# EEL 3470 Electromagnetic Fields

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16 Νοε 2013 (πριν από 4 χρόνια και 7 μήνες)

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Department

of Electrical
Engineering and Computer Science

EEL 3470 Electromagnetic Fields

Homework Set No. 5

Each group
should turn in ONLY one set of homework.

Names: __________________ ___________________ ___________________

Problem 1
.
A closed loop (
30cm 20cm
x y
   
) of wire passes through a nonuniform
time
-
independent magnetic field
y

z
B a
T with a constant velocity
=5
0 x
v a

m/s. At t=0, the loop

s lower left corner is located at the origin. Find an
expression for the voltage V, generated by the loop as a function of time.
You may neglect the magnetic field created by the current in the loop.

B
y
x
v
(
t
)

x

y

Problem
2
.
A conducting axle oscillates over two conducting parallel rails in a uniform
magnetic field
0
=
z
B
B a
(
B
0
=4 T). The
position

of the axle is given by
(/2)[1 cos ]
x x t

  

(
0.2m
x
 
,
1
500s

). Find and plot the current
I
(
t
)

if the resistance is R=10

and the distance between the rails is
0.1m
y
 

B
I
y
R
x

Problem
3
.

The time harmonic electric fields inside a source
-
free perfect magnetic

conducting (PMC) rectangular waveguide are given by

b
y
a
x
e
y
b
H
x
z
jk
z

0

,
0

,
)
sin(
0
a
H

Assume that the media inside the waveguide is free space, and four side walls at
a
x
,
0

and
b
y
,
0

are perfect magnetic conductor. For a section of the

waveguide of length
l
along z axis from
0

z

to
l
z

, find:

(1)

time domain

expression of electric fields;

(2)

z
k
;

x

y

Problem 4
.
An electromagnetic wave with a frequency f=1MHz propagates in a dielectric
material (
4,1
r r
 
 
) and has an electric
field

component
y
E (,) 1.3cos( ) V/m
z t t kz

 
. Find the velocity of the wave
,

the wave
vector
k
, the characteristic impedance of the
material

Z
c
, and the magnetic
field
H
(
z,t
).

Problem 5
.

The phasor

form expression for the electric field of a uniform plane wave

propagating in lossless isotropic medium is

E
(
x
) = 60

2

a
y
-
a
z
)
exp(j4
x
) V/m.

The magnetic field at point (0,0,0) when t = 0 is
2
A/m. The unit of t is

second and the unit of
x

is meter. The radian frequency is
8
10
3

(1)

the time domain expression of the electric field;

(2)

the wave impedance

of the media;

(3)

the relative dielectric constant

r
and relative permeability

r;

(4)

the time domain expression for magnetic field;

Problem 6
.
Find the attenuation constant

(Np/m), the phase constant

the phase velocity
v

if the conductivity

of the material is such that
 

(the material parameters are
1
r

and
2.5
r

). The
wavelength

in free space is
0
30

cm.

Problem 7
.

Find the skin depth

at a frequency 4MHz in aluminum, where
7
3.82 10 S/m

 

and
1
r r
 
 
, Also find the phase velocity
p
v

.