ¢ ¥!¤¡ ¢ ¤ ¡ " ¢ 122

¦ ¥ ¡ ¨ ¡ ¢ ¥¡ £ ¢¡ ¥¨ ¢ ¥ ¦ ¨ ¡ ¢6 Relativistische Formulierung6.1 Einführung in die spezielle RelativitätstheorieGalileiinvarianz bedeutet: physikalische Gesetze sind forminvariant unter Galileitransformationend. h. man betrachtet die Gesetze in zwei Koordinatensystemen, die sich relativ zueinander mitder Geschwindigkeit¢bewegen. ¨ ¢ ¥ " ¥ ¥Die Newton’sche Bewegungsgleichung ist galileiinvariant:¦ ¨ ¡ ¦ ¡ ¦ ¨ ¡ ¡ ¡ Übergang ins bewegte System: ¥ ¡ ¨ ¡ ¦ ¥ ¡ ¦ ¡ ¦ ¥ ¡ ¨ ¦¨£entsprechend für Teilchen 2in den gestrichenen Variablen erkennen wir dieselben Gesetzmäßigkeit, d. h. Forminvarianz.Dieses Gesetz erlaubt nicht, das eine Koordinatensystem vor dem anderen auszuzeichnen.Ist die Maxwelltheorie auch galileiinvariant?Betrachten wir die Wellengleichung für das skalare Potential :¥ ¡¤£Transformation:¨ ¡ ¨ ¡¡ ¨ " ¥ ¢ " ¥ es ergeben sich gemischte Terme in den gestrichenen Variablen , und die Wellengleichung istnicht forminvariant unter der Galileitransformation. Damit sind die Maxwellgleichungen nichtforminvariant unter Galileitransformationen.Es ergeben sich mehrere Alternativen:a) Die Maxwellgleichungen sind falsch, da sie nicht forminvariant sind unter Galileitransformationen.123

) Die Maxwellgleichungen beziehen sich auf das ausgezeichnete Koordinatensystem des ”ruhendenÄthers” und ändern sich, falls sie auf ein anderes System bezogen werden.c) Die Maxwellgleichungen sind richtig, und die Galileitransformation istfalsch.Es begann die Suche nach dem ”Äther”.Bei Wasserwellen mißt ein Beobachter verschiedene Wellengeschwindigkeiten, je nach seinemBewegungszustand zum Wasser:¢ ¢¡¨¤£ ¡ ¦¦¥¡¨§WelleBei Existenz eines ”Äthers”, dem Träger der elektromagnetischen Welle, erwartet man deswegenverschiedene Lichtgeschwindigkeit, je nach dem Bewegungszustand des Beobachters relativ zum”Äther”.Michelson – Morley – Versuch (1887)% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Der folgende Apparat bewege sich mit der Geschwindigkeit¢relativ zum ruhenden Äther: ¡%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % %% % % % %% % % % %% % %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % %% % % % %% % % % %% % % % %% %% % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% % % % % % % % % %% % %% %% % % % % % % % %%% % % % %%% % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Beobachtungsschirm¡ ¦% %%%%% %%% %%% % %%%% %%%%% % %%%%% % %%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% ¥Laufzeit zwischen halbdurchlässigem Spiegel und ¦und zurück:¦ ¢¥¨ ¢¡¢ ¥ ¡¢ ¢¢ ¢¡ ¨¥¡ ¥Laufzeit zwischen halbdurchlässigem Spiegel und ¡und zurück:¥ ¡ zurückgelegter Weg im Äther¡¡¢¤ ¢¡ ¥ ¡ ¢ ¢ ¥¡¦ ¡ ¤Der Unterschied in den Laufzeiten der beiden reflektierten Wellen ist ¨¥ ¥ ¡©¨ ¥ ¦ ¢ ¢¡¡¥¥ ¢£¨ 124¡¥¥¨ ¨¥¥ ¤¦

¢Unterschied im optischen Weg, zurückglegt im Äther:Bei einer Drehung um ändert das VorzeichenStreifenDer experimentelle Befund war Null.Einstein nahm den negativen Ausfall dieses Experimentes zum Ausgangspunkt seiner Überlegungen:der gleichförmige Bewegungszustand des Systems hat keinen Einfluß auf das Ergebnisdes Experimentes. Er leitete daraus die Vermutung ab, dass es grundsätzlich durch Messungennicht feststellbar sei, welches von zwei gleichförmig gegeneinander bewegten Systemen ruht undwelches sich bewegt.Er postulierte das Relativitätsprinzip:Die Naturgesetze und die Messergebnisse in einem gegebenen System sind unabhängig von dergleichförmigen Bewegung des Systems als Ganzem.Ferner besagt das Postulat der Konstanz der Lichtgeschwindigkeit:Die Lichtgeschwindigkeit ist unabhängig von der Geschwindigkeit der Quelle.Um Schlußfolgerungen aus diesen Postulaten zu ziehen, benötigt man die weitere Präzisierungder Begriffe nach Einstein:a) Die Definition der Gleichzeitigkeit innnerhalb eines Systems geschieht durch folgende Einregulierungder Uhren: ¨ ¡ £ ¡ ¥ ¢Änderung der Interferenzstruktur, entsprechend £ £ ¨ ¡ £¢¥!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % %% % % % %% % % % %% % % %%%% %%% % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% %%%%%%% %% % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % %% % % %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%£ ¥ ¥ Ein Lichtsignal, ausgesandt zur Zeit am Ort der linken Uhr, kommt zur Zeit amOrt der rechten Uhr an.Die beiden Uhren sind synchron, wenn die rechte Uhr beim Eintreffen des Lichtsignals dieZeit anzeigt. Alle Uhren innerhalb der Systeme sollen derart synchronisiert sein.b) Die Längenmessung eines bewegten Stabes muss an beiden Enden gleichzeitig erfolgen. Diessetzt also die Definition der Gleichzeitigkeit voraus.Nun zu der Transformation zwischen zwei relativ zueinander gleichförmig bewegten Systemen:%% % % % %% % % % %% % % % %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%£ £% ¢ ¥ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%125% % % %% % % % %% % % % %% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%¢ ¢ ¥

¢ ¢ Gesucht ist der Zusammenhang zwischenAnnahmen für diesen Zusammenhang:a) linear (Homogenität des Raumes und der Zeit)b) gleichförmige Relativbewegung von und (spezielle Relativitätstheorie)c) Die Messung der Lichtgeschwindigkeit ergibt in beiden Systemen den gleichen Wert(dies ist in der Definition der Zeiten und ¥ ¡, d. h. der Einregulierung der Uhren, bereitsenthalten)¥d) Es soll durch keine physikalische Messung möglich sein, einen Unterschied zwischen denbeiden Systemen festzustellen.Sei " ¢ "" ¥ und " ¢ " " ¥ Es" ¢liegt dasselbe vor , da gleichberechtigt sind in bezug auf .Ein Beobachter in K’ mißt den Einheitsstab in K mit der Länge . ¢Ein Beobachter in K mißt den Einheitsstab in K’ mit der Länge ¦¤ .Es wäre ein objektiver Unterschied feststellbar zwischen K und K’ , falls nicht ¢ ¦¤ .¢ ¢Die Position £ ¢ ¥.bewegt sich mit der Geschwindigkeit¢in K , ist also beschrieben durchDie Position £bewegt sich mit der Geschwindigkeit¨ ¢in K’ , ist also beschrieben durch ¤ ¨ ¢ ¥ ¨ ¢ ¥ ¤ ¢ ¥ Zu zeigen ist nun nach d) , dass ¤ :¤Stab der Länge l ruhend in K werde von K’ aus gemessen¥: £ ¢£Situation in K ¤£ £ ¥Situation in K’in K’ findet man die modifizierte Länge . Nun werde der Stab der Länge l ruhend in K’ vonK aus gemessen: ¤£ £ ¢¥Situation in¥K’ £ £Situation in K¨in K findet man die modifizierte Länge ¨ . Nach d) muss ¤ ¤ sein.Schließlich erfolgt die Festlegung von durch die Forderung der Konstanz der Lichtgeschwindigkeit:¤126

