This document provides instructions and questions for a final examination in electromagnetic field theory. It consists of 5 questions testing concepts such as electric and magnetic fields, Maxwell's equations, boundary conditions, wave propagation, and vector calculus identities. The examination is for a course taught in the 2009/2010 semester and covers topics including electrostatics, magnetostatics, and time-varying fields. Students have 2 hours and 30 minutes to answer 4 out of the 5 questions.

1. I CONFIDENTIAL I
FINAL EXAMINATION SEMESTER I
SESSION 2009/2010
COURSE CODE SEE 2523 / SZE 2523
COURSE ELECTROMAGNETIC FIELD THEORY
LECTURERS ASSOC. PROF. DR. NORAZAN BIN MOHD
KASSIM
DR. YOU KOK YEOW
MDM. FATIMAH BT MOHAMAD
PROGRAMME SEC / SEE / SEI / SEL / SEM !SEP / SET /
SEW
SECTION 01 - 03
TIME 2 HOURS 30 MINUTES
DATE 9 NOVEMBER 2009
INSTRUCTION TO CANDIDATE
ANSWER FOUR (4) QUESTIONS ONLY.
THIS EXAMINATION BOOKLET CONSISTS OF 9 PAGES INCLUDING THE FRONT
COVER

2. SEE 2523
2
QI (a) Determine the electric field intensity on the axis of a circular ring of uniform
charge Pi (C 1m) with radius a in the .xy plane. Let the axis of the ring be
along the z axis. (7 marks)
(b) At what distance along the positive z axis is the electric field from Ql(a)
maximum, and what is the magnitude of this field? (8 marks)
(c) Find the absolute potential at any point along the z axis referring to Q 1(a).
(5 marks)
(d) Compare the result of electric field using the gradient concept.
(5 marks)

3. SEE 2523
3
Q2 (a) With the aid of suitable diagrams and Maxwell 's equation s, develop the
boundary condition equations for two dielectric materials with different
permittivities.
(5 marks)
(b) Two concentric spheres are shown in Fig. Q2(b). The spheres are built from
two different dielectric materials . Each sphere has been covered with a thin
layer of conductor, in wh ich the th ickness can be ignored .
Fig. Q2(b)
By assuming negative charge on the inner sphere surface:
(i) Determine the electric field intensities in material # 1 and material #2.
(4 marks)
(ii) Obtain the energy stored in the structure.
(5 marks)
(iii) Obtain the resistance between the spheres if the conductivity factor for
the material #2 is assumed to be (Yl !1S/m .
(4 marks)
(iv) Given the radius r a"" 10 urn. Determine the range of structural radiu s
(range of rb) such that the minimum and maximum charge perunit volt
that can be stored in material #2 are 3 F and 10 F, respectively.
(7 marks)

4. SEE 2523
4
Q3 (a) Compare the usefulness of Ampere's Circuital Law and Biot Savart Law in
determining the magnetic field intensity B of a current carrying circuit.
(5 marks)
(b) A coaxial cable as shown in Fig. Q3(a) consists of a solid cylindrical inner
conductor having a radius a and an outer cylinder in the form of a cylindrical
shell having a radius b. If the inner conductor carries a current of I A in the
form of a uniform current density J == IIrra 2 i (Alm 2 ) and the outer
conductor carries a return current of 1 A in the form of a uniform current
density Js = -Il2rrb i (Aim). Determine:
(i) the magnitude of the force per unit length acting 10 split the outer
cylinder apart longitudinally. (7 marks)
(ii) the inductance per unit length of the coaxial cable .
(8 marks)
(e) An infinite cylindrical wire with radius a carries a uniform current density
J == l ltta? i (A/m 2 ) , except inside an infinite cylindrical hole parallel to the
wire's axis. The hole has radius c and is tangent to the exterior of the wire.
A short chunk of the wire is shown in the accompanying Fig. Q3(b) .
Calculate the magnetic field everywhere inside the hole, and sketch the lines
of B on the figure . (S marks)
· ·~· · : I
a
-r----r
Fig. Q3(a)
y
x
2D view
3D view
Fig. Q3(b)

5. SEE 2523
5
Q4 (a) Within a certain region, e = 1O- 1 1F [m and f.-l = lO-sHlm. If
B; = 2 x 10- 4 3
cos lOSt sin 10- y (Testa):
(i) Use the appropriate Maxwell's equation to find E. (5 marks)
(ii) Find the total magnetic flux passing through the surface
x = 0,0< Y < 40m, 0 < z < 2m, at t = lfiS. (4 marks)
(iii) Find the value of the closed line integral of E around the perimeter of
the given surface at t = lfiS. (5 marks)
(b) A voltage source is connected by means of wire to a parallel-plate capacitor
made up of circular plates of radius a in the z =0 and z = d planes, and
having their centers on the z-axis. The electric field between the plates is
given by
_ n:T
E = £0 sin 20 cos wt z
Find the total current flowing through the capacitor, assuming the region
between the plates to be free space, and that no field exists outside the region.
(7 marks)
(c) By analyzing the expression of current in Q4(b) above, give two suggestions
to increase the magnitude ofthis current. (4 marks)
Hint: Judv = uv - Jvdu

