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ELECTROMAGNETICS AND APPLICATIONS
Lecture 2
Electromagnetic Waves
in Homogenous Media
Luca Daniel
L2-2
Course Overview and Motivations
Maxwell Equations (review from 8.02)
in integral form
in differential form
EM waves in homogenous lossless media
EM Wave Equation
Solution of the EM Wave equation
Uniform Plane Waves (UPW)
Complex Notation (phasors)
Wave polarizations
EM Waves in homogeneous lossy media
Todays Outline
TodayToday
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L2-3
Second derivative in space second derivative in time,therefore
solution is any function with identical dependencies
on space and time (up to a constant)
Maxwells Equationsin l inear isotropic homogeneous lossless
media
0
0Faradays Law: = BE t
Amperes Law:
= D = D E
GausssLaw
= +D
H Jt
=B 0 = B H
=
22
2EM Wave Equation : E E 0
t
ConstitutiveRelations
( )E(z, t) E t a z= + z
E(z, t ) E t
=
or
L2-4
What are Electromagnetic Waves
A wave is a fixed disturbance propagating through a medium
Medium A B A energy B energy
String stretch velocity potential kineticAcoustic pressure
velocity potential kineticOcean height velocity potential
kineticElectromagnetic E H electric magnetic
0
A,B energy density
z
z
null
A
B
0
A,B
wave motion
zA(z, t ) A t
=
zB(z, t ) B t
=
zvelocity c
t
= =
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L2-5
Solutions of the Wave Equation
Possible solutions are many Try Uniform Plane Wave (UPW),e.g.
assume:
0
2 2 22
2 2 2E E
x y z
= + +
0
propagationt = t
t = 0
z
E+(t z/)
z=vt0
( , )yy E z t=
=
2 2y y
2 2
E E
z t
= y
zy E t
1 =
In air/vacuum waves moves at velocity
81 3 10 [ / ]
= = = = o o
zc m s
t
( , , , ) ( , )=E x y z t E z t1)
2)
3)
2
2Ez=
=
22
2E E 0
t
=
2 2y y
2 2 2
E E1
z t
L2-6
Sinusoidal Uniform Plane Wave (UPW) in +z direction
Ey(z,t) = E+(t - z/v) [V/m]
One special solution:
= =
k
To find the magnetic field:
Faradays Law: = HEt
General solution:
0 0= = E cos[ (t z / )] E cos( t kz)
where
0
0
0
0
x y z
x y z
E det x y z
E E E
= 0xkE sin( t kz)=
0
=
kH x E cos( t kz)
H E
E H in +z direction
= E / HNote:
= = = o
oo
377
( )0x E cos t kz z
=
0=
Ex cos( t kz)
In air/vacuum
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L2-7
Uniform Plane Wave: EM fields
EM Wave in z direction:
( ) ( ) ( ) ( ) ( )0 0 oE z,t y E cos t kz , H z,t x E cos t kz
= =
y
z
x
( )E z,0
( )H z,0
= = =
2 2k f
Wavelength
L2-8
Complex Notation (Phasors)
{ }= tE(r,t) Re E(r ) eComplex notation for a single frequency
(f = /2)
oE(z,t) yE cos( t kz )= + =
Phasor : contains all amplitude, vector,
spatial and phase information
UPW case
{ }= tRe E(z) e
oE(y,t) zE sin( t ky)=
Phasor E Time domain EExample:
{ } = jky j to( Re zjE e e )
jkyoE(r ) zjE e =
{ }jkz j toRe yE e e e
= jkzoE(z) yE e e
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L2-9
Uniform Plane Wave (UPW) in Complex Notation
= oE xE cos( t kz)
= =
8
o o
1c 3x10 m/ s
Example : x-polarized UPW traveling in +z direction
==
jkz
o
jkzo
E(z) xE eE
H(z) y e
E H : z direction of propagation
y
z
( )H z,0
( )E z,0
x
=
2
kwavelength
direction of propagation
=c
f
= = = =
o
o0
2(rads / s) 2 f k(rads /m)
c
=
oEH y cos( t kz)
L2-10
Uniform Plane Wave (UPW)
Linear vs. Circular vs. Elliptical Polarization
y
z
( )E z,0
x
Image source: http://en.wikipedia.org
Linear Polarization
Linear Polarization Circular Polarization
