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

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 ||RR , 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 ),,( zyxr . [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 .  