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www.GOALias.blogspot.comE l = j ρ lor, E = j ρ (3.11)The above relation for magnitudes E and j can indeed be cast in avector form. The current density, (which we have defined as the currentthrough unit area normal to the current) is also directed along E, and isalso a vector j (≡ j E/E). Thus, the last equation can be written as,E = jρ (3.12)or, j = σ E (3.13)where σ ≡1/ρ is called the conductivity. Ohm’s law is often stated in anequivalent form, Eq. (3.13) in addition to Eq.(3.3). In the next section, wewill try to understand the origin of the Ohm’s law as arising from thecharacteristics of the drift of electrons.3.5 DRIFT OF ELECTRONS AND THE ORIGIN OFRESISTIVITYAs remarked before, an electron will suffer collisions with the heavy fixedions, but after collision, it will emerge with the same speed but in randomdirections. If we consider all the electrons, their average velocity will bezero since their directions are random. Thus, if there are N electrons andthe velocity of the i th electron (i = 1, 2, 3, ... N) at a given time is v i, thenN1∑ vi= 0N(3.14)=i1Consider now the situation when an electric field ispresent. Electrons will be accelerated due to thisfield by–e Ea =(3.15)mwhere –e is the charge and m is the mass of an electron.Consider again the i th electron at a given time t. Thiselectron would have had its last collision some timebefore t, and let t ibe the time elapsed after its lastcollision. If v iwas its velocity immediately after the lastcollision, then its velocity V iat time t ise EVi = vi– ti(3.16)msince starting with its last collision it was accelerated(Fig. 3.3) with an acceleration given by Eq. (3.15) for atime interval t i. The average velocity of the electrons attime t is the average of all the V i’s. The average of v i’s iszero [Eq. (3.14)] since immediately after any collision,the direction of the velocity of an electron is completelyrandom. The collisions of the electrons do not occur atregular intervals but at random times. Let us denote byτ, the average time between successive collisions. Thenat a given time, some of the electrons would have spentCurrentElectricityFIGURE 3.3 A schematic picture ofan electron moving from a point A toanother point B through repeatedcollisions, and straight line travelbetween collisions (full lines). If anelectric field is applied as shown, theelectron ends up at point B′ (dottedlines). A slight drift in a directionopposite the electric field is visible.97