oder die Umkehrung durch¢ ¢ ¨ ¢¥ ¤ ¡ ¥ ¢ ¥ ¤ ¡ ¢ ¥ ¡¥ ¤ ¡ ¥ ¨ ¢ ¥ ¤ ¡ ¨ ¢ ¥¡¡ ¤¡ ¡¡ ¨ ¢ ¡ ¢ ¤ ¡¨ ¤ ¨ ¢ ¥ ¤ ¢ ¥ ¡¡ ¥¥ ¥¥ ¥¥ Ein Lichtsignal werde zur £Zeit ausgesandt . Es passiert zur Zeit die Stelle ¡ ¥und zur Zeit ¥ ¥ die Stelle ¡ ¥ . Somit folgt ¥ ¥#oderLösen nun innach x’ und t’ auf : ¤ ¨ ¢ ¥ ¢ ¥ ¤ ¢ ¥ ¨ ¤ ¤ ¥¤ ¢ ¤ ¨ ¥¥ ¥¤ ¥ ¨ ¤ ¨ ¢ ¡¡¡ ¤¡ ¤ ¥¢ ¤ ¥ ¨ ¢Somit¥¥ ¨ ¢ ¥¨¥¥ ¨ ¥ ¨¥ ¥ ¢ ¥ ¥ ¥ ¥ ¨127 ¨

Diese Beziehungen, zusammen mit ¢, nennt man Lorentztransformationen. Für¢ ¢ £reduzieren sie sich auf die Galileitransformationen ¨ ¢ ¥¥ ¥Das ins Auge springende neue Element ist die Transformation der Zeit !Betrachten wir jedoch zunächst nochmals die Ausbreitung der Lichtwelle in und :% % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% % % % %% % % % % %% % % % %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% % % % %% % % % %% % % % %% %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% %% % %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Diese Welle erscheint in beiden Systemen als Kugelwelle :In K : ¦ ¡ ¥ ¦In K ’ :£ ¨ ££¡ £¢££ £¦£ ¨ ¢£ £ £ £¥ ¨ ¥£¡ ¨¢££ £¦¥ ¨ ¢£ £ £ £¤¤¥¥¤£ ¥¦¥¥¤¥ ¥¦¤¥¥¤¥¥ ¡ ¨ ¡ ¥ ¦ £ ¤ £¤£ £ ££ ££ £ £ ¤ ¡ £ £ £¡££ £¤¤¤¤¥¥¤¤¥¥¥¥¥¥¢¥ ¨¨¨¨¨¦¨¨¨¨¨§¥ ¨¨¨¨¨¦¨¨¨¨¨§ ¦ ¡ ¥ ¦ ¡ ¨ ¡ ¥ ¦ Die Welle bewegt sich demnach auch in K’ mit ¡ ( wie ja eingebaut ) . Die Ereignisse jedoch ,die gleichzeitig in K sind , nämlich ¦ ¡ ¥ ¦sind nicht gleichzeitig in K’:¡ ¨ ¡ ¥ ¦ ¡ ¥ ¦ ¥128

und auch ¡ ¦ ¤¨¥ ¥¨% % %% % % % %% % % % %% % %%% %% % % % %% % % % %% % % % %%¡ ¥ ¡ ¥% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %¢¥¥ ¡ ©entspricht £ ¥ £¥ ¥ ¨¥ © ¥¥ % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %Nun zu den verschiedenen Zeiten in K und K’ :Laufe in K am Ort ¤£das Zeitintervall ¡ ¥ ab , dann gilt in K’Das heißt, das Zeitintervall ¡¤£Zeitintervall , in dem Ereignis das ruht. Man spricht von Zeitdilatation.Dieses Phänomen ist in der Teilchenphysik ¡ ¥ Routine:z.B. hat in Ruhe die mittlere Lebensdauer¨¨¥ , in dem das Ereignis £ bewegt erscheint, ist größer als dasfalls sich mit bewegt, würde dies nur eine Laufstrecke von ¡ £ ¢ © ¡ " ergeben.Tatsächlich findet man schnelle Mesonen auch noch mit größeren Laufstrecken, z.B. 300 m. DerGrund ist der ¡ ¨Zeitdilatationsfaktor¢ © ¢ ££ ¦für¢ ¡ ¢Die im Labor gemessene Lebensdauer für schnell bewegte ist größer !¨ Ein zweites Phänomen ist die Längenkontraktion eines bewegten Stabes .Sei ein Stab der Länge © in K ruhend :%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Wir wollen die Messung in K’ zur Zeit ¥ ¤£durchführen : ¤£¥¤£129 ¨

wir eleminieren ¥ zur Zeit ¥ ¤£¡ ¤ © ¨ ¥ ¨ ¢ ¤ ¢ ¤¡¡¡¡¡¥¥ © ¨ ¢ ¡ ¤ ¡¤¥¥¥ ¨ ¢ ¤¡¡ ¡¡¡ ¥¥¢ ¡¢ ¡¡ ¡ ¤¡¡¥¥¡¡ ¥¡ ¤¡¡ ¥¢¡¡ ¤ ¥¨¡¡ ¥ © ¨ ¢ ¥¨¥ ¥¢¨ ¢ ¡¡¡ © bewegter Stab erscheint reduziert .¢Nun kommen wir zurück zur Wellengleichung :¡ ¥ ¡¤£¡Bleibt dieses Gesetz nun unter Lorentztransformationen forminvariant ?Umrechnung ins bewegte System¡ ¨ ¡ ¨ ¡ ¥ ¤ ¡ ¤¨ ¢ ¡ ¤ ¡ ¤ entsprechend¡ ¨ ¢ ¢¡¡ ¤¡ ¥ ¡ ¢ ¡¤¡ ¡ ¨ ¢ ¢ ¤¡ ¡ ¨ ¡¡ ¡¡ ¥ ¡ ¤¡ ¨ ¢ ¡¡¡ ¡ ¤¡¢ ¡¡ ¢¨ ¥¡Dieser Operator für die Wellengleichung ist demnach forminvariant unter Lorentztransformationen.Im zweiten System findet dieselbe Wellenausbreitung mit derselben Geschwindigkeit ¡statt.Sind nun die Maxwellgleichungen insgesamt forminvariant unter Lorentztransformationen?Wir werden dies später untersuchen, nachdem wir zunächst die kovariante Notation einführen.Die Newton’schen Bewegungsgleichungen sind forminvariant unterGalileitransformation, aber nicht unter Lorentztransformation. Da die Lorentztransformationensich als gültig erwiesen haben, können die Newton’schen Gleichungen nicht allgemein richtigsein. Wir werden sehen, wie sie zu modifizieren sind. ¡ ¨ ¡¡ 130

¥ & © " ¦" ¡" & ¡ ¥ " " ¢ " ¡ ¥ " £ ¥ & © " ¦§" ¡ " & ¡ ¥ "¢¨ "¨ ¢ "¨ ¡ ¥ "¨ £©£ ¨ £££ £ ¨ £¤¤££¦£6.2 Vierertensoren und Kovarianz physikalischer GesetzeLorentztransformationen verknüpfen Punktereignisse, die in zwei gleichförmig relativ zueinanderbewegten Systemen " ¢ "" ¥ und " ¢ " " ¥ an stattfinden.Die Lorentztransformation ist eine lineare Transformation unter der Nebenbedingung der Konstanzder Lichtgeschwindigkeit:¡ ¢ ¡ ¡ ¨ ¡¡ ¥ ¡ ¡ ¢ ¡ ¡ ¨ ¡¡ ¥ ¡Wir führen kontravariante und kovariante Vektorkomponenten ein:Dann gilt: ¡¡ ¥ ¡ ¨ ¡"was invariant ist.Diese beiden Vektorkomponenten werden durch folgenden metrischen Tensor verknüpft:£ £ ££ ¨ £ £¢£¡ && ¢Offensichtlich gilt: ¢ ¢ Wir wollen im folgenden stets die Einstein’sche Summationskonvention benützen. Der invarianteAusdruck schreibt sich dann:¡ & ¢ Die lineare Lorentztransformation werde geschrieben als ¤£ 131

