chapter 2 wave in an unbounded medium maxwell’s equationsrd436460/ew/chapter2-1.pdftime-harmonic...
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Time-harmonic form Phasor form (Real quantity) (co
ˆ ˆ (1.11 )
ˆ (1.11 )
ˆ ˆ ˆ (
mplex quantity)
1.11 )
BE j at
D b
DH J j ct
B
ω
ρ ρ
ω
∂∇× = − ∇× = −
∂∇ ⋅ = ∇ ⋅ =
∂∇× = + ∇× = +
∂∇ ⋅
E B
D
H J D
ˆ0 =0 (1.11d)= ∇⋅B
{ }j tˆwith E(x,y,z,t)=Re (x,y,z)e ωE
Chapter 2 Wave in an Unbounded Medium
Maxwell’s Equations
Chapter 2.1 Plane Waves in a Simple, Source-Free, andLossless Medium
Maxwell’s Equations
(2.1 )
(2.1 )
(2.1 )
0 (2.1 )
BE at
D b
DH J ct
B d
ρ
∂∇× = −
∂∇ ⋅ =
∂∇× = +
∂∇ ⋅ =
2.1 Plane Wave in a Simple, Source-free, and Lossless Medium
22
2
22
2
The general equations for electric and magnetic fields:
0 (2.2)
0 (2.3)
EEtHHt
με
με
∂∇ − =
∂∂
∇ − =∂
1 p 2 p
the general solution of (2.5) is in the form of:=p (z-v t) + p (z+v t) (2.6)xE
x2 2
2 2
If we restric the electric field as the function z, and the electric fieldto one component E only, equation (2.2) becomes:
0 (2.5)x xE Ez t
με∂ ∂− =
∂ ∂
2.1.1 The Relation between E and H
1 p
Assume we have a electric field of uniform plane wave: =p (z-v t)
The corresponding H(z) is not independent, and must be resolvedfrom Maxwell e
xE
1 p 1 p
quations (2.1a) and /or (2.1c):1 H =( )p (z-v t)=
intrinsic
( )p (z-v
impedan
t)
where / is called " " of the medium (in unit ohm
ces).
yεμ η
η μ ε=
{ }1 1
1 2
The electric field of a uniform plane wave in a lossless medium is
( , ) Re ( ) ) (2.11)
( , ) cos( ) cos( ) (2.12)
j t j t j tx
x
E z t C e C e e
orE z t C t z C t z
β β ω
ω β ω β
− += +
= − + +
2.2 Time-Harmonic Uniform Plane Waves in a Lossless Medium
x 1
1
The electric and magnetic fields of a time-harmonic wave propagating in the + Z direction are
E ( , ) cos( )1( , ) cos( ) (2.13)
where t intrinsic impedanhe " " (in unit ohms)
/ = /
ce
y
z t C wt t
H z t C wt t
β
βη
η μ ε ωμ
= −
= −
=
(note: = ) .
β
β ω με
6
The instantaneous expression for the electric field component of an AM broadcast signal propagating is given by
Example 2-1: A
E(x,t
M broadc
)=z10c
ast Singn
os(1.
al
in air
5 10 )t x Vπ β× + 1
(a) Determine the direction of propagation and frequency f.(b) Determine the phase constant and wavelength.
(c) Find the instantaneous expression for the corresponding H(x,t).
m
β
−−
-3 -j0.68 y 1
An FM broadcast signal traveling in y direction has a magneticfield given by the phasor:
ˆ H(y)= 2.92
Example 2-2: FM broadcast Si
10 e (-x+z
ngnal
j)(a) Determin
i i
e
r
n a
A mπ −× −the frequency .
ˆ(b) Find the corresponding E(y).
(c) Find the instantaneous expression for the E(x,t) and H(x,t).
2.3 Plane Waves in Lossy Media
{ }1 2
1 2
The electric field of a uniform plane wave in a lossy medium is
( , ) Re ( ) )
( , ) cos( ) cos( ) (2.18)
z j t z j t j tx
z zx
E z t C e e C e e e
orE z t C e t z C e t z
α β α β ω
α αω β ω β
− − +
−
= +
= − + +
2 1/ 2 1
2 1/ 2 1
The propagation constant = +i :
[ 1 ( ) 1]2
[ 1 ( ) 1] (2 19)2
np m
rad m
γ α β
με σα ωωε
με σβ ωωε
−
−
= + − −
= + + − −
1
1
The waves propagate in the +Z directions are
( , ) cos( ) (2.21)1( , ) cos( ) (2.23)
zx
zx n
c
E z t C e t z
H z t C e t z
α
α
ω β
ω β φη
+ −
+ −
= −
= − +
1
c
c c
(1/ 2) tan [ /( )]
2 1/ 4
where intrinsic impedance is:
(2 22)[1 ( ) ]
nj
eff
j
ej
e
φ
σ ωε
η
μ μη η σε εω
μεσωε
−
= = =−
= −+
c-1
r
Find the complex propagation constant and theintrisic impedance η of a microwave signal in
muscle tissue at 915 MHz ( =1.6
Example
s-m
2-
1
5
5 )
γ
σ ε =
cr
Consier distilled water at 25 GHz ( 34 9.01).Calculate the attenuation constant , phase constant
, penetration depth d, and t
Exa
he wavelength
mple 2-6
.
jεα
β λ
= −
2.4 Electromagnetic Energy Flow and the Poynting Vector
22
Poynting Theorem:
1 1( ) ( )2 2
(2.31)
Poynting Vecor:
V V SE Jdv H E dv E H ds
t
S E H
μ ε∂⋅ = − + − × ⋅
∂
= ×
∫ ∫ ∫
Consider a long cylindrical conductor of conductivity and radius a, carrying a direct current as shown
in the
Example 2-12: Wire carryi
figure. Find the power di
ng direct cu
ssipated in
rrent
a p
.
ortiIσ
onof the wire of length , using (a) the left-hand side of (2.31); (b) he right-hand side of (2.31)
l
Consider a coaxial line delivering power to a resistor asshown in the figure. Assume the wire to be perfect conductor
Example 2-13 Power flow in a
,so that there is no power di
coaxial
ssipati
line
on in the wires, and the electric field inside them is zero. The configurations of the electric and magnetic field lines in the coaxial line are shown.Find the power delivered to the resistor R.
-2
A survey conducted in the United State indicated that ~50% of the population is exposed to average power densities of 0.0
Example 2-18 VHF/UHF broadcast
05W-(cm) due to VHF and UH
ra
F
diation
broadasμ t radiaction. Find the correcponding amplitudes of the electric and magnetic fields.
-0.218z 0.2
Consider VLF wave progation in the ocean. Find thetime-average Poynting flux at the sea surface (z=0) and at the depth of z=100m. Assume:
Example 2-19 VLF wa
E(z)=x2.
ves in the oc a
84e
e n.
je− 18 -1
-0.218z 0.218 / 4 -1
k V-m
H(z)=x2.84e k A-m
z
j z je e π− −