6. SEE 2523
6
Q5 (a) Derive the vector wave equation for electric field using Maxwell equation.
(4 marks)
(b) The propagation constant y can be written as y = a + jf3, where a is the
attenuation constant of the medium and f3 is its phase constant. Proof that a
and f3 can be expressed as
a=w T ( 1+"7 -1 )
j1E' (E")2
j1E' ( [ (E")2 +1 )
f3=WT~1+7
where OJ is the angular frequency. E;' and E;" are the dielectric constant and
loss factor, respectively. J.1 is the permeability.
(8 marks)
(c) A 20 V 1m electric plane wave with frequency 500 MHz propagates in the z
direction and polarized in the x direction in a medium. The properties of medium
has relative permittivity, e r = 4.5 - jO .02 and relative permeability, J.1 r =1.
(i) Write a complete time-domain expression for the electric field, E
(6 marks)
(ii) Determine the corresponding expression for the magnetic field.
(7 marks)

7. SEE 2523
7
ELECTROSTATIC FIELD I MAGNETOSTATIC FIELD
Coulomb's Law - J dQ
E::= ---2
"
aR I Biot-Savart Law H =
- f~ XClR
4;d(2
47!&oR
Gauss's Law db.ds=Qm Ampere Circuital law dH· a1 = len
Force on a point charge F =EQ. Force on a moving charge F = Q(u x B)
Force on a current element F = I dI x 7f
Electric field for finite line charge Magnetic field for finite current
E =.--t1..-F(Sina 2 + sinal ) + ~(cosa2 -cosa)}
- I .
H =-(sina 2 + sina1)¢
47lr
47l"&o r r
Electric field for infinite line charge Magnetic field for infinite current
- I·
E=J:L,. H=-¢
27l"&or 27lr
Electric flux density D = cE Magnetic flux density B=pH
Electric flux If/ =Q=QD. ds
E Magnetic flux If/ = JB . ds
»»
m
Divergence theorem Q ds = j (v
D. Stoke's theorem c}H.dI = j (vx H) ds
s , I s
Potential difference VAB ::= -1 E.
B
dI
Absolute potential V
=
J47l"~R
dO
Gradient of potential ~ E = -V' V Magnetic potential, (A) -1 B = V' x A
Energy stored in an electric field Energy stored in a magnetic field
WE =!
2
J . E)1v
v
(15 ::= !
w; ~ (B. H)1v
Total current in a conductor
l=fJ·ds where] = 0-£
Polarization vector P =D-8aE Magnetization vector M == XmH
where Xm = u; -1
Bound surface charge density Magnetized surface current density
pso=P.n J,m=Mxn
Volume surface charge density Magnetized volume current density
Pv
o::=-Y'P ]m =YxM
Electrical boundary conditions Magnetic boundary conditions
=
.L1" - Dg, p, and £'1 E21 = B...
= Bz,. and ~I - H2t = J,
--
Resistance R =-
. I Inductance L ::=!:.- where A= 11/,,/'/
as I
Capacitance C = g
Vao
Poisson's equation V'2V ::= _fr
E:
Laplace equation v'v = 0
Maxwell equation y. D = Pv. V' x E =0 Maxwell equation V'. B = 0, 'YxH=J

8. SEE 2523
8
TIME VARYING FIELD
Maxwell equation V'. D== p, Gauss's Law for electric field
I
- a8
x E = -- Faraday's Law
if
1·8=0 Gauss's Law for magnetic field
- - aD
lxH=J+- Ampere Circuital Law
if
Characteristics of wave propagation in lossy medium (a*O,/-l=AA,E=fioE,.)
Electric field, E(z,t) == Eoe-a: cos(a>t - /lz)x
Magnetic field, H(z,t) =~e-«" cos(aJt- fJz+ 8n )y
Attenuation constan! a"Q)~~ ~I+(:J -I
Phase constant
j;i&
Intrinsic impedance '7=11+(CT I (j)£)" FLBn where tan 28 =.!!...
n OJf:
I~ = Eo
n,
III "
Skin depth o==l/a
Poynting theorem .r~ -) - a
I..!.E xH ·ds=--
s a ,. 2
1
-6£ +-pH
2
2} v- I aE dv
.'
2
Poynting vector tJ=ExH
Average power Pavg =-e -2«" cos en
E;
217

9. SEE 2523
9
Kecerunan "Gradient"
Vj = xOf + Yoj + Z oj
ox By oz
Vj = Poj + i
Of + Z oj
or r or/! 8z
Vj=p a +~ Of +- L Of
j
ar r ae rsine or/!
Kecapahan "Divergence"
- aA oA, aA
V -A = - ' +-" +-'
ax By oz
«: =.!.[acrAJ] +.!. OAf +aA,
r or r or/! OZ
V -:4 =~ [O(r'A,)]+_I_ [O(Aa Sine)]+_I_OA f
r' or r sin e' 8e r sine' or/!
Ikal "Curl"
-
= x 8;- 8A + Y a;-a; ) +Z a;--8; J
_(OA, j _(8A, OA, _lOA, oA,
y
Vx A az
rA;)
VxA =r[ -8A, - - +r/! - -OA,) +- [oC - -8A,]
- _ 1 - oA; j "( 8A, - Z- -
r or/! oz oz or " r Or or/!
V x A = _P_[OCAf sine) _8Aa] +~[_1_8A, _ 8(rA;)] +i( 8crAa) - oA, J
rsine ae ar/! r sine or/! or r or oe
Laplacian