UNIVERSITY OF LIMPOPO
TURFLOOP CAMPUS
FACULTY OF SCIENCE AND AGRICULTURE
SCHOOL OF PHYSICAL AND MINERAL SCIENCES
2026 Problem Sets
SPHA082 – ELECTRODYNAMICS
Based on Textbook: Ǥ Introduction to Electrodynamics (3rd Ed), D. J. Griffiths (Prentice Hall)
Vector Analysis [Ǥ Chapter 1]
1. Ǥ Problem 1.11
2. Ǥ Problem 1.13
3. Ǥ Problem 1.15
4. Ǥ Problem 1.25
5. Ǥ Problem 1.26
6. Ǥ Problem 1.27
7. Ǥ Problem 1.28
8. Ǥ Problem 1.30
9. Show that for any vector field 𝐹⃗ (𝑟⃗), ∇ × ∇ × 𝐹⃗ = ∇(∇ ∙ 𝐹⃗ ) − ∇2 𝐹⃗ . [Alternatively: Verify the identity for the curl of the
curl of a vector function. Do this by writing it all out in component form.]
10. Show that the divergence of the curl of a vector function is zero for “normal” functions. Define “normal”.
Maxwell's Equations [Ǥ Chapter 7]
1. Consider Maxwell’s equations
(a) Write down these equations before Maxwell’s modified them
(b) Using vector analysis, describe the inconsistency in the formulas written in (a)
(c) With the aid of a suitable example, show that Ampere’s law is bound to fail for non-steady currents
(d) Explain how Maxwell resolved the inconsistency mentioned in (b)
(e) Write down Maxwell’s equations in free space (regions of space in which there are no charges or currents) and
demonstrate the symmetry about Maxwell’s equations that leads to the magnetic charge
(f) Using vector analysis, decouple the equations in (e) so as to have separate equations for E and B fields.
(g) State Maxwell’s equations in terms of free charges and currents
(h) Derive the units of 𝐽𝑚 and 𝑞𝑚 from Maxwell's Equations in the presence of magnetic charges
(i) If magnetic charges existed but electric charges did not, determine the source equations for static electric and
magnetic fields.
(h) Write down Maxwell’s equations inside matter, but in regions in which there are no free charges or currents.
(i) What do the equations in (h) reduce to if the medium is linear and homogenous?
2. Using Maxwell’s equations, explain the statement: “Just as a changing magnetic field induces an electric field, so
also, a changing electric field induces a magnetic field”
3. Maxwell’s equations is a set of four equations. Write down as many sets as you know, for various situations (in free
space, for free charges, in linear media, etc.). In each case, describe the situation.
4. Assume that magnetic charges existed. (a) Write down what Maxwell’s equations would be. (Be sure to explain the
symbols introduced) (b) What would the equations in (a) reduce to if electric charges did not exist and the fields were
static? (c) Given that the units of magnetic field, Tesla, can be written as ℎ𝑒𝑛𝑟𝑦 ∙ 𝐴 ∙ 𝑚2 , derive the units of the
magnetic charge
5. Ǥ Problem 7.47.
6. Ǥ Problem 7.55
∂2 𝐸⃗⃗
7. With respect to Maxwell’s equations in free space, the 2nd order “decoupled” equation for 𝐸⃗⃗ is given by ∇2 𝐸⃗⃗ = 𝜇𝑜 𝜖𝑜 2 .
∂𝑡
(a) Recover (derive) Faraday’s law from it (that is, “un-decouple” it). (b) Discuss the advantages and disadvantages of
decoupling Maxwell’s equations
2026
SPHA082 OCT/NOV EXAMINATION
Charge and Energy [Ǥ Chapter 8]
1. State Maxwell’s Equations
2. State the Poynting’s Theorem
3. A long coaxial cable carries current I (the current flows down the surface of the inner cylinder, radius a, and back along
the outer cylinder, radius b) as shown in the figure. Calculate the power (energy per unit time) transported down the
cable, assuming the two conductors are held at potential difference V.
⃗⃗ = 𝜇𝑜 𝐼 1 𝜙̂, and the electric field inside the
Note: The magnetic field outside a long straight wire carrying a steady current I is 𝐵
wire is that due to an infinite straight wire: 𝐸⃗⃗ =
𝜆
1
2𝜋𝜖𝑜 𝑠
2𝜋 𝑠
𝑠̂
4. Ǥ Problem 8.15
5. The integral form of Poynting’s Theorem may be written as
d𝑊
d𝑡
=−
𝜕
1
1
1
⃗⃗) ∙ d𝑎⃗. (a) Explain the terms on both sides of the expression. (b) Derive its
∫ (𝜖𝑜 𝐸 2 + 𝜇 𝐵2 ) 𝑑𝜏 − 𝜇 ∮𝑆 (𝐸⃗⃗ × 𝐵
𝜕𝑡 𝑉 2
𝑜
𝑜
differential version (c) Show how it leads to the definition of the Poynting Vector.
6. Starting with Maxwell’s equation “in media”, derive the generalized Poynting Theorem for electromagnetic fields from
𝜕𝑈
⃗⃗ is the Poynting vector and 𝑈 =
Maxwell’s equations in a linear dielectric: ∇ ∙ 𝑆⃗ + = −𝐽⃗𝑓 ∙ 𝐸⃗⃗ , where 𝑆⃗ = 𝐸⃗⃗ × 𝐻
𝜕𝑡
1
2
⃗⃗ + 𝐵
⃗⃗ ∙ 𝐻
⃗⃗) is the total energy density in the electromagnetic field. (Note that the ohmic heating is due to the free
(𝐸⃗⃗ ∙ 𝐷
current.)
Electromagnetic Waves [Ǥ Chapter 9]
1. Ǥ Problem 9.1
2. Ǥ Problem 9.2
3. Ǥ Problem 9.3
4. Consider the reflected and transmitted waves at a boundary between two media. (a) Show that at normal incidence, the amplitudes
𝜐 −𝜐1
are related to the incident wave as 𝐴𝑅 𝑒 𝑖𝛿𝑅 = ( 2
𝜐1 +𝜐2
) 𝐴𝐼 𝑒 𝑖𝛿𝐼 and 𝐴 𝑇 𝑒 𝑖𝛿𝑇 = (
2𝜐2
𝜐1 +𝜐2
) 𝐴𝐼 𝑒 𝑖𝛿𝐼 . Here, the incident wave travels with a
velocity 𝜐1 on a string that is connected to a second infinitely massive string. Thus, 𝜐2 represents the velocity of the transmitted
waves. (b) Give a brief account of the nature of the transmitted wave in the following cases: (i) 𝜐2 > 𝜐1 (ii) 𝜐2 < 𝜐1 (iii) The
junction is fixed.
5. Ǥ Problem 9.5
6. Ǥ Problem 9.6
7. Ǥ Problem 9.7
8. Consider a plane wave whose electric field is 𝐸⃗⃗ (𝑟⃗, 𝑡) = (𝑥̂ + 2𝑦̂ + 𝐸𝑧 𝑧̂) cos(−3𝑥 + 𝑦 + 𝑧 − 𝜔𝑡) in MKS-SI units.
Determine:
(a) The direction of propagation
(b) The direction of polarisation
(c) The wavelength
(d) The frequency (in Hz)
(e) The magnetic field
9. Ǥ Problem 9.8
10. Ǥ Problem 9.9.
11. A plane wave solution to Maxwell’s equations in a linear homogeneous dielectric is given as 𝐸⃗⃗ (𝑧, 𝑡) =
𝐸⃗⃗𝑜 cos(√6𝑧 − 6 × 1010 𝑡), where t is in seconds, z is in cm, and 𝐸⃗⃗𝑜 is a constant vector.
(a) What part of the electromagnetic spectrum is this wave (radio, microwave, x-ray, etc.)?
(b) What is the index of refraction of the medium?
(c) What would the wavelength be if this wave travelled in free space?
(d) Give an expression for the associated magnetic field.
12. Consider a plane wave whose electric field is 𝐸⃗⃗ (𝑟⃗, 𝑡) = (𝑥̂ + 2𝑦̂ + 𝐸𝑧 𝑧̂ ) cos(−3𝑥 + 𝑦 + 𝑧 − 𝜔𝑡) in MKS-SI units. Determine: (i)
The direction of propagation (ii) The direction of polarisation (iii) The wavelength (iv) The frequency (in Hz) (v) The magnetic
field.
