Download Final Exam SEE2523 20080901

SEE 2523
2
PART A
Q1. (a)
It is a known fact that the formula for the electric field intensity of a point
charge was given by: E  rˆE  rˆ
(i)
Q
4o r 2
V /m.
Why was it necessary to stipulate that Q is a boundless free charge.
(3 marks)
(ii)
Why did we not construct a cubic or a cylindrical surface around Q.
(3 marks)
(b) Two concentric spherical shells form a capacitor given by radii Ri and Ro. The
space between them is filled with a dielectric material of relative permittivity
 r from Ri to b (Ri < b < Ro) and another dielectric material of relative
permittivity 2 r from b to Ro. Assume that there exists a total charge Q on the
surface Ri. Determine:
(i)
the electric field intensity everywhere in terms of an applied voltage V.
(14 marks)
(ii)
the capacitance.
(5 marks)
SEE 2523
3
Q2 (a)
Starting from Gauss’s law, find the mathematical expressions that constitute
the divergence theorem. Then in your own words, state the meaning of the
divergence theorem.
(5 marks)
(b) A finite length of coaxial cable with length L = 50 cm is greater than the outer
radius b = 4 mm and the inner radius a = 1 mm. Such a device is also termed
a coaxial capacitor. The space between the conductors is assumed to be filled
with air. The total charge on the inner conductor is 30 nC. Determine:
(i)
Surface charge density on each conductor.
(4 marks)
(ii)
Electric field density, D and electric field intensity, E everywhere in
the regions.
(6 marks)
(iii)
Plot the graph of electric field intensity, E versus radius.
(4 marks)
(iv)
Find the total energy stored in the coaxial cable.
(6 marks)
SEE 2523
4
Q3
(a)
A finite length wire carrying current I is shown in Fig. Q3 below.
z
I
y
α2
α1
A
x
r
Fig. Q3
(i)
Show that the alternative expression of magnetic field intensity at A can
be of the form:
H
I
4r
sin  2  sin  1 ˆ
( A / m)
(6 marks)
(ii)
From the given expression in Q3a(ii) above, obtain the magnetic field
intensity at point A due to an infinitely extended wire current. Prove
your answer by applying the Ampere Circuital Law.
(5 marks)
(b) A filamentary conductor on the z-axis carries a current of 10A in the +z
direction, a conducting cylindrical shell at r = 6 cm carries a total current of
12A in the –z direction and is at potential Vo. A grounded uniform cylindrical
current sheet at r = 10 cm carries a current of 2A/m in the +z direction.
(i)
Find the magnetic field intensity, everywhere.
(6 marks)
(ii)
Electric field intensity for the region 6 cm < r < 10 cm.
(8 marks)
Given:


xdx
c
2
x

2 3/2

1
c
2
x

2 1/ 2
,

dx
c
2
x

2 3/2

x
c c  x 2 
2
2
1/ 2
SEE 2523
5
Q4
(a)
An infinite planar conductor carries a uniform surface current of J s (A/m).
Show that the magnetic field intensity on each side of the conductor can be
written as H 
Js
appropriate sign for
 aˆ n and indicate how to determine the 
2
the normal to the plane aˆ n .
(b)
(6 marks)

One planar conductor is defined by the equation 3x  2y  z  1 and a second

(parallel) planar conductor is defined by the equation 6x  4 y  2z  1. On the
first conductor, the magnitude of the surface current density is 2 A/m, the x

component of the surface current density is positive, the y component is

negative, and the surface current density has no z component. The surface
current density on the second conductor is equal in magnitude and opposite in
direction when compared to the surface current density on the first conductor.
(i)
Calculate the magnitude and direction of the magnetic field intensity at
a point halfway between the two conductors.
(8 marks)
(ii)
Calculate the distance between the two planes.
(6 marks)
(c)
Explain the differences in the magnetic field intensity if the surface current
density on the second conductor has the same direction and magnitude as on
the first conductor.
(5 marks)
SEE 2523
6
PART B
Q5
(a)
By using divergence and Stoke’s theorem, derive the integral form for all the
Maxwell equations in time varying fields.
(6 marks)
(b)
Consider the region defined by x, y and z  1. Let
r 5,r 4,0.Given
Jd 20cos1.510
 8t bxyˆ(A/m2);
(i)
(ii)

find D and E ; 
(4 marks)


use the point form of Faraday’s Law and integration with respect to
time to find B and H ;


(iii) use   H  J  Jd to find J d ;
(5 marks)
(iv) what is thenumerical value of b?

(2 marks)
(3 marks)

(c)

With the help of Lenz’s Law, explain the significance of the negative sign in
Faraday’s Law.
(5 marks)
SEE 2523
7
Q6
(a)
State 2 similarities and 2 differences between the propagation of plane waves
in free space and conductive medium.
(6 marks)
(b)
A 1 MHz
plane wave is propagating in a conductive medium of
r 8,4.8102S/m,0.
(i)
Find the ratio between the magnitudes of the conduction to the
displacement currents.