& ¡ ¡& ¡ £ £Ebenso die Folge aus der InvarianzforderungDer Vergleich zeigt:¡ ¡ ¤£ ¢ ¢ ¡ ¤£¢Dann ergibt die Invarianzforderung ¡ £ £ £ £ ¢ £ £ £ £ ¢£ £die folgende Bedingung an die Matrix £ :£ £ £ ¡ £ £ £Wie sieht die Umkehrtransformation aus?Sei ¡ ¡ £ ¡ £ Dabei ist das vierdimensionale Kroneckersymbol.heben und senken:Die Indizes lassen sich mittels ¡ ¢Somit£ ¡ ¤ £¤¡¤ £¤ ¡ ¢ £¡Die obige Bedingung lässt sich demnach auch schreiben als¡ ¢ £ £ £ ¢ £ £££Somit ¤£ Hiermit lässt sich wiederum die Invarianzforderung formulieren: £¢¢ £ £ ¢ ¡132

¢£¢££ £ ¤¡ ¤¨¨ ¢ ¢ ¢£¢£¨¤¤£££ £ £ £ ¢ £ £ £ ¢ ££ ¨ £££ £ ¨ £¤¤££¦¦ £ £ £¥§££¤¦££¢£¦£¤¤££¦¤¡£¤¤££¦£¤¤££¦£¡Man spricht von Orthogonalitätsbedingungen an £ , modifiziert durch .Die uns bekannte Matrix der Lorentztransformation erfüllt natürlich diese Bedingungen:Sei die Relativbewegung längs der – Achse:¢ £ £ ¢ ¤ ¨ ¢ ¥ ¥ ¡ ¥ ¨ ¢¡ ¡¡ ¤¤ £ £ ¨ ¤£ £££££ ¤ £ £ ¤¤ £ £ ¤£ £ £££ £Wie zu erwarten war, wird bei der Rücktransformation einfach¢durch¨ ¢ersetzt.Nun zu den Orthogonalitätsbedingungen: ¤ £ £ ¤¤ £ £ ¨ ¤¤ £ £ ¨ ¤£ £££££¨ £ £££ ¨ ££ ¤ £ £ ¨ ¤ ¤ £ £ ¤ ¥¨ £ £££ ¨ ££¥¥ ¨ ¦£ £ £ ¤£ £ ££ ¨ £ £¢£ £ £Entsprechend verifiziert man die zweite Orthogonalitätsbedingung. "Einen Satz von £ "" ¢ "vier Größen, , die sich wie unter der Lorentztransformationtransformieren, nennt man Vierervektor:%% £ %133

¡ ¢ ¡ & ¡¡ ¥ ¡ ¨ ¡¡¡ ¥ ¡¢£ ¡ ¢"" "¡¥¥ §Ein Skalar liegt vor, fallsDas Skalarprodukt von zwei Vierervektoren % " ¥, definiert alsist ein Skalar:% £ ¥ & % ¥ % ¥ ¡ "Betrachten wir den Vierervektor ¥ % ¥ ¡ £%£% £ % £ % ¥££ £ £ ¢% £ ¥ £¡ ¥ " ¦" ¡ ¦gebildet aus den differentiellen Raum – Zeit – Verschiebungen, z. B. eines Massenpunktes, dannist & ¤ ¥¡¡ eine Lorentz–Invariante.¡¡ ¤¦ ¡¡ ¥ ¡ ¨ ¢ ¡ ¨ ¢ ¥Diese Relation hat¢offensichtlich folgende Bedeutung: ist das Zeitintervall in dem System, indem die £räumliche Verschiebung ist (das Teilchen ruht). Man spricht von der Eigenzeit.Das Zeitintervall , in dem eine räumliche Verschiebung erfolgt, , ist gegenüber ¢ ¥ ¨verlängert — Zeitdilatation.Wie lässt sich der übliche Vektor der Geschwindigkeit mit drei Komponenten auf die Vierergeschwindigkeitverallgemeinern? £ ¢ " ¢ "Dies sind nicht die Raumkomponenten eines Vierervektors, da nicht invariant ist.Jedoch sind ¥ " ¥¢ die Raumkomponenten eines Vierervektors.Verallgemeinerung der Geschwindigkeit auf die Vierergeschwindigkeit 134

ein Skalar ist, ist % ¤ ¡ ¢ " ¤ ¦¡ %%"" ¢ " ¤ ¡ " % ££ ¡ & ¤ £ £¡ %%£ £ %£%¢ ¤ ¡£¢¢¡£¢ £ ¢£¢ ¥ ¡¢ & ¥ ¢ ¥Falls ein Vierervektor:Die Viererdivergenz eines Vierervektor ist ein Skalar: ¡ ¤£ £ £Der vierdimensionale Laplace – Operator ist ein Skalar:& £ Daraus folgt, dass¤ %ein Vierervektor ist.Man nennt einen Vierervektor Tensor vom Rang 1, einen Skalar Tensor vom Rang 0.Tensoren vom Rang 2 sind definiert durch£ £etc. ¡ £ £ £ £Kontraktion:Diese Kontraktion generiert den Vierervektor:¥ &£ £ £ ¢ %¥ ¢ % ¤£ £ £ £135

Dabei " ist" "¨ ¡" £" "¤¡" " """"""¡ © " ¡ ¡ ¡©¡¡ ¡ © ¤ " © ¤ ¢ ¢ Das vierdimensionale Volumenelement ¢ ¦ ¡ © " © ¡ ¥ist invariant unter Lorentztransformationen: ¢ " £die FunktionaldeterminanteNun gilt:£ £ £Bilde davon die£ ¡ ¢ £ ¡ DeterminanteDa die Lorentztransformation stetig aus der 1 hervorgeht, ist ¨ ausgeschlossen und££¢¥¤ £ , somit ¢ ¢ .Die spezielle Relativitätstheorie besagt, dass physikalische Gesetze in verschiedenen, gleichförmigrelativ zueinander bewegten Koordinatensystemen (Inertialsystemen) dieselbe Form haben(Forminvarianz). Physikalische Gesetze müssen deswegen Relationen zwischen gleichen Tensorensein. Eine gültige Relation kann also z. B. nicht einen Skalar einem Vektor gleichsetzen.Beispiele für forminvariante Gesetze: £ ££ ¤£ ¤ ¡wobei ¡ein Tensor 2.Stufe und ¡ein Vierervektor ist. 6.3 Die Kovarianz der MaxwellgleichungenWir werden zeigen, dass man in konsistenter Weise Stromdichten, Felder etc. als Tensoren imvierdimensionalen Raum einführen kann, so dass die Maxwellgleichungen eine kovariante Formerhalten und damit forminvariant sind unter Lorentztransformationen. und sind verschiedene Aspekte des gleichen Objekts. Es ist eine Frage des Bezugssystems,ob man eine ruhende oder eine bewegte Ladung sieht. Deswegen erwarten wir, dass sich und zu einem Vierervektor zusammenfassen lassen.Sei © die Ladungsdichte in dem System, in dem sie ruht. Dann betrachten wir den Vierervektor& ¡ " & ¡ " ¢ ¢136

¡ ¡ ¡ " ¢£© © £ ¨ %¡ ¨ ¤ © ¡¨ ¤% ¢ ©"¨ ¥ ¡ ¨ ¡ ¤wobei © ¤ © ¨¥ ¥ .Für £ © wird nach erhöht, und eine Stromdichte stellt sich ein. Das heißt, eine ruhende¢ Ladungsdichte © " £ ¦¡repräsentiert sich von einem¢mit bewegten System aus als eine veränderte Ladungs – undStromdichte¡ ¤¥¥ ¥¥ ¢ ¤¦Ist dieses Transformationsverhalten von verträglich mit der Ladungserhaltung?¢¡ ¤¢£ ¤ ¦¥¨§ £ ©¤¨ ¢ ¢¨ ¢ ¡¡¡ Der Längenkontraktionsfaktor in – Richtung kompensiert exakt die Vergrösserung der La- © ¢ ¤ ©dungsdichteWir erwarten demnach, dass die Kontinuitätsgleichung sich in einer kovarianten (forminvarianten)Weise schreiben lässt: £ £¥Dies ist dasselbe wie £ ¤£ ¤¥ ¡ ¡ ©Offensichtlich ¡ ¤£ist forminvariant.Die Einführung der Viererstromdichte ist der Einstieg zur kovarianten Formulierung der Maxwellgleichungen:Mit der Lorentzbedingung¥ £%sind die Wellengleichungen für und :¡ % ¡¡¡¡ ¡ ¥ ¡ ¨ Dies lässt sich zusammenfassen zu¡¡ 137