2|Page
2026
SPHA082 OCT/NOV EXAMINATION
13. The reflection and transmittance coefficients for waves polarized parallel to the plane of incidence are respectively
given by 𝑅 = (
𝛼−𝛽 2
2
𝛼+𝛽
𝛼+𝛽
) and 𝑇 = 𝛼𝛽 (
2
) , where 𝛼 =
cos 𝜃𝑇
cos 𝜃𝐼
and 𝛽 =
𝜇1 𝑣1
𝜇2𝑣2
=
𝜇 1 𝑛2
𝜇2𝑛1
. Here, the subscripts 1 and 2 refer to the
media containing the incident and transmitted waves respectively. Construct plots of these coefficients as functions of
the angle of incidence for the air-diamond interface. Assume that the index of refraction of diamond is 2.42 and use
the permeability of free space for air, as well as diamond. (Hint: First express 𝛼 in terms of 𝜃𝐼 only, i.e. show that 𝛼 =
√1−[(𝑛1 /𝑛2 ) sin 𝜃𝐼 ]2
cos 𝜃𝐼
. This will be useful in calculating the coordinates of some of the seven points required for sketching
the graph.)
∞…
14. The reflection and transmittance coefficients for waves polarized parallel to the plane of incidence are respectively
given by 𝑅 = (
𝛼−𝛽 2
𝛼+𝛽
) and
𝑇 = 𝛼𝛽 (
2
2
𝛼+𝛽
) , where 𝛼 =
cos 𝜃𝑇
cos 𝜃𝐼
=
√1−[(𝑛1 /𝑛2 ) sin 𝜃𝐼 ]2
cos 𝜃𝐼
and 𝛽 =
𝜇1 𝑣1
𝜇2𝑣2
=
𝜇 1 𝑛2
𝜇2𝑛1
. Here, the
subscripts 1 and 2 refer to the media containing the incident and transmitted waves respectively.
1
Coefficients
0.8
0.6
R
0.4
T
0.2
0
0
10
20
30
40
50
60
70
80
90
I
In the diagram, plots of these coefficients, as functions of the angle of incidence 𝜃𝐼 are constructed for the air-diamond
interface. From it estimate the index of refraction of diamond with the assumption that 𝜇1 = 𝜇0 = 𝜇2 . (Hint: First
estimate 𝑅 or 𝑇 values from the graph, for a specific 𝜃𝐼 . This will be useful in calculating 𝛼 and 𝛽.)
15. Ǥ Problem 9.13
16. Show that tan   n for Brewster's angle. (Start with n cos  cos   0 , eliminate all cosines using Pythagoras, 
using Snell, and solve for sin  as a function of n only. Use Pythagoras to calculate cos  , then take the ratio to get
the tangent.)
17. Ǥ Problem 9.17.
18. For waves propagating down a hollow rectangular wave guide of cross-sectional dimensions a and b, the wave number is given by
𝜔 2
𝑚 2
𝑛 2
𝑐
𝑎
𝑏
𝑘 = √( ) − 𝜋 2 [( ) + ( ) ]
(a) Determine the cut off frequency 𝜔𝑚𝑛 for any given mode. (b) Rewrite the wave number expression above in terms of the
cutoff frequency 𝜔𝑚𝑛 . (c) What will be the lowest cutoff frequency for the given wave guide? (d) Given that TEM waves cannot
occur in a hollow wave guide, prove that the TE00 mode cannot occur in the wave guide.
19. Ǥ Problem 9.20.
20. Ǥ Problem 9.27.
21. Ǥ Problem 9.28
22. Ǥ Problem 9.29
23. Ǥ Problem 9.30
24. Ǥ Problem 9.31
25. Ǥ Problem 9.33
Potentials and Fields [Ǥ Chapter 10]
1. Ǥ Problem 10.8
2. Ǥ Problem 10.9
3. Ǥ Problem 10.10
𝜕
1
4. The following expressions contain all the information in Maxwell’s equations: ∇2 𝑉 + (∇ ∙ 𝐴⃗) = − 𝜌, and
𝜕𝑡
𝜕2 𝐴⃗
𝜖𝑜
𝜕𝑉
(∇ 𝐴 − 𝜇𝑜 𝜖𝑜 2 ) − ∇ (∇ ∙ 𝐴⃗ + 𝜇𝑜 𝜖𝑜 ) = −𝜇𝑜 𝐽⃗. (a) What do these equations reduce to in the Coulomb gauge? (b)
𝜕𝑡
𝜕𝑡
Mention an advantage and a disadvantage of the Coulomb gauge. (c) What do these equations reduce to in the
2 ⃗
3|Page
2026
SPHA082 OCT/NOV EXAMINATION
Lorentz gauge? (d) Demonstrate that the Lorentz gauge treats the potentials 𝑉 and 𝐴⃗ on equal footing by expressing
the equations in terms of the d’Alembertian.
5. Ǥ Problem 10.14
6. Ǥ Problem 10.18
7. Ǥ Problem 10.22
8. Ǥ Problem 10.23.
Radiation [Ǥ Chapter 11]
1. Ǥ Problem 11.1
2. Ǥ Problem 11.2
3. Ǥ Problem 11.3
4. Ǥ Problem 11.4
5. Consider that you are tasked with deriving an expression for the power radiated by an oscillating electric dipole: (a)
Construct a well-labelled diagram that will in finding the potentials 𝑉 and 𝐴⃗ for the dipole. (b) Discuss the
approximations that will be useful in finding the potentials 𝑉 and 𝐴⃗. (c) If the scalar potential of the oscillating perfect dipole
𝑝 cos 𝜃
𝜔
1
is found to be, 𝑉(𝑟, 𝜃, 𝑡) = 𝑜
{− sin[𝜔(𝑡 − 𝑟/𝑐)] + cos[𝜔(𝑡 − 𝑟/𝑐)]}, deduce what the expression will reduce to in
4𝜋𝜖𝑜 𝑟
𝑐
𝑟
the case of a static dipole. (d) Explain [only] how you will proceed from the potentials to finding the power radiated by
the oscillating electric dipole. (e) Using the formula for the intensity ⟨𝑆⃗⟩, obtained by averaging (in time) over a
complete cycle, demonstrate that the oscillating electric dipole will have no radiation along its axis. (f) Sketch the
intensity profile. (g) What is meant by radiation resistance? Show that the radiation resistance of the radiating dipole is
𝑅 = 790(𝑑/𝜆)2 ohms, where 𝜆 is the wavelength of the radiation. (h) If the power radiated by a magnetic dipole is 𝑃 =
𝜇0 𝑚02 𝜔4
, demonstrate that, for configurations of comparable dimensions, the power radiated by an oscillating electric
12𝜋𝑐 3
dipole is much greater than that of an oscillating magnetic dipole.
6. Ǥ Problem 11.6
𝜇 𝑝̈ 2
7. The power radiated by an arbitrary point source is given by 𝑃 = 0 . From this, deduce the power radiated by (a) an
6𝜋𝑐
oscillating electric dipole (b) a point charge (Larmour formula).
8. Ǥ Problem 11.10.
9. Ǥ Problem 11.11.
10. Ǥ Problem 11.12
11. Ǥ Problem 11.13.
Suppose an electron decelerated at a constant rate a from some initial velocity 𝑣𝑜 down to zero. Determine the fraction of
its initial kinetic energy that is lost to radiation? (The rest is absorbed by whatever mechanism keeps the acceleration
constant.) Assume 𝑣𝑜 ≪ 𝑐.
12. Ǥ Problem 11.14
13. Ǥ Problem 11.15
14. Ǥ Problem 11.17
15. Ǥ Problem 11.21
16. Ǥ Problem 11.22
17. Ǥ Problem 11.23
18. Ǥ Problem 11.24
19. Ǥ Problem 11.25
…
∞…
4|Page
2026
SPHA082 OCT/NOV EXAMINATION
Past Test Questions
2025
Test 1:
1. The 2nd order “decoupled” equation for the electric field 𝐸⃗⃗ is given by the expression
⃗⃗
∂2 𝐵
⃗⃗ = 𝜇𝑜 𝜖𝑜
∇2 𝐵
.
∂𝑡 2
(i) Derive Ampere’s law from this expression (i.e., “un-decouple” it).
(ii) Mention one advantage and one disadvantage of the decoupled equation over Maxwell’s equations.
