• Joule’s law in differential and integral form• Time-varying current: Inductance • Time-varying currents: capacitance • Method of phasors (complex amplitudes)• Complex impedances• Maxwell’s equations for field phasors• Time-dependent wave equation• Helmholtz’ wave equation• Plane wave

Elements of electromagnetic field theory and guided waves

Joule’s law

JE

lF

nquEnpdvdPqEu

tw

p

qEudtqEdlFdlddw

qq

q

q

,

,

J

Area S

Elemental volume dV=Sdl

RIIVJdSdlEdvEJPSlV

l2

12 1

2

l

Differential form

Integral form

Time-varying current. Capacitance

)()()()()( 12 tIVEMFtQtItEMF exttext

CQVIRVEMFext 1212 ,

12 definition of C,Q dQV IC dt

R

))/1(

sin(),cos()/1(

,)/1(

/),)/1(

cos(),cos()1(

)sin(1)cos(),cos()(

)!(.cos,1

2222

22022

220

000

RC

RatRC

AdtdQI

RCAQ

RCRatR

CI

tC

QtRQEMFtQtQ

tIFindtAEMFLetEMFQCdt

dQR

ext

extext

EMFext

1 2

Process of recharging:

Time-varying current. Inductance

)()()()()( .. tIeetBtIte indselfextItransext

teIRee isisext

,

totS

tot LId SBDefinition of L

22022

220

000

)(,)

)(cos(

),cos()(

)sin()cos(),cos()(

)!(.cos,

RLAI

RLRa

wheretRLI

tLItRIetItI

tIFindtAeLeteRIdtdI

L

ext

extext

Process of self-inductance

R

.BI

eext

I(t)

Method of phasors

Series LRC-circuit with external EMF e:

Finally, one can find both amplitude and phase of the real current i(t)

Let it be cos t

So simple!

Complex impedances, Ohm’s and Kirchhoff laws for current and voltage phasors

ZL=jL

ZC=1/jC

ZR=R or

Impedance Notation

General notation

V=IZEs Z

r

)(

,

ZrIE

ZIΕ

s

mmm

nn

Maxwell equations for field phasors

Wave equation for source-free regions (=J=0)

phase velocity

For phasors it is the so-calledHelmholtz’ equation

Time-harmonic case: k=/u - wave number

axbxc=b(ac)-c(ab)

(Recall 3d Maxwell’s equation)

Wave equation

Apply

Plane wave in free space

z

y

z=ray

E=Ex

t t

Wave number in free space

Forward wave Backward wave

z=ct - wave front