¢£% % ©%% ¨ £¦ %¡ %%% ¡ %¥% © £ ¥' % % ¦££¤¦£¤¡ ¨ ¢ ¤ %¡ mit " % . Dies ist eine kovariante Form, und ist deswegen ein Vierervektor.Wir lernen hieraus, wie sich die Potentiale unter Lorentztransformationen transformieren!% %Die Lorentzbedingung ist ebenfalls kovariant:¤£Betrachten wir nun die Feldstärken ¥:'und ' ¨ ¡ ausgeschrieben:¥ ¢' ¦ ¨ ¦ ¨ ¦ ¨ © ©¥¦ ¡ ¨ ¨ ¡Es liegt auf der Hand, den Feldtensor vom Rang 2 einzuführen:Dann findet man ¢ ¨ ' ¦ ' ¡ ' £' ¦ £ ¥ ¨ ¥¡¨ ' ¡ ¨ ¥ £ ¥¦ ¨ ¢ Damit lassen sich die inhomogenen Maxwellgleichungen formulieren:¨ ' ¥¡ ¨ ¥¦ £' ¥ ¨ ¡ ¡¡Die rechte Seite ist ein Vierervektor, demnach ist zu hoffen, dass die linke Seite auch einer ist.Wir raten:138

§¤¤ © ¡ ¡ '¤ ¨ oder¢ ¤ ¤¦ '¥¥£ ¥ ¤¢ £ ¦¤£ ¦¤£¡ ¡¡¥ ¡¤£¤¡ ¡¦ ¤Nachweis: ¢' ¦ © © ¦¤ ¡ ¨ ' ¦ ©¡¦ ¦ entsprechend für ¢ ".Schließlich lauten die homogenen Gleichungen: ¨ ¡ ¥ £ £ ' ¡¥ £Diese vier Gleichungen lassen sich zusammenfassen zu¡ ¡Offensichtlich sind die Fälle, wo zwei Indizes gleich sind, ohne Aussage und trivial erfüllt. Esmüssen demnach alle drei Indizes verschieden sein: © ¦ ¦ ©¥¡ ©¨ ¨ ©' ¦' ¡ ¤£ ¦ ¦¡ ¡ ¦¥ ¥¦¥¡ ¡¤£ ¨£¥ ¤£oderDamit ist gezeigt, dass sich die Maxwellgleichungen kovariant schreiben lassen und somit dieElektrodynamik der Grundforderung der speziellen Relativitätstheorie genügt, forminvariant gegenüberLorentztransformationen zu sein. ¨ ¦ ¨ 139

£ ¡¤¢£¨¡¡¡¡¥¦¡¡¤ £ ¨ ¢¤¡¡¡ ' ¡¡¤ ¦¤ ¡¤ ¡ © ¤ ¡ ¤ © ¦ ¤ ¦¤¤££¦£¡¥¡6.4 Lorentztransformation des elektromagnetischen Feldes und Anwendungauf das Feld eines geladenen TeilchensDa 'und ¥im Tensor ¡ enthalten sind, ist das Transvormationsverhalten festgelegt. Offensichtlichwerden 'und ¥Komponenten dabei ineinander umtransformiert, was die Wesensgleichheitvon ¥haben keine unabhängige Existenz. Ein rein elektrisches'und 'und Feld z.B. im Koordinatensystem K erscheint als Mischung ausspricht deswegen vom elektromagnetischen Feld.¥zeigt.¨ ¡ £ £ £ £ £ £ Betrachten wir speziell LT von K K’ längs der -Achse :¢'und ¥im System K’. Man¤ £ £ ¨ ¤£ £££££ ¤ £ £ ¤' ¦ © ¦ ¤£ © £ ¦ ¡ £ © ¦ ¤ © ¦ ¨ ¢¤ ¥¡ ¤ §' ¦ ¨ ¢ ¤ ' ¦ ¨ ¢' ¡ © ¡ ¤£ © £ ¡ ¡ £ © ¡ ¤ © ¡ ¨ ¢ ¤ ' ¡¢' © ¤£ © £ ¡ ¤ £ © ¨ ¢¤ © ¤ £ ¤ © ¨ ¢ ¤¡ ' ¨ ¢ ¡¥ ¦ ¡ ¤£ ¡ £ ¡£ ¡ ¨ ¢ ¨ ¢ ¤¨ ' ¡ ¦ ¥¦ ¤ ¥¦¢¥¤ ' ¡ ¡¥ ¡ ¦ ¤£ £ ¡ ¡ £ ¦ ¨ ¢¤ ¨ ¢¡' ¦ ¥¡140

¦¡ ¤£ ¦ £ ¡ ¡ ¥ ¥Wir sehen, die 3-Komponenten ändern sich nicht. Wir können dieses Ergebnis verallgemeinernbezüglich LT in Richtung ¢:' ' ¥ ¥' ¡ ¥ ¡ ¤ ¥ ¡ ¨ ¤ ¢ '¥ ¡¡¡ ' ¡Wir lesen hieraus z.B. ab, dass eine mit¢bewegte Ladung eine magnetische Induktion erzeugt.Sei Ladung in K ruhendDann gilt in K’, das mit¨ ¢sich bezüglich K bewegt' ¡ ¥ ¡ ¢ ¥ '(¤¤¡¡Somit in nichtrelativistischer Näherung ¤ ¡ '¡¢¥ ¤£ ' ' "" ¥ £ '' ' ¥ '(¥ ¢ Das ist der uns aus der Magnetostatik bekannte Ausdruck des Magnetfeldes für langsam bewegteLadungen.Wir wollen dies nun genauer betrachten :%% % % % %% % % % %% % % % %% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % %% % % % %% % % % %% % % % %% ¦ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%¡¢ £¦¡ ¡ ¢¡ ¡% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % %%% % % % % % % % % %%% %%%%% % %%%% %% %%%%%%% % % % % % % % %%%%%% %% %% %% %%%%%%%%%%%% %%% %%% % %%% %%% % %%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% %%%%% %% %% % %% %% % %%%% %%% %% %% % % %% % % % %% %% %%% % % %% %%%%%%%% % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % %¥% % % % % % % % % % % % % % % %% %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%¡ % % %141¢ "

¡ £¦ ¨ ¢ ¥ ' ¨ ¥ ¤ ¢ ¥' ' ¨ ¥ ¤ ¢ ¥¥ ¡¤¤¡ ¢ ¡ ¥ ¡ £¢¡§ ¦¡¥¡¢ ¦¦ £ ¥¡ £ ¥¡An ¥© ¥ £sollen die Ursprünge zusammenfallen und der Abstand von¥und Beobachtungspunkt¢ sei . In K’ hat ¢ die Koordinatenund damit den Abstand ¤ ¡ ¢ ¡ ¥¡von¥.Das elektromagnetische Feld in K’ am Punktist¥ £' ¦ ¥' ¡ £Das Ziel ist es, das elektromagnetische Feld am Punkt P im System K auszudrücken. müssen und durch Koordinaten von K ausdrücken :¥ ¨ ¥ ¢ ¥ ' ¤ ¥ ¨ ¢¥¡ ¤ ¥¡da £für P in K' ¦ ¥ ¡ ¤¡ ¢ ¡ ¥ ¡ £¢¡¢¡ ¤¡ ¢ ¡ ¥ ¡ £¢¡Nun sind diese Feldkomponenten noch nach K zu transformieren. Dies entspricht der Umkehrtransformationvon vorher :' ¦ ¤ ' ¦ ¤ ¥ ¡¥¤¡ ¢ ¡ ¥ ¡ £¢¡ ¡¥¤¡ ¢ ¡ ¥ ¡ £¢¡¡ ¤¢¥ ' ¦ ¢¡' ¦" ¥¦ ¥ £¡Der Beobachter in P sieht demnach zusätzlich ein magnetisches Induktionfeld B :¡ ¢¥¡¢¡ ¥Betrachten wir nun extrem relativistische Verhältnisse¢ ¢ ¡¥¡ ' ¦ ¢142:

Der Maximalwert tritt für auf : £ ¦ ¤¥¡ 'Wegen ¥ tritt eine erhebliche Verstärkung gegenüber dem nichtrelativistischen Wert auf!Nach dem Ausdruck für ¤ ¦ist ein typisches Zeitintervall, in dem dieser Wert auftritt' ¨¡ ¥ ¡¢ ¢ £¤ ¢ , d.h. geht mit ¤ ¢ £ gegen Null# ¥ ¦' ¦¡%¥¡% % % %% % % % %% % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% % % %%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%¢ ¢ % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%¡ ¥ ¡ # ¦ %%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Für ¢ & ¢ sieht der Beobachter ein transversales elektromagnetisches Feld mit gleicherStärke'von und , wie bei einer ebenen Welle in -Richtung. Aber es ist nur ein kurzerImpuls.Daneben sieht er noch ein longitudinales elektrisches Feld¥' % % % %% % % % %% % % % %% % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%¢%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%¢ ¢ £¢ ¢ %Da das Vorzeichen wechselt, ist die Wirkung von ' reduziert. Somit ist der wesentliche Effekteines sehr schnellen vorbeifliegenden relativistischen geladenen Teilchens der eines kurzenPulses einer transversalen elektromagnetischen Welle.Wollen uns das elektrische Feldlinienbild der bewegten Ladung noch mehr veranschaulichen:143¥¥

¢ ¦% % %% % % % % %% % % % %% % % %%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% %% %% %% % %% %% %%%% % % % % % % % % % % % % % % % % % % % ¡'zeigt von momentaner Lage von¥in radialer Richtung zum Aufpunkt.Für ¢ £für ¢¥¢ ¢¡ ¡ ¨ ¡ ¤¥ ¡ ¨ ¡ ¦ ¥ ¡ ¡ ¢ ¢ ¢¦ ¡ ¤ ¨¤¥ ¥ ¢ ¡ ¢¦£¥und gilt ' ¥ ¡ ¤¡ "¢' ¥ ¤% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %¦ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%¢ ¥' ¦' ¨¢ ¥' ¡ ' ¡' ¡¦ ¤¡ ¥ ¡ ¡ ¥ ¡¤¡ ¢ ¡ ¥ ¡¤¡ ¥ ¡ ¡¤ £ ¤ ¡ ¨¥¤ ¢ £ ¤ ¨¥¥ #¥ ¥¡¥¤¡ ¢ ¡ ¥ ¡ ¤¡ ¥ ¡ ¢ ¡ ¡ ¥¥ ¡¤¨¥ ¥ ¤ ¢ ¢' ¥ ¡ 144

Feldstärke in transversaler Richtung zu¢extrem verstärkt% %% % % % %% % % % %% % % % %% % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%% % % % % %% % % % % %% % % % %% % % % % % % % % % % % % % % % % % % % %%% %% % %% %% % %% % %% % %% % %% % %% % %% % %% % %% %% % %% % %% % %% % %% % %% % %% % %% %% % %% % %% % %% % %% % %% % %% % %% %% % %% % %% % %% % %% % %% % %% % %% %% % %% % %% % %% % %% % %% % %% % %% %% % %% % %% % %% % %% % %% % %% % %% % %% %% % %% % %% %%%%%% %% %%%%%%%%%%%%%%%%%%%%%%%%% %%%%%% %%%%%%% %%%%%% % % % % % % % % % % % % % % % % % % % % % % '%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%6.5 Die Erhaltungssätze für Energie und ImpulsDie Kraft des elektromagnetischen Feldes auf Ladung und Strom kann in Form einer Kraftdichteformuliert werden:in Komponenten¥ ' ¡¡ ¡¥¦ ¥ ' ¦ ¡ ¡¡¥ ¨ ¡ ¥¡ ¦ © © ¦ ¡¡ ¦¡ ¡ ¦ ¦¨ ¡¨ ¡¤¡ ¦¡¤ ¦ ©¡¤¡ © ¦¡¡ ¡ ¦¡ ¦Die rechte Seite ist offensichtlich die Raumkomponente eines Vierervektors ¨ ¥ ¡¡Was ist die Bedeutung der £ -ten Komponente ?© ¨ ¥ ©¡¤¡ ¤¡¡Dies ist die Arbeitsrate des Feldes an den Quellen pro Volumen, was gleichbedeutend ist mitder zeitlichen Änderung der mechanischen Energie der Quelle pro Volumen. Entsprechend ¥istgleich der zeitlichen Änderung des mechanischen Impulses der Quellen pro Volumen.Dieser Einstieg kann zum Erhaltungssatz für Energie und Impuls des Gesamtsystems TeilchenFeld ausgearbeitet werden.145¡' £ ¢

Wir eleminieren ¡ ¨ ¥ ¢ ¡ ¡¢£ ¨ ¡ ¡ ¥ ¨ ¨ ¡ ¡ ¢ ¢ ¨ ¡ £ ¨ ¢¡ §zugunsten des Feldes: und erhalten¡ ¡ ¢¨ Definieren den Spannungs - Energie - Impuls Tensor ¢ ¢ ¢ ¡Nun gelten die homogenen Maxwellgleichungenoder ¡ £ Auf passende Indizes umgeschrieben: ¡¢ ¢ ¡¡ Somit gilt ¡ und¡Betrachten nun die Struktur des Tensors ¢¡ ¢ ¢ ¢146

¡ ©¤ ' ¡ ¡ ¡¡¡ '¡ '¢©¤¤ © © ¤ ¦ ¨ ©¤¤ ¢ ¡ ¢©¥¡¢ ¤¤¤© ¤¤ ' ¡© ¤¥¡ ¦ ' ¡ ¨ ¥ ¤ ' ¥¡ ¦ ¤ '¡ ¡¡Somit ist die Struktur des symmetrischen Tensors¡¤ ¥¢¥¤¤¡ ¢£¤¦¡¡¡ ¡¤¤¥¤ '¡¦¡¡¡ ¡¤¤¤© ¦¢¤¤¤¤££¦¤£¦¤ '¡¨ ¤¤¢©¨© ¤ ¨ ¢ ' ¡ ¥ ¡ ¥¡ ¡ ¥ ¡ ¥¦ ¡ ¥¡ ¡ ¥¦ ¡ ¦¤¢ ist die Energiedichte des elektromagnetischen Feldes¤ ' © ¨ ¡ ¥ ¦ ©Impulsdichte des Feldes¥ ©¥ § ¡Der Vergleich mit dem Maxwellschen Spannungstensor zeigt, dassDie Erhaltungssätze für Energie und Impuls des Gesamtsystems sind nun abzulesen: ¨ ¨ ¨ ¨ © ¨ ¥¨ ¥ ¢ " ¨ ¥¡£¢ ¤Impuls des Feldes 147