[25]
2. Starting with Maxwell’s equation “in media”, derive the generalised Poynting Theorem for electromagnetic fields from
𝜕𝑈
⃗⃗ is the Poynting vector and 𝑈 =
Maxwell’s equations in a linear dielectric: ∇ ∙ 𝑆⃗ + = −𝐽⃗𝑓 ∙ 𝐸⃗⃗ , where 𝑆⃗ = 𝐸⃗⃗ × 𝐻
𝜕𝑡
1
⃗⃗ + 𝐵
⃗⃗ ∙ 𝐻
⃗⃗) is the total energy density in the electromagnetic field. (Note that the ohmic heating is due to the free
(𝐸⃗⃗ ∙ 𝐷
2
current.)
[25]
3. Consider a plane wave whose electric field is
𝐸⃗⃗ (𝑟⃗, 𝑡) = (𝑥̂ + 2𝑦̂ + 𝐸𝑧 𝑧̂ ) cos(−3𝑥 + 𝑦 + 𝑧 − 𝜔𝑡)
in MKS-SI units. Determine:
(a) The direction of propagation
(b) The direction of polarisation
(c) The wavelength
(d) The frequency (in Hz)
(e) The magnetic field.
[25]
4. Consider a hollow rectangular wave guide of cross-sectional dimensions 𝑎 and 𝑏. The wave number of the waves
propagating down the wave guide is given by
𝜔 2
𝑚 2
𝑛 2
𝑘 = √( ) − 𝜋 2 [( ) + ( ) ]
𝑐
𝑎
𝑏
(a) Determine the cut off frequency 𝜔𝑚𝑛 for any given mode.
(b) What will be the lowest cutoff frequency for the given wave guide?
(c) If the wave guide has dimensions 2.28 cm x 1.01 cm, determine the TE modes that will propagate in it if the driving
frequency is 1.70 x 1010 Hz?
[25]
Test 2
1. The following expressions contain all the information in Maxwell’s equations:
𝜕
1
𝜕2 𝐴⃗
𝜕𝑉
∇2 𝑉 + (∇ ∙ 𝐴⃗) = − 𝜌,
(∇2 𝐴⃗ − 𝜇𝑜 𝜖𝑜 2 ) − ∇ (∇ ∙ 𝐴⃗ + 𝜇𝑜 𝜖𝑜 ) = −𝜇𝑜 𝐽⃗
𝜕𝑡
𝜖𝑜
𝜕𝑡
𝜕𝑡
(a) What do these equations reduce to in the Coulomb gauge?
(b) Mention an advantage and a disadvantage of the Coulomb gauge.
(c) What do these equations reduce to in the Lorentz gauge?
Demonstrate that the Lorentz gauge treats the potentials 𝑉 and 𝐴⃗ on equal footing by expressing the equations in terms
of the d’Alembertian.
[25]
2. Write down the expressions for the retarded potentials. Proceed to prove that the retarded potentials satisfy the Lorentz
Gauge condition.
[25]
3. Consider that you are tasked with deriving an expression for the power radiated by an oscillating electric dipole:
(a) Construct a well-labelled diagram that will in finding the potentials 𝑉 and 𝐴⃗ for the dipole.
(b) Discuss the approximations that will be useful in finding the potentials 𝑉 and 𝐴⃗.
(c) If the scalar potential of the oscillating perfect dipole is found to be,
𝑝𝑜 cos 𝜃
𝜔
1
𝑉(𝑟, 𝜃, 𝑡) =
{− sin[𝜔(𝑡 − 𝑟/𝑐)] + cos[𝜔(𝑡 − 𝑟/𝑐)]},
4𝜋𝜖𝑜 𝑟
𝑐
𝑟
deduce what the expression will reduce to in the case of a static dipole.
(d) Explain how you will proceed from the potentials to finding the power radiated by the oscillating electric dipole.
(Explain/describe only. No need to show how it was done.)
(e) If the intensity of the electric dipole given by:
𝜇𝑜 𝑝𝑜2 𝜔4 sin2 𝜃
⟨𝑆⃗⟩ = (
) 2 𝑟̂ ,
32𝜋 2 𝑐
𝑟
demonstrate that the oscillating electric dipole will have no radiation along its axis.
5|Page
2026
SPHA082 OCT/NOV EXAMINATION
(f) Sketch the intensity profile.
[25]
4. Consider an oscillating magnetic dipole – made of a circular wire loop of radius b, around which we drive an
𝜇 𝑚2 𝜔4
alternating current of frequency 𝜔. Given that the total power radiated by this dipole is 𝑃𝑚 = 𝑜 0 3 .
12𝜋𝑐
(a) First, explain what is meant by radiation resistance.
(b) Show that the radiation resistance of the oscillating magnetic dipole, at a wavelength 𝜆, is
𝑏 4
𝑅 = 3 × 105 ( ) Ω.
𝜆
(c) Demonstrate that, for configurations of comparable dimensions, the radiation resistance of an oscillating electric
𝑑 2
dipole [which is given by 𝑅 = 790 ( ) Ω] is much greater than that of an oscillating magnetic dipole.
𝜆
[25]
2024
Test 1:
1. Consider Maxwell’s equations in regions in which magnetic charges existed.
(a) Given that the units of magnetic field, Tesla, can be written as henry∙A∙m-2, derive the units of the magnetic
charge.
(b) If magnetic charges existed but electric charges did not, determine the source equations for static electric and
magnetic fields.
2. State Maxwell’s equations inside matter in terms of free charges and currents. Reconstruct these equations in their
integral form.
3. By expressing charge as 𝑄(𝑡) = ∫𝑉 𝜌(𝑟⃗, 𝑡)𝑑𝜏 , derive the Continuity Equation in its differential form.
4. Consider that some charge and current configuration produces fields E and B at time t. How will you derive an
expression, akin to the “work-energy theorem”, for the rate at which work is done by the electromagnetic forces acting
on these charges in an interval dt? What is this expression called?
5. Show whether the following function represent waves.
𝑓(𝑧, 𝑡) =
𝐴
𝑏(𝑧 + 𝜐𝑡)2 + 1
Test 2
1. Write down Maxwell’s equations in vacuum for regions of space in which there are no charges or currents. Decouple
the equations (to have separate equations for E and B fields) so as to demonstrate that “in vacuum, each Cartesian
component of E and B satisfies the 3-dimensional wave equation. Mention one advantage and one disadvantage of
the decoupled equation over Maxwell’s equations.
[15]
2. Consider a plane wave whose electric field is 𝐸⃗⃗ (𝑟⃗, 𝑡) = (𝑥̂ + 2𝑦̂ + 𝐸𝑧 𝑧̂) cos(−3𝑥 + 𝑦 + 𝑧 − 𝜔𝑡) in MKS-SI units.
Determine the magnetic field.
[10]
3. Consider a normal incidence of an electromagnetic wave that is reflected and transmitted waves at a boundary
between two media. The real amplitudes of the reflected and transmitted wave are related to that of the incident wave
respectively as:
𝜐 −𝜐
2𝜐
𝐸0𝑅 = | 2 1 | 𝐸0𝐼 and 𝐸0𝑇 = ( 2 ) 𝐸0𝐼
𝜐1 +𝜐2
𝜐1 +𝜐2
(a) Determine these amplitudes in terms of indices of refraction. (Convert the v’s to n’s.)
𝐸
𝐸
(b) Show that at grazing incidence, ratio 0𝑅 = 0 and ratio 0𝑇 = 1, for any material.
[5]
[10]
(c) Calculate R and T, respectively the fraction of the intensity that is reflected and transmitted.
(d) Show that the coefficients R and T in (b) above, satisfy 𝑅 + 𝑇 = 1.
[10]
[10]
𝐸0𝐼
𝐸0𝐼
4. Monochromatic waves propagating down a tube (or wave guide) has the generic form
6|Page
2026
SPHA082 OCT/NOV EXAMINATION
In the interior of the wave guide, these fields satisfy Maxwell’s equations:
Since confined wave are not (in general) transverse, show one of the following:
[20]
5. For waves propagating down a hollow rectangular wave guide of cross-sectional dimensions a and b, the wave
𝜔 2
𝑚 2
𝑛 2
𝑐
𝑎
𝑏
number is given by 𝑘 = √( ) − 𝜋 2 [( ) + ( ) ]. Consider such a guide having dimensions 1.01 cm by 2.28 cm.
(a) Determine the lowest possible cut-off frequency for transverse electric waves. (Be sure to explain why you are
sure of your answer.)
[5]
(b) If the driving frequency is 3.00 x 1010 Hz, find all the possible TE modes.
