King Fahd University of Petroleum & Minerals
Electrical Engineering Department
Spring 2006 (062)
EE 340 – Final Exam
Tuesday, June 5, 2007
7:00-10:00 PM
Name
ID

Solve all problems.

Total exam time is 3 Hours.
Problem
1 (20 points)
2 (20 points)
3 (20 points)
4-1 (10 points)
4-2 (10 points)
5-1 (10 points)
5-2 (10 points)
Total (100 points)
Grade
Useful Equations: Coulomb’s law: E 
Q
4o r 2
b
D  εE
,
Vab    E  dl
a
E
E Field of infinite line and infinite plane:
D1n  D2 n   s
Boundary conditions:
dH  I
Biot-Savart law:
BN 1  BN 2
I
2d
S
Phase velocity:
γ
 
vp 
ω
β
,
jωμ(σ  jωε)
 [ {1  (
λ
 2
) }  1]

2
For good conductors: α  β 
For good Dielectrics:
,
 H  dl  I
(ai xan )
B
 ds
t
enclosed
1 
Kxan
2
H
M 
,
wm 
1
H2
2
jωμ
σ  jωε
 
πμσf
1/ 4
Z   /  [1  (  ) 2 ]
,
 [ {1  (
 2
) }  1]

2
μπf
Z
(1  j ) ,
2σ
,
,
δ
tan( 2 z )   
1
α
 1  σ 2 
μ 
σ 
, β  ω με 1  
1  j

  , Z
ε 
2ωε 
 8  εω  
σ μ
α
2 ε
Reflection and Refraction: For normal incidence take all angles to be ZEROS.
η 2 cos θ i  η1 cos θ t
η 2 cos θ i  η1 cos θ t
η cos θ t  η1 cos θ i
Γ//  2
η 2 cos θ t  η1 cos θ i
Γ 
Φ
I
2π
β
Z
,
s
an
2
aˆ12  ( H 2  H1 )  K
,
emf   (v  B )  dl  
Faraday’s law :
v
E
, Ampere’s Law
H Field of infinite line and infinite plane: H 
Boundary conditions:
l
an
2d
2 2 cos  i
, 1     
 2 cos  i  1 cos  t
2 2 cos i
 cos t 

,  // 
, 1  //   // 
 2 cos i  1 cos t
cos

i 

,
enclosed
We   we dv
,
E1t  E2t
dl  aˆ r
4πr 2
 D  ds   q
Gauss’s law:
1
we  ε E 2
2
Q
,
V
C
,
ar
 
θi  θ r
,
sin θ t 
β1
sin θ i
β2
Problem 1
Consider two very thin concentric conducting cylindrical shells of radius a and b and length L
(L b). The inner shell carries a total fixed charge of +Q, and the outer shell carries a total
fixed charge –Q, as shown in the figure. The material filling the space between the two shells
has r = 1, r = 3o. The rest of the space is air.
a)
b)
c)
d)
Find expressions for the electric field intensity in all regions of the structure. Ignore edge
effects.
Find the electric potential at any point outside the structure ( >b) (assume that V=0 at
infinity).
Find the electric potential at =0.
Find the capacitance of the structure.

Problem 2
Consider two infinite thin wires both
parallel to the y-axis as shown in the
figure. Wire 1 carries a constant current
of 10 A flowing in the positive ydirection whereas wire 2 carries a
constant current of 3 A flowing in the
negative y-direction. The spacing
between the two wires is 5 meters
a)
b)
Write an expression for the total
magnetic field intensity due to both
wires at any point along the x-axis.
Find the position(s) along the x-axis
at which the magnetic field is zero.
z
5m
I1
I2
x
y
Problem 3
A conducting circular ring of radius a = 8 cm is centered at the origin in the x-y plane. The total
resistance of this ring is 10 Ohms.
Find the current flowing in the ring if the magnetic flux density is given by:
a)
 1

B  cos(120t )a z mW/m2
b)
 1

B  cos(120t  0.5 )a z mW/m2
c)
Calculate the power dissipated in the ring for case (a) above.


Problem 4-1
The electric field of an EM wave is given by:


E  ax 10cos(108t  y  )  az 10cos(108t  y  )
4
4
a)
b)
c)
d)
What is the direction of wave propagation?
Is the medium lossless or lossy?
The relative permeability of the medium r=1, calculate its relative permittivity r.
What is the wave polarization?
Problem 4-2
An electromagnetic wave has the following electric field:
E  ay E o cos( x )cos(t   z )
where E o ,  ,  , and  are constants.
Use Maxwell's equations to derive an expression for the magnetic field vector H ,
associated with this electromagnetic wave. Assume the permittivity and permeability of
the medium are  and  , respectively.
[Warning: For this type of EM waves HE/].
Problem 5-1
The figure shows a uniform plane wave
normally incident from a lossless dielectric
[region (i)] of (  o ,   4 o ) onto another
lossless dielectric [region (2)] of
(  o ,   9 o ). The electric field vector is
given by


Ei  30 e  j 7.5 z ax
x
 Region (i)
E i
 
Hi
Incident
V/m
Find,
a) The phasor expressions of the reflected
electric and magnetic fields.
b) The phasor expressions of the
transmitted electric and magnetic fields.

ak

ak

Er
 
Reflcted H r
Region (ii)

Et
 
Ht

ak
Transmitted
z
Problem 5-2
Consider a uniform plane wave obliquely
incident from free space on a perfectly
conducting medium as shown in the figure. The
electric field vector is expressed as


Ei  25 e  j (3 y4 z ) ax
d)
Find the angular frequency  , and the
incident angel  i .
Find the reflection coefficient.
Write the expression of the reflected
electric field in phasor form.
Find the simplified expression of
e)
use the identity e  jx  e  jx  2 j sin x ]
Find the nearest distance to the interface
a)
b)
c)

 
Etot  Ei  Er . [Hint: for simplification

along z for which Etot  0 .

Ei 

Hi

Hr
z
i

Er

r
y
Perfect Conductor