Student Name:____________________________________ Student Number________________________________
Page 1 of 4
ELEC ENG 3FK4
DAY CLASS Dr T R FIELD
DURATION: 2.5 HOURS
MCMASTER UNIVERSITY FINAL EXAMINATION DECEMBER 2019
THIS EXAMINATION PAPER INCLUDES 4 PAGES (includes the cover page) AND 6 QUESTIONS. YOU ARE
RESPONSIBLE FOR ENSURING THAT YOUR COPY OF THE PAPER IS COMPLETE. BRING ANY
DISCREPANCY TO THE ATTENTION OF YOUR INVIGILATOR.
Special
Instructions:
CLOSED BOOK
8.5”x11” crib sheet one side is allowed
Use of any calculator is allowed
Answer any FOUR questions
All questions carry equal weight
Explain all reasoning carefully
1. Consider a static volume current density )J(r' where 'r is the position vector of a point in the current
distribution. Show that the field generated at a point with position vector r , according to the Biot-Savart law,
vd
R
3
3
)'(
4
)(
RrJ
rB
,
in which 'rrR and ||RR , satisfies Maxwell’s magnetostatic equation JB ( should be
considered as constant).
Consider the magnetic vector potential defined by
vdR
3
4
J
A
.
Show that this potential satisfies the equation AB and the Lorenz gauge condition 0 A .
[Hint: see Useful formulae and apply the equation of continuity and anti-symmetry under interchange of ',rr ,
where appropriate.]
Student Name:____________________________________ Student Number________________________________
Page 2 of 4
2. Explain the meaning of dipole moment for a pair of equal and opposite charges q separated by a
displacement vector d , and the limiting case of a perfect dipole with moment p .
State the expression for the electrostatic potential for a single point charge q situated at 0r . By considering a
pair of equal and opposite charges in superposition, derive the following expression for the electrostatic
potential for a perfect dipole situated at the origin:
3
04
1
r
V
rp
.
where ),,( zyxr . [Hint: use the binomial approximation
2
12/1 1)1( for small values of , and the
cosine rule (see Useful formulae) applied to a triangle with vertices situated at the two charges and the field
point.]
Suppose that a dielectric material consists of dipoles with dipole moment P per unit volume (the polarization
density). Write down an expression for the contribution to the potential at a general point due to a volume
element '
3vd of the material in terms of the vector 'rrR where r , 'r are the position vectors of the field
point and material element respectively. Using the relation
3
1
'
RR
R
and the vector identity
fff '' ' AAA show that the potential due to the entire material occupying volume 'V with
boundary 'S is given by
'
'
'
'
4
1 3
'
2
'0
vd
R
Sd
R
V
V
pV
S
pS
where
pS' and pV' should be expressed in terms of the polarization density P , and whose physical
significance should be explained.
3. For a time dependent electromagnetic field show that, in the Lorenz gauge, its electric scalar and magnetic
vector potentials V and A satisfy the wave equations with source:
/
2
2
2
t
V
V , J
A
A
2
2
2
t
and identify the required Lorenz gauge. [You may assume Maxwell’s equations and the potential field relations
tV AE , AB .] Show that the retarded potentials
v vdR
V 3
4
][
, v vdR
3
4
][
J
A
are solutions to the above wave equations, where ][ denotes evaluation at the retarded time which should
be explained. [Hint: apply your argument to a general function f that satisfies the wave equation with
source g .]
Student Name:____________________________________ Student Number________________________________
Page 3 of 4
4. Given the total electromagnetic energy
1
2
W dv E D H B
show from Maxwell's equations that
( )
S V
W
d dv
t
E H S E J
Interpret this equation physically in terms of the Poynting vector and dissipation.
5. Starting from first principles, derive expressions for the Brewster angles pertaining to reflection of a plane
wave at oblique incidence, in the cases of parallel and perpendicular polarization. In this context, compare and
contrast the different polarizations for different types of electromagnetic media. [You may assume Snell's law
without proof.]
6. Explain the physical meaning of the reflection coefficient and standing wave ratio for electromagnetic
waves given, respectively, by the following expressions:
12
12
,
||1
||1
s .
Explain what is meant by the dominant mode for a rectangular waveguide. A rectangular waveguide with
cross sections shown in the figure below has dielectric discontinuity. Calculate the standing wave ratio
(using the above expressions) if the guide operates at 8 GHz in the dominant mode.
[Relevant physical constants: speed of light in vacuum
18 ms103 c ; intrinsic impedance of free space
Ohms 1200 .]
Student Name:____________________________________ Student Number________________________________
Page 4 of 4
Useful formulae
)B(A)A(BABBAB)(A
vv)(v 2
3
3
4 ( )
r
r
r
( ) ( ) ( ) A B B A A B
vv)(v 2
THE END
cosine rule:
Abccba cos2222
for triangle with side lengths cba ,, and
opposite angle A .