2023
Test 2
1. Consider a normal incidence of an electromagnetic wave that is reflected and transmitted waves at a boundary between two
media. The real amplitudes of the reflected and transmitted wave are related to that of the incident wave as:
𝜐 −𝜐1
𝐸0𝑅 = | 2
𝜐1 +𝜐2
| 𝐸0𝐼 and 𝐸0𝑇 = (
2𝜐2
𝜐1 +𝜐2
) 𝐸0𝐼 , respectively.
(e) Determine these amplitudes in terms of indices of refraction.
(f) Calculate the fraction of the intensity that is (i) reflected, R, and (ii) transmitted, T.
(g) Show that the coefficients R and T in (b) above, satisfy 𝑅 + 𝑇 = 1.
2. Demonstrate that TEM waves cannot occur in a hollow waveguide.
[5]
[10]
[5]
[20]
3. Demonstrate that the gradient of any scalar function 𝜆, can be added to the vector potential 𝐴⃗, without affecting the physical
⃗⃗, provided the time derivative of the same function is simultaneously subtracted from the scalar potential 𝑉.
quantities 𝐸⃗⃗ and 𝐵
[15]
4. Consider that you are tasked with deriving an expression for the power radiated by an oscillating _electric_ dipole.
(a) Sketch well-labelled diagrams that will aid you in finding the potentials 𝑉 and 𝐴⃗ for the dipole.
[5]
⃗
(b) Discuss the approximations that will aid you in finding the potentials 𝑉 and 𝐴.
[10]
(c) If you obtained the scalar potential of the oscillating perfect dipole to be:
𝑝𝑜 cos 𝜃
𝜔
1
𝑉(𝑟, 𝜃, 𝑡) =
{− sin[𝜔(𝑡 − 𝑟/𝑐)] + cos[𝜔(𝑡 − 𝑟/𝑐)]},
4𝜋𝜖𝑜 𝑟
𝑐
𝑟
Deduce what the expression will reduce to in the case of a static dipole.
[5]
(d) Explain how you would proceed from the potentials to finding the power radiated by the oscillating electric dipole.
[5]
(e) If the intensity (or the time average of the Poynting vector over a complete cycle), is deduced to be:
𝜇𝑜 𝑝𝑜2 𝜔4 sin2 𝜃
⟨𝑆⃗⟩ = (
) 2 𝑟̂
32𝜋 2 𝑐
𝑟
(i) Demonstrate that the oscillating electric dipole will have no radiation along its axis
[5]
(ii) Express this intensity in “coordinate-free” form.
[5]
(f) Sketch the intensity profile.
[5]
5. The power radiated by an arbitrary point source is given is expressed as 𝑃 =
by
7|Page
𝜇0 𝑝̈ 2
6𝜋𝑐
. From this, deduce the power radiated
2026
SPHA082 OCT/NOV EXAMINATION
(a) an oscillating electric dipole.
(b) a point charge (Larmor Formula).
[5]
[10]
2014
Test 1:
1.1 Determine gradient of 𝑟 𝑛 , where 𝑟⃗ is the position vector of a fixed point (𝑥, 𝑦, 𝑧) and 𝑛 is a number .
1.2 Show that for any vector field 𝐹⃗ (𝑟⃗), ∇ × ∇ × 𝐹⃗ = ∇(∇ ∙ 𝐹⃗ ) − ∇2 𝐹⃗ .
1.3 Define the Laplacian of a scalar function T.
[25 marks]
2. Consider Faraday’s Law in its differential form
I.
Write down this expression.
II.
What does this equation suggest or imply?
III.
If magnetic charge were to exist such that 𝐽⃗𝑚 is its current density, write down an expression for this law based
on symmetry of Maxwell’s equations.
IV.
If 𝜌𝑚 were the volume charge density of the magnetic “charge” mentioned in (iii) write down its continuity
equation.
V.
Derive the integral form of this Law.
[15 marks]
3. Consider Ampere’s Law in its differential form.
I.
Write down this expression before Maxwell’s modification.
II.
Write down this expression after Maxwell’s modification.
III.
Comment on the implication or consequence of this expression after Maxwell’s modification.
IV.
Write down what the expression in (ii) will reduce to in free space?
V.
Write down what the expression in (ii) will reduce to in regions of free currents (inside matter)?
[20 marks]
4.1 The Poynting’s Theorem may be written as
d𝑊
𝜕
1
1
1
⃗⃗) ∙ d𝑎⃗
=− ∫
(𝜖𝑜 𝐸 2 + 𝐵2 ) 𝑑𝜏 − ∮ (𝐸⃗⃗ × 𝐵
d𝑡
𝜕𝑡 𝑉 2
𝜇𝑜
𝜇𝑜 𝑆
Explain the terms on both sides of the expression and show how it leads to the definition of the Poynting Vector. [20
marks]
4.2 A long coaxial cable carries current I (the current flows down the surface of the inner cylinder, radius a, and back
along the outer cylinder, radius b) as shown in the figure. Calculate the power (energy per unit time) transported down
the cable, assuming the two conductors are held at potential difference V.
⃗⃗ =
Note: The magnetic field outside a long straight wire carrying a steady current I is 𝐵
inside the wire is that due to an infinite straight wire: 𝐸⃗⃗ =
𝜆
1
2𝜋𝜖𝑜 𝑠
𝜇𝑜 𝐼 1
2𝜋 𝑠
𝜙̂, and the electric field
𝑠̂ .
[20 marks]
Test 2
1. Consider a plane wave whose electric field is
𝐸⃗⃗ (𝑟⃗, 𝑡) = (𝑥̂ + 2𝑦̂ + 𝐸𝑧 𝑧̂ ) cos(−3𝑥 + 𝑦 + 𝑧 − 𝜔𝑡)
in MKS-SI units. Determine:
(a) The direction of propagation
(b) The direction of polarisation
(c) What part of the electromagnetic spectrum is this wave? (Hint: First calculate the frequency.)
[20 marks]
2. With respect to Maxwell’s equations in free space, the 2nd order “decoupled” equation for 𝐸⃗⃗ is given by
8|Page
2026
SPHA082 OCT/NOV EXAMINATION
∇2 𝐸⃗⃗ = 𝜇𝑜 𝜖𝑜
∂2 𝐸⃗⃗
∂𝑡 2
(a) Derive Faraday’s law from it (that is, “un-decouple” it).
(b) Mention one disadvantage of the decoupled equation over Maxwell’s equations.
[15 marks]
3. The reflection and transmittance coefficients for waves polarized parallel to the plane of incidence are respectively given by
𝑅=(
where 𝛼 =
cos 𝜃𝑇
cos 𝜃𝐼
and 𝛽 =
𝜇1 𝑣1
𝜇2𝑣2
=
𝜇 1 𝑛2
𝜇2𝑛1
𝛼−𝛽 2
𝛼+𝛽
)
and
𝑇 = 𝛼𝛽 (
2
𝛼+𝛽
2
) ,
. Here, the subscripts 1 and 2 refer to the media containing the incident and transmitted waves
respectively.
Construct plots of these coefficients as functions of the angle of incidence for the air-diamond interface. Assume that the index of
refraction of diamond is 2.42 and use the permeability of free space for air, as well as diamond.
(Hint: First express 𝛼 in terms of 𝜃𝐼 only, i.e. show that 𝛼 =
√1−[(𝑛1 /𝑛2 ) sin 𝜃𝐼 ]2
cos 𝜃𝐼
. This will be useful in calculating the coordinates of
some of the seven points required for sketching the graph.)
[45 marks]
4. Consider a rectangular wave guide with dimensions 1.01 cm by 2.28 cm.
(a) Determine the lowest possible cut-off frequency for transverse electric waves. (Be sure to demonstrate why you are sure of your
answer.)
(b) If the driving frequency is 3.00 x 10 10 Hz, find all the possible TE modes.
[20 marks]
Test 3:
1. The following expressions contain all the information in Maxwell’s equations:
𝜕
1
𝜕2 𝐴⃗
𝜕𝑉
∇2 𝑉 + (∇ ∙ 𝐴⃗) = − 𝜌,
(∇2 𝐴⃗ − 𝜇𝑜 𝜖𝑜 2 ) − ∇ (∇ ∙ 𝐴⃗ + 𝜇𝑜 𝜖𝑜 ) = −𝜇𝑜 𝐽⃗
𝜕𝑡
𝜖𝑜
𝜕𝑡
𝜕𝑡
(a) What do these equations reduce to in the Coulomb gauge?
