082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

344 Chapter 9 Complex numbers
Electric field
Magnetic field
x
y
z
Figure9.16
An electromagnetic wave travelling in freespace.
where E and H are vector quantities. Note that S is also a vector quantity as this
expression isavector product.
It is possible to specify magnitude and phase of the field quantities using complex
numbers; that is, in terms of phasors. Consider the case when E and H are the
following:
E=(2+j0.5)i+0j+0k
H = 0i + (3.5 +j0.25)j +0k
HereCartesiancoordinateshavebeenusedtodefinethetwofieldquantitiesandiisa
unitvectorinthexdirection,jisaunitvectorintheydirectionandkisaunitvector
inthezdirection.Examiningthesetwotermswenotethattheelectricfieldisaligned
inthexdirection and the magnetic field isaligned intheydirection.
Calculating the Poynting vector we obtain
i j k
S=E×H=
2+j0.5 0 0
∣0 3.5 +j0.25 0 ∣
= 0i −0j + (2 +j0.5)(3.5 +j0.25)k = (6.875 +j2.25)k
Weseethattheonlynon-zerocomponentisalignedinthezdirection.Thisisconsistentwithourunderstandingofatransversewaveinthatthedirectionofpropagation
is at right angles to the two field components. This implies that the energy flow of
the electromagnetic wave isinthezdirection.
9.9 DEMOIVRE’STHEOREM
AveryimportantresultincomplexnumbertheoryisDeMoivre’stheoremwhichstates
that ifn ∈ N,
(cosθ +jsinθ) n = cosnθ +jsinnθ (9.8)
Example9.17 VerifyDe Moivre’s theorem whenn = 1 andn = 2.
Solution Whenn = 1,the theorem states:
(cosθ +jsinθ) 1 =cos1θ +jsin1θ