¤ ¢ ¥ © ¨ Wiederum integriert über ein Volumen ¢¥ ¡¥ ¨ ¡¦ © ¡ ¥ ¥ ¤ ' " ¡¤ £ ¨ ¤ © £¤ £ ¡ ¡ £ ¢ ¥ ¢ " Dies ist der Impulserhaltungssatz, wie schon gesehen©¨© ¨ ©¨ ¡ ¥ ¨ ¨ ¡ hierbei ist die Energiestromdichte.Somit¡§¢ " ¨ ¥ ¡ ¥ ¥ ©¥ ' ¨Dies ist der Energieerhaltungssatz, wie wir ihn bereits kennen.6.6 Relativistische Teilchenkinematik und TeilchendynamikEin geladenes Teilchen kann als lokalisierte Verteilung von Ladung und Masse betrachtet werden.Die Auswirkung auf das Teilchen ergibt sich dann zu ¡Die -te Komponente ist gleich ¦ ¡ der zeitlichen Änderung der Energie des Teilchens, dieräumlichen Komponenten gleich der zeitlichen Änderung des Impulses des Teilchens. Führen£ ¤einwobei ¤ ¦¤Daraus lesen wir ab, dass ein Vierervektor ist, ¥ ¥da ein solcher ist. Wir erhaltendamit die sehr wichtige Aussage, dass Energie und Impuls eines Teilchens zusammen einenVierervektor bilden:¤" ¤ ¦ '148

¢¢¤ ¥ ¢ ¡ & ¡ ¥ ¡ ¤¡¡¤ ¥ ¤ ¢ ¤ ¢ ¦¡ © ¡¤ ¢ ¢ ¥ ¤ ¢ ¤ ©¦¡ ¡ ¡' " ¢ ¤Es ergibt sich dann ¢¨ ¡ ¡Dies ist noch keine manifest kovariante Form, da weder noch Skalare sind.Die Eigenzeit, das Zeitintervall im Ruhesystem des Teilchens, ist ein Skalar, wie wir gesehen¥haben: oderDemnach ¢ ¥¤¤ ¢¡ ¤¨ ¡Das Teilchen hat eine stark lokalisierte Verteilung von Ladung und Masse. Es giltund demnach die Gesamtladung" ©¡¡ ¢ ¢ ¡¡ ¥ ¦¢¤ ¡ ¡¦ ¢ §¥ ¦¡ ©Über den Bereich des Teilchens können wir ¡als konstant ansehen ¤ ¨ ©¡ ¡ ¨Wie hängt der Viererimpuls und die Vierergeschwindigkeit zusammen ?¤ Invariante Länge von p: ¤ ¡ " ¤ ¡ & ¤ ¤ & £ ¡¤ ¡149¡ ' ¡¡¡ ¨ ¡

¥¢¡ ¡und ¥¥ ¤ ¤ ¢ 'Dabei haben wir der invarianten Länge¡den Namen¤ £¤Im Ruhesystem istWas ist demnach ?Betrachten eine Lorentztransformation des Viererimpulses aus dem System K, wo das Teilchenruht, in ein System, in dem das Teilchen eine Geschwindigkeitskomponente v in der 3-Richtunghat:£ ¤ ¤£ ¢ ' ¥¥ gegeben.¤ ¨¦ ¤ ¦ £ ¤ ¨¡ ¤ ¡ £¡¡ ¤£¡ ¢¡¨ ¤ ¤ ¢ ¤¡¡ ¤ ¢'£¡¡' ¤ '¤ ¤¤ ¤£¡ ¢¡Nun ist der nichtrelativistische Ausdruck für den Impuls¢ ' ¤ £ ¤£ ¡¡,wobei m die Masse des Teilchens ist. Somit¢¢ ¢ ¤'( ¤ £ ¡¡£¡ ¢¡£¡ ¡ ¡¡ ¢ ¢ ¡ £ £¨ Die Energie besteht demnach aus Ruheenergie, gegeben durch die Masse des Teilchens, und&'kinetischer Energie. ¡ ¤ ¨ ¡¡ ¡¡ ¨©¢ ¢ ¡ ¢ £Diese Masse nennt man präziser die Ruhemasse des Teilchens, da sie den Energiegehalt desTeilchens in Ruhe darstellt. Man findet also wieder allgemeinSomit ergibt sich der gesuchte Zusammenhang zwischen und : ¤ ¤¤ ¤ ¦ ¤ ¡ " ¤ ¢ "Weitere wichtige Relationen sindund¤ ¡ ¡¡¡ ¤ ¥ ¤ ¡ ¢ '( ¤ ¡¡ ¢ ¤ ¡¡¡ ¨150

¤ ¦ ¢ ¤ © ¢ ¤ ¢ ¦¦¦ ¨ ' ¦ '¦ ¤ ¥ ¦' ¥ ¦¦¤¤¦ ¢¤ ¥¡'¦¤ ¦ © ¤© ¦¤¥¦ ¦¦¥¢ ¡¡ 'Nun zurück zur Bewegungsgleichung eines geladenen Teilchens im elektromagnetischen Feld(Ruhemasse m , Ladung e ) ¨ ¡ ¢ ¤Diese Form ist manifest kovariant. In Komponenten besagt diese Gleichung:¡ ¡ ¨¡ ¦ ¤ ¨¤¤¦ ¨ ¡ ¨ ¡¡¨ ¥ ¤ ¡¦¥¡ ¤ ¡ ¤ ' ¦¦¤ ¤ ¡¥ ¨ ¤ ¥¡ ¦ ¡ ¡ ¦ ' ¦¤¤ ¡oder£ ¢ ¦ §' ¦¦¤ ¦ ¦¢¡¢¡' ¢£¡ ¤¦Dabei ist natürlichSchließlich¤ ¤ ¢! ¡ ¨ ¡ ©¤¤ ' £ ¤ ¦ ¡¤ ¢ £Somit ist¢ £ ¦ 'die wohlbekannte Form der zeitlichen Änderung der Energie E des Teilchens auf Grund deselektrischen Feldes . E ist natürlich gegeben als'( ¤ ¡¡.151

¤ ¢ ¢¢£¢£¡¢ ¦ © ¨ ¦ © ¨ '¦ © ¨ ¥¦ © ¨ ¡ ¡ ¡ ¡ ¦ © ¨ £££¤¦£¤¤££¦£6.7 Relativistische Elektrodynamik materieller KörperDie Größen, die sich auf das Ruhesystem des materiellen Körpers beziehen, seien mit Index ( 0) gekennzeichnet. Damit lauten die Maxwellgleichungen im Ruhesystem des Körpers¦ © ¨ ¦ © ¨ ¦ ¦ © ¨ ¢£¡ ¦ © ¨ ¦ © ¨ ¦ © ¨ ¨ ¡ ¦ © ¨ £ ¢£¡¤ ¦ © ¨¦ © ¨ £¡' ¦ © ¨ Nun treten 4 Feldvektoren auf benötigen 2 Feldtensoren.Für die beiden homogen Gleichungen wie bisher¢' ¦ ' ¡ ' £' ¦ £ ¥ ¨ ¥¡¨ ' ¡ ¨ ¥ £ ¥¦ ¨ ¢ ¨ ' ¥¡ ¨ ¥¦ £und für die beiden inhomogenen analog ¦ ¡ £ ¦ £ ¤ ¨ ¤ ¡¨ ¡ ¨ ¤ £ ¤ ¦ ¨Der Index ( 0 ) ist der einfacheren Notation wegen weggelassen worden . Außerdem natürlichdie ¦ © ¨ ¦ © ¨Stromdichte¨ ¤ ¡ ¨ ¤ ¦ £Die Maxwellgleichungen lauten dann¡"¤ ¡ ¦ © ¨¦ © ¨¦ © ¨¢ ¦ © ¨ ¦ © ¨ 152