(b) Mention an advantage and a disadvantage of the Coulomb gauge.
(c) What do these equations reduce to in the Lorentz gauge?
(d) Demonstrate that the Lorentz gauge treats the potentials 𝑉 and 𝐴⃗ on equal footing by expressing the equations in terms of the
d’Alembertian.
[20 marks]
2. Discuss briefly what is meant by the following:
(a) Retarded time
(b) Advanced time.
[10 marks]
3. Summarise, as best as you can, the laws of electrodynamics in a “triangle diagram” that relates the sources 𝜌 and 𝐽⃗ (constrained by
⃗⃗, and potentials 𝑉 and 𝐴⃗ (constrained by the Lorentz gauge condition).
the continuity equation), fields 𝐸⃗⃗ and 𝐵
[10
marks]
4. Consider that you are tasked with deriving an expression for the power radiated by an oscillating electric dipole.
(a) Construct a well-labelled diagram that will aid you in finding the potentials 𝑉 and 𝐴⃗ for the dipole.
(b) Discuss the approximations that will aid you in finding the potentials 𝑉 and 𝐴⃗.
(c) If you obtained the scalar potential of the oscillating perfect dipole to be
𝑝𝑜 cos 𝜃 𝜔
1
𝑉(𝑟, 𝜃, 𝑡) =
{− sin[𝜔(𝑡 − 𝑟/𝑐)] + cos[𝜔(𝑡 − 𝑟/𝑐)]}
4𝜋𝜖𝑜 𝑟
𝑐
𝑟
deduce what the expression will reduce to in the case of a static dipole.
(d) Explain how you will proceed from the potentials to finding the power radiated by the oscillating electric dipole.
(e) Using the formula for the intensity ⟨𝑆⃗⟩, obtained by averaging (in time) over a complete cycle, demonstrate that the oscillating
electric dipole will have no radiation along its axis.
(f) Sketch the intensity profile.
[30 marks]
5. What is meant by radiation resistance?
[5 marks]
6. Demonstrate that, for configurations of comparable dimensions, the power radiated by an oscillating electric dipole is much greater
than that of an oscillating magnetic dipole.
(Hint: Determine the powers radiated from the Poynting vector.)
[15 marks]
7. Suppose an electron decelerated at a constant rate a from some initial velocity 𝑣𝑜 down to zero. What fraction of its initial kinetic
energy is lost to radiation? (The rest is absorbed by whatever mechanism keeps the acceleration constant.) Assume 𝑣𝑜 ≪ 𝑐.
[10 marks]
End-of-Questions.
9|Page
2026
SPHA082 OCT/NOV EXAMINATION
Past Exam Questions
[2025]
QUESTION 1: MAXWELL’S EQUATIONS; CHARGE AND ENERGY [25 MARKS]
1.1
Demonstrate how the symmetry about Maxwell’s equations in free space leads to the concept of the ”magnetic
charge”.
1.2
[10]
The Poynting’s Theorem may be written as:
d𝑊
𝜕
1
1
1
⃗⃗) ∙ d𝑎⃗
=− ∫
(𝜖 𝐸 2 + 𝐵2 ) 𝑑𝜏 − ∮ (𝐸⃗⃗ × 𝐵
d𝑡
𝜕𝑡 𝑉 2 𝑜
𝜇𝑜
𝜇𝑜 𝑆
1.3
(a)
Explain all the terms on both sides of the expression
[3]
(b)
Show how this expression leads to the definition of the Pointing Vector.
[2]
The following expressions contain all the information in Maxwell’s equations:
∇2 𝑉 +
𝜕
𝜕𝑡
1
(∇ ∙ 𝐴⃗) = − 𝜌, and
𝜖𝑜
(∇2 𝐴⃗ − 𝜇𝑜 𝜖𝑜
𝜕2 𝐴⃗
𝜕𝑡 2
𝜕𝑉
) − ∇ (∇ ∙ 𝐴⃗ + 𝜇𝑜 𝜖𝑜 ) = −𝜇𝑜 𝐽⃗
𝜕𝑡
(a) What do these equations reduce to in the Coulomb gauge?
[3]
(b) Mention one advantage and one disadvantage of the Coulomb gauge.
[2]
(c) What do these equations reduce to in the Lorentz gauge?
[3]
(d) Demonstrate that the Lorentz gauge treats the potentials 𝑉 and 𝐴⃗ on equal footing by expressing the
equations in terms of the d’Alembertian.
[2]
QUESTION 2: ELECTROMAGNETIC WAVES; GUIDED WAVES [15 MARKS]
2
Consider the reflection and transmittance of electromagnetic waves at oblique incidence for the air/diamond
interface. By taking the index of refraction of diamond to be 2.42, sketch a graph of the transmitted and reflected
amplitudes as functions of the incidence angle for the air/diamond interface. (Be sure to show all the calculations
involved.)
[15]
QUESTION 3: POTENTIALS AND FIELDS [15 MARKS]
3
Show how a set of potentials (𝑉, 𝐴⃗) can undergo gauge transformations such that when you add the gradient of a
scalar function, 𝜆, to 𝐴⃗, and simultaneously subtract
(magnetic field) remain the same.
𝜕𝜆
𝜕𝑡
⃗⃗
from 𝑉, the physical quantities 𝐸⃗⃗ (electric field) and 𝐵
[15]
QUESTION 4: RADIATION [45 MARKS]
4.1
Consider a circular wire loop of radius b, around which an alternating current of frequency 𝜔 is driven. Assume that
the radiating loop is uncharged. For this configuration,
(a)
Construct a clear, well-labelled diagram that will aid you in finding the potentials. [3]
(b)
Discuss the approximations that will aid you in finding the potentials.
[4]
(c)
Write down the potentials and deduce the total power radiated.
[10]
(d)
Sketch the intensity profile.
(e)
Demonstrate that, for configurations of comparable dimensions, the power radiated by the electric dipole
is much greater than that of this configuration.
(f)
[2]
[5]
𝑏 4
Show that the radiation resistance of this configuration is 𝑅 = 3 × 105 ( ) ohms, where 𝜆 is the
𝜆
wavelength of radiation.
4.2
An electron decelerates at a constant rate 𝑎 from some initial velocity 𝑣0 down to zero.
10 | P a g e
[6]
2026
SPHA082 OCT/NOV EXAMINATION
(a)
What fraction of its initial kinetic energy is lost to radiation? (The rest is absorbed by whatever mechanism
keeps the acceleration constant.) Assume 𝑣0 ≪ 𝑐 so that Larmor formula can be used.
[10]
(b)
Suppose the initial velocity is thermal (around 105 m/s) and the distance the electron travels is 30Å. What
can you conclude about radiation losses for the electrons in an ordinary conductor?
[5]
[2023]
QUESTION 1: MAXWELL’S EQUATIONS; CHARGE AND ENERGY [25 MARKS]
1.1
(a)
Write down Maxwell’s equations in free space (regions of space in which there are no charges or currents.
(4)
(b)
With respect to Maxwell’s equations in free space, the 2nd order “decoupled” equation for the electric field
𝐸⃗⃗ is given by the expression
∇2 𝐸⃗⃗ = 𝜇𝑜 𝜖𝑜
∂2 𝐸⃗⃗
.
∂𝑡 2
(i) Derive Faraday’s law from this expression (i.e., “un-decouple” it).
(6)
(ii) Mention one advantage and one disadvantage of the decoupled equation over Maxwell’s equations. (2)
(c)
Demonstrate the symmetry about Maxwell’s equations which leads to the concept of the ”magnetic charge”.
(8)
1.2
The Poynting’s Theorem may be written as
d𝑊
𝜕
1
1
1
⃗⃗) ∙ d𝑎⃗
=− ∫
(𝜖𝑜 𝐸 2 + 𝐵2 ) 𝑑𝜏 − ∮ (𝐸⃗⃗ × 𝐵
d𝑡
𝜕𝑡 𝑉 2
𝜇𝑜
𝜇𝑜 𝑆
(a)
Explain all the terms on both sides of the expression
(4)
(b)
Show how this expression leads to the definition of the Poynting Vector.
(4)
QUESTION 2: ELECTROMAGNETIC WAVES [25 MARKS]
2.1
Write down the (real) electric and magnetic fields for a monochromatic plane wave of amplitude 𝐸𝑜 , frequency 𝜔,
and phase angle zero that is
(a)
Travelling in the negative x direction and polarised in the z direction;
(4)
(b)
Travelling in the direction from the origin to the point (1, 1, 1), with polarization parallel to the x z plane.
