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− −