(ii)
(2 marks)
From the result obtained in Q6b(i), derive the expression for attenuation
constant and its value.
(4 marks)
(iii) If the maximum magnitude of the sinusoidal variation of the x-directed
electric field is 100V/m at t 0, z0.3, determine the expression for
the electric field in real-time.
(6 marks)
(iv) Determine the corresponding magnetic field.
(4 marks)

(c)
The velocity of propagation of a plane wave is given by v p   /  . Discuss
this velocity in terms of velocity of light, c, for a conductive medium as
compared to free space.

(3 marks)
SEE 2523
8
ELECTROSTATIC FIELD
Coulomb’s Law E 
Gauss’s Law
 4 R
dQ
 D  ds  Q
2
MAGNETOSTATIC FIELD
Idl  aˆ R
 4R
Ampere Circuital law  H  dl  Ien
aˆ R
Biot-Savart Law H 
0
en
2
Force on a moving charge F  Qu  B 

 a current element F  Idl  B
Force on
Electric field for finite line charge
Magnetic field for finite current
  rˆ
I 

zˆ
ˆ
 sin  
H
sin  2
E  l  sin  2  sin 1   cos  2  cos1 
1
4r

40 r 
r
Electric field for infinite line charge
Magnetic field for infinite current
I ˆ
l
H

E
rˆ

2r
20 r
Force on a point charge

F  EQ
Magnetic flux density B  H
Electric flux density D  E
Electric flux E  Q 

Divergence
theorem
 D  ds
 D  ds     D dv
s

Magnetic flux m 
Stoke’s theorem

v
A
Potential
difference VAB    E  dl

 H  dl     H  ds
l
s

B

V
Absolute potential
 B  ds
 4 R
dQ

0
Gradient ofpotential  E  V
Magnetic potential, (A)  B    A
Energy stored in a magnetic field
Energy stored in an electric field
1
W E   D
 E dv
2 v
Total current in a conductor I 

Polarization vector P  D  0 E
Wm 

 J  ds

Bound surface charge
 density
sb  P  nˆ

Volume surface charge density
vb    P



Electrical boundary conditions

D1n  D2n  s and E1t  E2t
Q
l
Resistance R 
Capacitance C 

Vab
s

Poisson’s equation  2V   v



2

V

0
Laplace
equation




1
2
 B H dv
v
Magnetization vector M   m H
where  m  r 1
Magnetized surface current density
Jsm  M  nˆ

Magnetized volume current density
Jm    
M
Magnetic boundary conditions
B1n  B2n and H1t  H2t  Js

Inductance L 
where   m N
I



SEE 2523
9
Maxwell equation   D  v ,   E  0
TIME VARYING FIELD
Maxwell equation  D  v

E 
 B  0



Gauss’s Law for electric field
B
t
Faraday’s Law
Gauss’s Law for magnetic field
D
H J 
t


Maxwell equation   B  0,   H  J
Ampere Circuital Law
Characteristics of wave propagation in lossy medium
(  0,  0r,  0r )

E (z,t)  E 0ez cos(t  z) xˆ
E
Magnetic field, H (z,t)  0 ez cos(t  z   ) yˆ


Electric field,
2



     
1   1
Attenuation constant   
2 
   


Phase constant

2


    
 
1   1
2 
   

Intrinsic impedance  

Skin depth 
Poynting theorem
 /
E0
H0
 1/



1

 E  H  ds   t  2 E
 E  H

Average power
Pavg 

where tan 2 



Poynting vector


1  / 
1/ 4
s
v
E 02 2z
e cos
2
2

1
 H 2dv   E 2 dv

2
v
SEE 2523
10
Kecerunan “Gradient”
f
f
f
 yˆ
 zˆ
x
y
z
f ˆ f
f
f  rˆ 
 zˆ
r r 
z
f ˆ f
ˆ f
f  rˆ 

r r  r sin  
f  xˆ
Kecapahan “Divergence”
Ax Ay Az


x
y
z
1   (rAr )  1 A Az
 A  


r  r  r 
z
2
1   (r Ar ) 
1   ( A sin  ) 
1 A
 A  2 


 r sin  
r  r  r sin  


 A 
Ikal “Curl”
 A Ay 
 A A   A A 
  yˆ  x  z   zˆ y  x 
  A  xˆ z 
z 
x   x
y 
 z
 y
 1 Az A  ˆ Ar Az  zˆ   (rA ) Ar
   
  A  rˆ



  
r



z

z

r


 r  r





rˆ   ( A sin  ) A  ˆ  1 Ar  (rA )  ˆ   (rA ) Ar 

 
 
 A 




r sin  

  r  sin  
r  r  r
 
Laplacian
2 f 
2 f 2 f 2 f


x 2 y 2 z 2
2 f 
1   f  1  2 f  2 f

r  
r r  r  r 2  2 z 2
2 f 
1   2 f 
1
 
f 
1
r
 2
 sin 
 2
2
r r  r  r sin   
  r sin 2 
 2 f
 2
 