(4)
⃗⃗ and 𝒏
In each case, sketch the wave, and give the explicit Cartesian components of 𝑘
̂.
2.2
Consider the reflection and transmittance of electromagnetic waves at oblique incidence for the air/diamond
interface. The index of refraction of diamond is 2.42.
(a)
Calculate the reflected and transmitted amplitudes at normal incidence
(b)
Determine the Brewster’s angle
(c)
Calculate the “crossover” angle, at which the reflected and transmitted amplitudes are equal.
(d)
Using the values obtained in (a), (b) and (c) above, sketch a graph of the transmitted and reflected
amplitudes as functions of the incidence angle for the air/diamond interface.
(4)
(2)
(2)
(4)
QUESTION 3: POTENTIALS AND FIELDS; GUIDED WAVES [25 MARKS]
3.1
In conductors, the free current density 𝐽⃗𝑓 is generally not zero but actually proportional to the electric field. Thus,
the (real) electric and magnetic fields can be shown to be respectively
11 | P a g e
2026
SPHA082 OCT/NOV EXAMINATION
⃗⃗(𝑟⃗, 𝑡) = 𝐵𝑜 𝑒 −𝜅𝑧 cos(𝑘𝑧 − 𝜔𝑡 + 𝛿𝐸 + 𝜙) 𝒚
̂ and 𝐵
̂
𝐸⃗⃗ (𝑧, 𝑡) = 𝐸𝑜 𝑒 −𝜅𝑧 cos(𝑘𝑧 − 𝜔𝑡 + 𝛿𝐸 ) 𝒙
(a)
Calculate the (time averaged) energy density of an electromagnetic plane wave in a conducting medium.
(4)
(b)
Show that the magnetic contribution always dominates.
(c)
Show that the intensity is
𝐼=
(3)
𝑘
𝐸 2 𝑒 −2𝜅𝑧
2𝜇𝜔 2 𝑜
(5)
3.2
Show that the TE00 cannot occur in a rectangular wave guide.
(Note that in this case
3.3
𝜔
𝑐
(8)
= 𝑘.)
The generalisations for the retarded potentials for non-static sources are:
𝑉(𝑟⃗, 𝑡) =
1
∫
4𝜋𝜀
𝜌(𝑟⃗ ′,𝑡𝑟 )
𝑟
𝑜
⃗ ′
𝐽(𝑟⃗ ,𝑡𝑟)
𝜇
𝑑𝜏′ and 𝐴⃗(𝑟⃗, 𝑡) = 𝑜 ∫
𝑑𝜏′
4𝜋
𝑟
These potentials can be derived by making the assertion that electromagnetic waves travel with the speed of light.
Show that they satisfy Lorentz gauge condition.
(8)
QUESTION 4: RADIATION [25 MARKS]
4.1
Given that the total power radiated by an electric dipole is
𝑃 = ∫⟨𝑆⟩ ∙ 𝑑𝑎⃗ =
𝜇𝑜 𝑝02 𝜔4
12𝜋𝑐
𝑑 2
Show that the radiation resistance of the radiating dipole is 𝑅 = 790 ( ) ohms, where 𝜆 is the wavelength of the
𝜆
radiation and d is the charge separation.
4.2
(8)
Demonstrate that, for configurations of comparable dimensions, the power radiated by an oscillating electric dipole
is much greater than that of an oscillating magnetic dipole.
(8)
An electron decelerates at a constant rate 𝑎 from some initial velocity 𝑣0 down to zero.
4.3
(a)
What fraction of its initial kinetic energy is lost to radiation? (The rest is absorbed by whatever mechanism
keeps the acceleration constant.) Assume 𝑣0 ≪ 𝑐 so that Larmor formula can be used.
(b)
[10]
Suppose the initial velocity is thermal (around 105 m/s) and the distance the electron travels is 30Å. What
can you conclude about radiation losses for the electrons in an ordinary conductor?
[5]
[2022]
QUESTION 1: MAXWELL’S EQUATIONS; CHARGE AND ENERGY [25 MARKS]
1.1
Demonstrate how the symmetry about Maxwell’s equations in free space leads to the concept of the ”magnetic
charge”.
1.2
(10)
The Poynting’s Theorem may be written as:
d𝑊
𝜕
1
1
1
⃗⃗) ∙ d𝑎⃗
=− ∫
(𝜖 𝐸 2 + 𝐵2 ) 𝑑𝜏 − ∮ (𝐸⃗⃗ × 𝐵
d𝑡
𝜕𝑡 𝑉 2 𝑜
𝜇𝑜
𝜇𝑜 𝑆
1.3
(a)
Explain all the terms on both sides of the expression
(3)
(b)
Show how this expression leads to the definition of the Pointing Vector.
(2)
The following expressions contain all the information in Maxwell’s equations:
∇2 𝑉 +
𝜕
𝜕𝑡
1
(∇ ∙ 𝐴⃗) = − 𝜌, and
𝜖𝑜
(∇2 𝐴⃗ − 𝜇𝑜 𝜖𝑜
(a) What do these equations reduce to in the Coulomb gauge?
12 | P a g e
𝜕2 𝐴⃗
𝜕𝑡 2
𝜕𝑉
) − ∇ (∇ ∙ 𝐴⃗ + 𝜇𝑜 𝜖𝑜 ) = −𝜇𝑜 𝐽⃗
𝜕𝑡
[3]
2026
SPHA082 OCT/NOV EXAMINATION
(b) Mention one advantage and one disadvantage of the Coulomb gauge.
[2]
(c) What do these equations reduce to in the Lorentz gauge?
[3]
(d) Demonstrate that the Lorentz gauge treats the potentials 𝑉 and 𝐴⃗ on equal footing by expressing the
equations in terms of the d’Alembertian.
[2]
QUESTION 2: ELECTROMAGNETIC WAVES; GUIDED WAVES [25 MARKS]
2.1
In conductors, the free current density 𝐽⃗𝑓 is generally not zero but actually proportional to the electric field. Thus,
the (real) electric and magnetic fields can be shown to be respectively
⃗⃗(𝑟⃗, 𝑡) = 𝐵𝑜 𝑒 −𝜅𝑧 cos(𝑘𝑧 − 𝜔𝑡 + 𝛿𝐸 + 𝜙) 𝒚
̂ and 𝐵
̂
𝐸⃗⃗ (𝑧, 𝑡) = 𝐸𝑜 𝑒 −𝜅𝑧 cos(𝑘𝑧 − 𝜔𝑡 + 𝛿𝐸 ) 𝒙
(a)
Calculate the (time averaged) energy density of an electromagnetic plane wave in a conducting medium.
[6]
(b)
2.2
Show that the magnetic contribution always dominates.
[4]
Consider the reflection and transmittance of electromagnetic waves at oblique incidence for the air/diamond
interface. The index of refraction of diamond is 2.42.
(a)
Calculate the reflected and transmitted amplitudes at normal incidence
(6)
(b)
Calculate the reflected and transmitted amplitudes at grazing incidence
(2)
(c)
Determine the Brewster’s angle
(d)
Calculate the “crossover” angle, at which the reflected and transmitted amplitudes are equal.
(e)
Using the values obtained in (a), (b) and (c) above, sketch a graph of the transmitted and reflected
(2)
amplitudes as functions of the incidence angle for the air/diamond interface.
(3)
(2)
QUESTION 3: POTENTIALS AND FIELDS; RADIATION [50 MARKS]
3.1
Consider that you are tasked with deriving an expression for the power radiated by an oscillating electric dipole.
(a)
Sketch well-labelled diagrams that will aid you in finding the potentials 𝑉 and 𝐴⃗ for the dipole.
(b)
Discuss the approximations that will aid you in finding the potentials 𝑉 and 𝐴⃗.
(c)
If you obtained the scalar potential of the oscillating perfect dipole to be:
𝑉(𝑟, 𝜃, 𝑡) =
(4)
If the intensity (or the time average of the Poynting vector over a complete cycle), is deduced to be: ⟨𝑆⃗⟩ =
𝜇 𝑝2𝜔4 sin2 𝜃
( 𝑜 𝑜2 )
32𝜋 𝑐
(f)
3.2
(2)
Explain how you will proceed from the potentials to finding the power radiated by the oscillating electric
dipole.