¡¡¢¡¤¡£ ' ' ¦ ¨ £¡ ¡¡ ¢ ¤ ¢ ¤ ¤ ¤ Schließlich¤¤©In dieser kovarianten Form lassen sie sich auf ein Koordinatensystem transformieren, in dem dermaterielle Körper bewegt ist . Bevor wir dies tun, betrachten wir noch die Verknüpfungsgleichungen:Wie lassen sie sich kovarinant schreiben?Wir betrachten nun zunächst die erste Beziehung und setzen folgende Form an:¥ wobeidie Vierergeschwindigkeit der bewegten Materie ist ¤ ¡ " ¤ ¢ " ¢Geschwindigkeit der bewegten MaterieIm Ruhesystem¦ © ¨¦ ¨ £¡ © ¦ © ¨ ¡© ¨¦ ¦ '£¦ © ¨¡¢£'© ¦ © ¨¦¦ £D.h. diese Viererstromdichte resultiert nur aus Stromdichte im Ruhesystem. L steht für Leitungsanteilder Stromdichte, der durch das elektrische Feld erzeugt wird. Wir werden später nochzusätzlich den Konvektionsanteil der Stromdichte einführen.Nun zur zweiten Beziehung:besagt im Ruhesystem :¤ ¡ ¢© ¦ © ¨ ¡ © ¦ © ¨ ¡ ¢ ¦ © ¨ ' ¦ © ¨¤besagt im Ruhesystem : ¢ ¦ © ¨ ¦ © ¨© ¦ © ¨©153

triviale Aussage £ £¡¡ ¡© ¤ ¡¨¡¡¡¡¨ £¡¡£ ¢£© ¨ ¡¦&¨ ¡¦ ¨ ¢ ¨¡© ¨ ¨ ¢ ¦¡¨¡ ¤¡ © ¨¦© ¨ " £ ¦¡"" ¢ "¤¡ ¦ © ¨ ¤ ¡ ¦ © ¨ ¤¡ ¦ © ¨ ¤ ¨ ¢ ¦ © ¨ " £ " £ "¨ ¤ ¢ © ¤ © ¦ © ¨©££¤¦¡£¦ © ¨ © © ¨ ¨ ¢ ¦¡ ¦ © ¨ ¡¦ © ¨ "¦ © ¨ ¡ ¦ © ¨ © ¤ ¡ ¦ © ¨© ¤ ¦ © ¨© ¤ ¦ © ¨© © ¦ © ¨ £ ¡ ¦ © ¨ © ¦ © ¨oder ¤ ¡ ¦ © ¨ ¤ © ¦ © ¨ £¤£¤Damit sind manifest kovariante Formen erreicht und man kann die Felder, Stromdichten, Magnetisierungetc. für den bewegten Körper durch LT bestimmen : ¢ ¥ ¤ ¢ ¥ Betrachten LT längs z-Achse mit Geschwindigkeit¢, d.h. Körper bewegt sich in K’ mit Geschwindigkeit¨¢. ¨ ¡ £ £ £ £ £ £etc.¤ £ £ ¨ ¤£ £££££Viererstromdichte ¤ £ £ ¤¦ © ¨¡£¢ ¤¡£¢ ¤£ " " ¤ ¢ " ¦¦ © ¨¨¦ ¡¦ ¦ © ¨¨¡ ¡¡ ¦ © ¨ ¨ ¢¡¨¤ ¡ Dieser wird zerlegt in den Konvektionsanteild.h. die Ladungsdichteund dem Leitungsanteil¦ © ¨¤ ¡ führt im bewegten System auf Stromdichte,¡ ¦ © ¨ "¡¦ ¦ © ¨ "¡¡ ¦ © ¨ " ¤¡ ¦ © ¨ ¤154

Dies zeigt den interessanten Effekt, dass eine reine ¡ ¦ © ¨Stromdichte im bewegten System auchals Ladungsdichte erscheint :¦ ¨ ¤¢ ¨¡ ¡ ¦ © ¨ ¨ ¡¡¡¡Dies sei nun veranschaulicht. Sei ein ruhender Metallstab von Strom durchflossen.ruhende positive Ionen¢ £% % % % % %% % % % % % % % % %%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%% % %%%%%%%%%%%%%%%%%%%%% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %%%% %%%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % %% % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%¡ ¥ ¦ © ¨¡ ¥ ¦ © ¨¦¨bewegte ElektronenVom bewegten Koordinatensystem aus betrachtet sind auf bestimmtem Abschnitt der x’-Achsenicht gleichviel positive wie negative Teilchen Stab erscheint lokal geladen (In dieser Figursind die + Abschnitte kürzer als die - Abschnitte).Betrachten nun die Maxwellgleichungen für langsam bewegte Materie:¢¤ ¡ ¢rot ¡ ¡¤ mit¡ ¡ ¦ ¤¡¡¡¦ © ¨¨ ¤ ¢ ¡¥¡ ¦ ¡¡ © ¨ ¨ ¢¦( beachte, der Körper bewegt mit¨ ¢sich )div 155¡¡

¡¤¤£ ¡rotdiv ¡¤¡ ¦ £¤ ' ' £ ¢ ¤ ¢ ¢¡¡¢ ¥¥¡¡ ¡ ¦ ¨ ¤ ¦ ©¢ ¥ ¢ ¡ ¨ ¥¡ ¢ ¤¢ ¤¡ ¡ ¢ ¥ ¤ ¡ ¨ ' ¡¤ ¢ ¦ ' ¦¤ ¢ ¡ ¤ ¡ ¥ ¨¥ ¨ '¡ ¢¢¡ ¡¤ ¡¡ ¦¢ £¡ ¦¦ © ¨¡¡' ¨ ¡ ¥ £Verknüpfungsgleichungen¡ ¦ ¨ £ ¡ " ¤ ¡ "¡¤ ¡ ¦ © ¤ ¡ ¨ ¤ ¦¤¡¢ ¡¦¦ ¨ £¤ ¡ ' ¦ ££¡¥zur elektrischen Kraft kommt noch magnetische Kraft hinzu.¦ ¨ £ © ©¡£¤¡¡ ' £ ¢ "dies ist nur die Leitungsdichte©¦¢ ¡¤¨ ¤ ¦¤©© ¨ ¦¤ ¤¤ ¦ ©¨ ¦¤ ¡ ¨ ¤ ¤ ¢ ¡ ¤ ¡¤ ¢ ' ¦ ¤ ¡ ¨ ¥ ¤ ¢ ¡ ¥¡¤ ¢ ¤ ' ¥d.h. Magnetisierungs- und Polarisationsphänomene werden vermischt.¢ ¤ ¤ ¡ ¤ ¤ £ ¡ ¤ ¡ © ©ist trivial für oder £ £ £¢¡' ¦ ¤ ¨ ¢¡156

Lösen diese Beziehungen nach oder ¥¥ £¢¤ ¢¡und ¥ ¨ ¢ ¡¡ ¥¡¡¡ ¥ ¢¡¥¥ ¢ ¢¢¢ ¥ ¡ ¢¡ ¡¡¢ ¢ ¡¢¡£ #¨ ¢ ¤££¤ ¤¢¡ ¢¡£¤ ¢¡£ ¢¡¡ ¤¡¢¡ ¢' ¡¢¡ ¡£ ¢ ¤¡ ¡'¢¡¡¢¡¢¡¡¡¡ ¨ ¤ ¢ ¡£ ¡ ¨ ¡¥ ¨ ¢auf:' ¨ ¢¡ ¡ ¢ ¢ ¨¥ ¤ ' ¨ ¨ ¢¡£¥ ¨ ¥¢ ¡¡¡ ¡ ¡¥ £¡ ¡¡¡ ¥¨ ' ¨ ¤ ¤ ¢ ¡¢¡¡ ¡¡¡¢ ¢ ¡¥ ¨ ¥ £¡¡ ¨ ¢¡ #¨ ' ¨ ¤ ¤ ¢ ¡¢¡¡ ¡¢ £¢ ¨ ¨ ¢ ¡¡¡ ¡ ¨ ¢¡, was direkt aus folgt.¢ £¢ £¢ £¨ ¢¡¤ ¢ ¡¡¡ ¨ § ¥ ¡ ¤ § £ ¨ ¡¡ ¨ #¨ ¡ ¡¤ ¨ ¢ ¡' ¨ ¢ ¡ ¢¡157

