• 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
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