(e)
(4)
𝑝𝑜 cos 𝜃
𝜔
1
{− sin[𝜔(𝑡 − 𝑟/𝑐)] + cos[𝜔(𝑡 − 𝑟/𝑐)]},
4𝜋𝜖𝑜 𝑟
𝑐
𝑟
deduce what the expression will reduce to in the case of a static dipole.
(d)
(4)
𝑟2
𝑟̂ ,
(i) Demonstrate that the oscillating electric dipole will have no radiation along its axis
(1)
(ii) Express this intensity in “coordinate-free” form.
(3)
Sketch the intensity profile.
(2)
The electric and magnetic fields of a point charge 𝑞 moving with constant velocity is given by:
𝐸⃗⃗ (𝑟⃗, 𝑡) =
𝑞
1−𝑣 2 ⁄𝑐 2
𝑅⃗⃗̇
4𝜋𝜖0 (1−𝑣 2 sin2 𝜃 ⁄𝑐 2 )3/2 𝑅 2
, and
⃗⃗ = 1 (𝓇̂ × 𝐸⃗⃗ ) = 12 (𝑣⃗ × 𝐸⃗⃗ ),
𝐵
𝑐
𝑐
where 𝑅⃗⃗ ≡ 𝑟⃗ − 𝑣⃗𝑡 is the vector from the present location of the particle to 𝑟, and 𝜃 is the angle between 𝑅⃗⃗ and 𝑣⃗
(Figure 1). If a particle carrying such a charge is travelling along the 𝑥-axis, calculate the total power passing through
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SPHA082 OCT/NOV EXAMINATION
the plane located at 𝑥 = 𝑎, at the moment the particle itself is at the origin.
(15)
Figure 1
3.3
An electron decelerates at a constant rate 𝑎 from some initial velocity 𝑣0 down to zero.
(a)
What fraction of its initial kinetic energy is lost to radiation? (The rest is absorbed by whatever mechanism
keeps the acceleration constant.) Assume 𝑣0 ≪ 𝑐 so that Larmor formula can be used.
(b)
[10]
Suppose the initial velocity is thermal (around 105 m/s) and the distance the electron travels is 30Å. What
can you conclude about radiation losses for the electrons in an ordinary conductor?
[5]
[2019]
1.1
1.2
1.3
Demonstrate how the symmetry about Maxwell’s equations in free space leads to the concept of the ”magnetic
charge”.
(10)
The Poynting’s Theorem may be written as:
d𝑊
𝜕
1
1
1
⃗⃗) ∙ d𝑎⃗
=− ∫
(𝜖𝑜 𝐸 2 + 𝐵2 ) 𝑑𝜏 − ∮ (𝐸⃗⃗ × 𝐵
d𝑡
𝜕𝑡 𝑉 2
𝜇𝑜
𝜇𝑜 𝑆
(a)
Explain all the terms on both sides of the expression
(3)
(b)
Show how this expression leads to the definition of the Pointing Vector. (2)
The following expressions contain all the information in Maxwell’s equations:
𝜕
1
𝜕2 𝐴⃗
𝜕𝑉
∇2 𝑉 + (∇ ∙ 𝐴⃗) = − 𝜌, and
(∇2 𝐴⃗ − 𝜇𝑜 𝜖𝑜 2 ) − ∇ (∇ ∙ 𝐴⃗ + 𝜇𝑜 𝜖𝑜 ) = −𝜇𝑜 𝐽⃗
𝜕𝑡
2.1
𝜖𝑜
𝜕𝑡
𝜕𝑡
(e) What do these equations reduce to in the Coulomb gauge?
[3]
(f) Mention one advantage and one disadvantage of the Coulomb gauge.
[2]
(g) What do these equations reduce to in the Lorentz gauge?
[3]
⃗
(h) Demonstrate that the Lorentz gauge treats the potentials 𝑉 and 𝐴 on equal footing by expressing the
equations in terms of the d’Alembertian. [2]
In conductors, the free current density 𝐽⃗𝑓 is generally not zero but actually proportional to the electric field. Thus,
the (real) electric and magnetic fields can be shown to be respectively
̂
𝐸⃗⃗ (𝑧, 𝑡) = 𝐸𝑜 𝑒 −𝜅𝑧 cos(𝑘𝑧 − 𝜔𝑡 + 𝛿𝐸 ) 𝒙
and
⃗⃗(𝑟⃗, 𝑡) = 𝐵𝑜 𝑒 −𝜅𝑧 cos(𝑘𝑧 − 𝜔𝑡 + 𝛿𝐸 + 𝜙) 𝒚
̂
𝐵
(a)
Calculate the (time averaged) energy density of an electromagnetic plane wave in a conducting medium.
[6]
(b)
Show that the magnetic contribution always dominates.
[4]
2.2
Consider the reflection and transmittance of electromagnetic waves at oblique incidence for the air/diamond
interface. The index of refraction of diamond is 2.42.
(a)
Calculate the reflected and transmitted amplitudes at normal incidence(6)
(b)
Calculate the reflected and transmitted amplitudes at grazing incidence(2)
(c)
Determine the Brewster’s angle
(2)
(d)
Calculate the “crossover” angle, at which the reflected and transmitted amplitudes are equal.
(3)
(e)
Using the values obtained in (a), (b) and (c) above, sketch a graph of the transmitted and reflected
amplitudes as functions of the incidence angle for the air/diamond interface.
(2)
3.1
Consider that you are tasked with deriving an expression for the power radiated by an oscillating electric dipole.
(a)
Sketch well-labelled diagrams that will aid you in finding the potentials 𝑉 and 𝐴⃗ for the dipole.
(4)
⃗
(b) Discuss the approximations that will aid you in finding the potentials 𝑉 and 𝐴.
(4)
(c) If you obtained the scalar potential of the oscillating perfect dipole to be:
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SPHA082 OCT/NOV EXAMINATION
𝑝𝑜 cos 𝜃
𝜔
1
{− sin[𝜔(𝑡 − 𝑟/𝑐)] + cos[𝜔(𝑡 − 𝑟/𝑐)]},
4𝜋𝜖𝑜 𝑟
𝑐
𝑟
deduce what the expression will reduce to in the case of a static dipole.
(2)
Explain how you will proceed from the potentials to finding the power radiated by the oscillating electric dipole. (4)
If the intensity (or the time average of the Poynting vector over a complete cycle), is deduced to be: ⟨𝑆⃗⟩ =
𝑉(𝑟, 𝜃, 𝑡) =
(d)
(e)
𝜇 𝑝2 𝜔4 sin2 𝜃
( 𝑜 𝑜2 ) 2 𝑟̂ ,
32𝜋 𝑐
𝑟
(i) Demonstrate that the oscillating electric dipole will have no radiation along its axis(1)
(ii) Express this intensity in “coordinate-free” form.
(3)
(f) Sketch the intensity profile.
(2)
3.2
The electric and magnetic fields of a point charge 𝑞 moving with constant velocity is given by:
𝐸⃗⃗ (𝑟⃗, 𝑡) =
1−𝑣 2 ⁄𝑐 2
𝑞
𝑅⃗⃗̇
4𝜋𝜖0 (1−𝑣 2 sin2 𝜃 ⁄𝑐 2 )3/2 𝑅 2
, and
⃗⃗ = 1 (𝓇̂ × 𝐸⃗⃗ ) = 12 (𝑣⃗ × 𝐸⃗⃗ ),
𝐵
𝑐
𝑐
where 𝑅⃗⃗ ≡ 𝑟⃗ − 𝑣⃗𝑡 is the vector from the present location of the particle to 𝑟, and 𝜃 is the angle between 𝑅⃗⃗ and 𝑣⃗
(Figure 1). If a particle carrying such a charge is travelling along the 𝑥-axis, calculate the total power passing through
the plane located at 𝑥 = 𝑎, at the moment the particle itself is at the origin.
(15)
3.3
Figure 1
An electron decelerates at a constant rate 𝑎 from some initial velocity 𝑣0 down to zero.
(a)
What fraction of its initial kinetic energy is lost to radiation? (The rest is absorbed by whatever mechanism
keeps the acceleration constant.) Assume 𝑣0 ≪ 𝑐 so that Larmor formula can be used.
[10]
(b)
Suppose the initial velocity is thermal (around 105 m/s) and the distance the electron travels is 30Å. What
can you conclude about radiation losses for the electrons in an ordinary conductor?
[5]
[2018]
1.1
Write down Maxwell’s equations in free space and decouple these equations so as to have separate equations for
⃗⃗ fields.