¨ ¥ ¥£¢£ ¥ ¤¥ ¨ ¢ ¡¡¡¡¢¨ ¢ ¤¤¤¤££¦£¡ ' § £Entsprechend findet man ¢¡¡¡ ¨ ' ¨ ¢ ¡ ¡ ¡Im Vakuum reduziert sich dies auf .Für diese bewegte Materie finden wir eine komplizierte Verknüpfung zwischen PolarisationsundMagnetisierungsphänomenen. Dies rührt daher, dass bewegte Polarisationsdichte als Ma-gnetisierungsstrom und umgekehrt erscheint.Dies lässt sich präzisieren : 'und ¥ ¡ & ¤ ¡ ¥ ¢Mit der Bezeichnung :¨ ¢ ¦ ¨ ¢ ¡ ¨ ¢ £¦ £ ¥ ¨ ¥ ¡¢¡ ¨ ¥ £ ¥ ¦¢¥ ¢ ¢ ¥ ¡ ¨ ¥ ¦ £gelten die bekannten BeziehungenIn dieser kovarianten Form kann man ¥ ¢ vom Ruhesystem auf ein bewegtes System transformieren.SeienRichtung:¢ ¦ © ¨¥ ¦ © ¨unddie Werte im Ruhesystem des Körpers, dann gilt für LT in z-' ¢ ¦ ¤ ¢ ¦ ¦ © ¨ ¨ ¢¥ ¡ ¦ © ¨¢ ¡ ¤ ¢ ¡ ¦ © ¨ ¢¥ ¦ ¦ © ¨¢ ¢ ¦ © ¨¢ ¡ ¦ © ¨¥ ¦ ¤ ¥ ¦ ¦ © ¨ ¢¢ ¦ ¦ © ¨¥ ¡ ¤ ¥ ¡ ¦ © ¨ ¨ ¢¥ ¥ ¦ © ¨158

Wir erkennen, dass eine reine Polarisation im Ruhesystem im bewegten System auch auf Magnetisierungführt und umgekehrt. In niedrigster Ordnung gilt¢ ¢ ¥ ¥ ¦ © ¨ ¨ ¦ © ¨ ¢ ¢¡ ¥ ¦ © ¨¡¢¡¡Veranschaulichen uns den Term¡ ¥ ¦ © ¨als elektrische Dipoldichte. Betrachten die Kreisbahn eines Elektrons in der ¢-Ebene¡%% %% % %% % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%§¡ ¥ ¦ © ¨% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % %%%%% % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%%%%%%%%%% %%% %% % %%% % % % % % % % % % % % % % % % % % ¢ %%%%%%%%%%%%%%%%%%%%%%%%% % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%¢Materie ist unpolarisiert, wohl aber liegt eine Magnetisie-Der positive Kern liege bei ¤£rungsdichte vor.¢¦ © ¨% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%%%%%%%%%%%%%%¢ ¦ © ¨1,2,3,4,... sind die Durchstoßpunkte der Kreisbahndurch die¥ £ ; dazwi- ¢-Ebene£schen ist oder . Die Aufenthaltsdauerim Bereich ¢ und ¢ ££ £sind gleich:¡ ¢ ¡ ¢¢ ¥©¡ ¥ ¢ ¥ £159

Übergang ins bewegte System, bewegt in x-Richtung§¢%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%¡ ¥ ¦ © ¨ ¦ © ¨¡ ¥ kürzerlänger%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Die Verweildauer in den Gebieten ¢ £ und ¢ ¡ £sind verschieden¢ ¥¢ ¡ ¨© ¥ ¢ ¥ £Da der Kern in der ¥ -Ebene bleibt, erscheint ein elektrisches Dipolmoment.6.8 Abstrahlung einer relativistisch bewegten PunktladungViererpotential herrührend von Ladungs- und Stromdichte fanden wir zu160

" ¥ %¡ ¥ ¡ " ¥ ¨ & ¡ ¥ " & © " ¢ ¨ © ¨ © ¨ ¥ ¨ ¨ ¥ ¡ © ¨ © © ¨ ¤ ¨ ¡ ¦ ¢ ¨ © ¨ © ¨ ¨ © ¨ ¥ ¤¨ & & ¥% ¡ ¡¡¦¨ ¨ ¢ ¢ © ¨ ¤ ¨ ¡ ¦©¨ ¦ ¨ Dies lässt sich manifest kovariant schreiben: ¤ ¨ ¡ ¦ ¤ © ¨ © ¡ ¨ ¨ ¡ ¦ ¨ ¢ ¡Sei ferner & © ,dann giltDies ist manifest kovariant.Nun ist die Viererstromdichte eines Punktteilchens, das sich längs einer Weltlinie ¢ , ¢ Eigenzeit © ¨ © ¨ ¡ bewegt:Dabei ist die Vierergeschwindigkeit¡ ¦ ¢¢ ¢ ¨ ¢ Eingesetzt¢ ¢ ¢ ¢%¡ ¡ ¢ ¢ Was besagt die-Funktion? ¢ ¦ ¢ ¢ © ¨ © ¢ ¤ ¨ ¢ ¡ ¦ ¨ ¢ ¡ © ¨ © ¢ ¡ ¨ ¨ ¢ ¡ £161

d.h. der Raum-Zeit Punkt © " ¢% %% % % % %% % % % % %% % % % %% % % % %% % % % %% % % % %% % % % % %% % % % %% % % % %% % % %% % % % %%%% %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% % % % % % % % % % % % %% % % % % %% % % % % %% % % % % % % % % %% % % % % % % % % % % % % % % % % % %%%%¨ ¡ ¡ mit ¥ ¡ £¥ ¢ ¢ ¨ ¢ ¨ ¢ £ ¢ ¦ ¢ ¢ © ¨ © ¢ %% ¦ ¢ © © £ ¨ ¢ © ¢¢ & ¨ ¢ © ¨ ¢ © ¨ # ¤ ¡ © ¨ © ¨ ¤ ¢ £ ¨ #£ ¢% % % % % % % % % % % % % % % % % % % % % % % % % %¨ ¢ © ¢ ¨ ¢ £ ¢ ¢¥ ¢ ¤ ¡ " ¢ und die Position des geladenen © ¢ " ¢ Teilchenssind genau durch die Lichtgeschwindigkeit verknüpft. In anderen Worten der Lichtkegel durchschneidet die Weltlinie des Teilchens ¢ an . Da fest vorgegeben, ist ¢ ¢ © dadurch festgelegt ZeitWeltlinie ¢ ¢ © %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LichtkegelRaum%%%%%%%%%%%%%%%%%%%%%%%%%%%Genau von diesem ¢ © Punkt auf der Weltlinie kann Licht an ankommen. Vorausgesetzt ist ¢ © . ©entsprechend der -Funktion, dass ©Allgemein gilt ¥ ¡einfachSpeziell¢ ¨ ¢ ¡ ¨ ¢ ¨ ¢ £ ¢ Dies ist das Lienard-Wichert PotentialFühren einfachere Notation ein : ¨ ¢ © " ¡ & ¥ ¢ ¢ 162

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% ¨%¢ ¢ ¨ ¢ ¢ £¨ $ £ ¨ ¢ ¨ ¨ ¢ £¨ $ £ ¨ ¨ ¤ ¡¡£ ¨ ¤ £¤ ¨ $ £ ¦¡¤ ¨ # £ ¦¡ ¨ £ ¦¡ ¤£ ¨ $ £ ¦¡ ¤© ¨ ¡§ ¨ ¡ ¢¡§ ¨ $ £¦¡¡ ¥Damit lassen sich die Felder angeben ¢ ¨ £¦¡¡ ¥¦ ¨ £ ¨ $ £ ¢ ¨ $ £ §Somit¨ ¨ ¨ ¨ ¡¡ ¥¦ © ¦ ¦ ¨ $ £ '¡¢£ ¤¨ ¨ $ ¦ ¤ ¥¡ ¡ ¨ # £ ¢ ¨ $ £ §oder¨¡ ¦ ¨ ¨ ¦¡¡ ¥' ¦ ¨ £¡ & ¤ £¨ ¨ ¡¡ ¡ ¨ £¡ ¨ ¨ ¡¤ ¢ ¨ £ §entsprechend¡¡ ¥¥ ¦ ¨ # £¤ &¡ ¨ $ £ ¨ $¡ ¢ ¨ $ £ §Wir lesen ab:¨ ¡¡ ¥¡Schreiben dieses in übliche Notation um:¥ ¢ 'wobei 164

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