elements of electromagnetic field theory and guided waves
DESCRIPTION
Joule’s law Elemental volume dV=Sdl J 2 1 l Differential form Area S Elemental volume dV=Sdl J Differential form 2 1 l Integral formTRANSCRIPT
![Page 1: Elements of electromagnetic field theory and guided waves](https://reader036.vdocuments.site/reader036/viewer/2022062302/5a4d1aed7f8b9ab05997c485/html5/thumbnails/1.jpg)
• 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
![Page 2: Elements of electromagnetic field theory and guided waves](https://reader036.vdocuments.site/reader036/viewer/2022062302/5a4d1aed7f8b9ab05997c485/html5/thumbnails/2.jpg)
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
![Page 3: Elements of electromagnetic field theory and guided waves](https://reader036.vdocuments.site/reader036/viewer/2022062302/5a4d1aed7f8b9ab05997c485/html5/thumbnails/3.jpg)
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:
![Page 4: Elements of electromagnetic field theory and guided waves](https://reader036.vdocuments.site/reader036/viewer/2022062302/5a4d1aed7f8b9ab05997c485/html5/thumbnails/4.jpg)
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)
![Page 5: Elements of electromagnetic field theory and guided waves](https://reader036.vdocuments.site/reader036/viewer/2022062302/5a4d1aed7f8b9ab05997c485/html5/thumbnails/5.jpg)
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!
![Page 6: Elements of electromagnetic field theory and guided waves](https://reader036.vdocuments.site/reader036/viewer/2022062302/5a4d1aed7f8b9ab05997c485/html5/thumbnails/6.jpg)
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
![Page 7: Elements of electromagnetic field theory and guided waves](https://reader036.vdocuments.site/reader036/viewer/2022062302/5a4d1aed7f8b9ab05997c485/html5/thumbnails/7.jpg)
Maxwell equations for field phasors
![Page 8: Elements of electromagnetic field theory and guided waves](https://reader036.vdocuments.site/reader036/viewer/2022062302/5a4d1aed7f8b9ab05997c485/html5/thumbnails/8.jpg)
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
![Page 9: Elements of electromagnetic field theory and guided waves](https://reader036.vdocuments.site/reader036/viewer/2022062302/5a4d1aed7f8b9ab05997c485/html5/thumbnails/9.jpg)
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