𝐸⃗⃗ and 𝐵
[6]
1.2
(a)
By using the analogy between Faraday’s law and Ampere’s law, together with the Biot-Savart law, show
⃗⃗(𝑟⃗′,𝑡)×𝓇̂
1 𝜕
𝐵
that for Faraday-induced electric fields 𝐸⃗⃗ (𝑟⃗, 𝑡) = −
𝑑𝜏′
[4]
∫
2
4𝜋 𝜕𝑡
(b)
𝓇
Since the vector potential may be similarly derived as 𝐴⃗(𝑟⃗, 𝑡) =
1
4𝜋
∫
⃗⃗(𝑟⃗′,𝑡)×𝓇̂
𝐵
𝓇2
𝑑𝜏′
𝜕𝐴⃗
1.3
(i)
Show that 𝐸⃗⃗ = −
[2]
𝜕𝑡
(ii)
Check the result in (i) by taking the curl of both sides
[4]
Starting with Maxwell’s equation “in media”, derive the generalised Poynting Theorem for electromagnetic fields
from Maxwell’s equations in a linear dielectric:
𝜕𝑈
∇ ∙ 𝑆⃗ + 𝑒𝑚𝑡𝑜𝑡 = −𝐽⃗𝑓 ∙ 𝐸⃗⃗ ,
𝜕𝑡
⃗⃗ is the Poynting vector and 𝑈𝑒𝑚𝑡𝑜𝑡 = 1 (𝐸⃗⃗ ∙ 𝐷
⃗⃗ + 𝐵
⃗⃗ ∙ 𝐻
⃗⃗) is the total energy density in the
where 𝑆⃗ = 𝐸⃗⃗ × 𝐻
2
electromagnetic field. (Note that the ohmic heating is due to the free current.)
[9]
2.1
Consider a monochromatic plane wave of amplitude 𝐸𝑜 , frequency 𝜔, and phase angle zero.
(a) If the wave is travelling in the negative x direction and polarized in the z direction
⃗⃗ and 𝒏
(i) Determine the explicit Cartesian components of 𝑘
̂.
[2]
(ii) Determine the (real) electric and magnetic fields of the wave [4]
(iii) Sketch the wave.
[2]
(b)
If the wave is travelling in the direction from the origin to the point (1, 1, 1), with polarization parallel to the
x-z plane
⃗⃗ and 𝒏
(i) Determine the explicit Cartesian components of 𝑘
̂.
[4]
(ii) Determine the (real) electric and magnetic fields of the wave [3]
2.2
Consider two long pieces of string (medium 1 and medium 2) that are connected together by a massless knot. The
amplitudes of the reflected (𝐴𝑅 ) and transmitted (𝐴 𝑇 ) waves on the strings, are related to the amplitude (𝐴𝐼 ) of the incident
wave as
𝜐 −𝜐
2𝜐
𝐴𝑅 𝑒 𝑖𝛿𝑅 = ( 2 1 ) 𝐴𝐼 𝑒 𝑖𝛿𝐼
and
𝐴 𝑇 𝑒 𝑖𝛿𝑇 = ( 2 ) 𝐴𝐼 𝑒 𝑖𝛿𝐼
𝜐1 +𝜐2
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𝜐1 +𝜐2
2026
SPHA082 OCT/NOV EXAMINATION
2.3
3.1
Here, the incident wave travels with a velocity 𝜐1 in medium 1, and thus, 𝜐2 represents the velocity of the transmitted
waves in medium 2. The phase constants are denoted by 𝛿.
(a)
In the following cases, comment briefly on the velocities, phases, amplitudes and nature of the outgoing
waves (i.e. the reflected and the transmitted waves).
(i) When the second string is lighter than the first,
[5]
(ii) When the second string is heavier than the first,
[5]
(iii) When the second string is infinitely massive.
[3]
(b)
Explain the implication on the transmitted wave if its velocity is infinitely larger than that of the incident
wave.
[2]
Consider a rectangular wave guide with dimensions 2.28 cm x 1.01 cm.
(a)
Determine the TE modes that will propagate in this wave guide, if the driving frequency is 1.70 x 10 10 Hz?
[6]
(b)
Suppose you wanted to excite only one TE mode, determine the range of frequencies you would use?
[2]
(c)
What are the corresponding wavelengths (in open space)?
[2]
The electric and magnetic fields of a point charge 𝑞 moving with constant velocity is given by
𝑞
1 − 𝑣 2 ⁄𝑐 2
𝑅⃗⃗̇
𝐸⃗⃗ (𝑟⃗, 𝑡) =
4𝜋𝜖0 (1 − 𝑣 2 sin2 𝜃⁄𝑐 2 )3/2 𝑅2
1
1
⃗⃗ = (𝓇̂ × 𝐸⃗⃗ ) = (𝑣⃗ × 𝐸⃗⃗ ),
𝐵
𝑐
𝑐2
where 𝑅⃗⃗ ≡ 𝑟⃗ − 𝑣⃗𝑡 is the vector from the present location of the particle to 𝑟, and 𝜃 is the angle between 𝑅⃗⃗ and 𝑣⃗
(Figure 1). If a particle carrying such a charge is travelling along the 𝑥-axis, calculate the total power passing through
the plane located at 𝑥 = 𝑎, at the moment the particle itself is at the origin.
(15)
Figure 1
3.2
[2013]
1.1
Figure 2
A particle of mass 𝑚 and charge 𝑞 is attached to a spring with force constant 𝑘, hanging from the ceiling (Figure 2).
Its equilibrium position is a distance ℎ above the floor. It is pulled down a distance 𝑑 below equilibrium and released,
at time 𝑡 = 0.
(a)
Under the usual assumptions (𝑑 ≪ 𝜆 ≪ ℎ), calculate the intensity of the radiation hitting the floor, as a
function of the distance 𝑅 from the point directly below 𝑞. [Note: The intensity here is the average power
per unit area of floor.)
(6)
(b)
From the results in (a), determine the distance 𝑅 for which the radiation most intense. (Ignore the radiative
damping of the oscillator.)
(4)
(c)
As a check on your results in (a), assume the floor is of final extent, and calculate the average energy per
unit time striking the entire floor. Comment on whether this result is what you would expect.
(5)
(d)
Because it is losing energy in the form of radiation, the amplitude of oscillation will gradually decrease.
After what time 𝜏 has the amplitude been reduced to 𝑑/𝑒? Assume that the fraction of the total energy lost
in one cycle is very small.)
(5)
Write down Maxwell’s equations in free space and decouple these first-order equations so as to have separate equations
for E and B fields
1.2
(a) Using the analogy between Faraday’s law and Ampere’s law, together with the Biot-Savart law, show that for
Faraday-induced electric fields
⃗⃗(𝑟⃗′,𝑡)×𝓇̂
1 𝜕
𝐵
𝐸⃗⃗ (𝑟⃗, 𝑡) = −
𝑑𝜏 ′ .
[4]
∫
2
4𝜋 𝜕𝑡
𝓇
(b) The vector potential may be derived in the same manner as in (a) above to give
⃗⃗(𝑟⃗′, 𝑡) × 𝓇̂
1 𝐵
𝐴⃗(𝑟⃗, 𝑡) =
∫
𝑑𝜏′.
4𝜋
𝓇2
(iii)
Find an expression that relates 𝐸⃗⃗ to 𝐴⃗.
[2]
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2026
SPHA082 OCT/NOV EXAMINATION
(iv)
Check that your result in (i) is valid. [Hint: Take the curl of both sides.]
[2]
1.3 (a) Starting with Maxwell’s equation “in media”, derive the generalized Poynting Theorem for electromagnetic fields
𝜕𝑈
⃗⃗ is the Poynting vector and
from Maxwell’s equations in a linear dielectric: ∇ ∙ 𝑆⃗ + 𝑒𝑚 = −𝐽⃗𝑓 ∙ 𝐸⃗⃗ , where 𝑆⃗ = 𝐸⃗⃗ × 𝐻
𝜕𝑡
2.1
2.2
1
⃗⃗ + 𝐵
⃗⃗ ∙ 𝐻
⃗⃗) is the total energy density in the electromagnetic field. (Note that the ohmic heating is due
𝑈𝑒𝑚 = (𝐸⃗⃗ ∙ 𝐷
2
to the free current.)
[8]
Write down the (real) electric and magnetic fields for a monochromatic plane wave of amplitude 𝐸𝑜 , frequency 𝜔,
and phase angle zero that is
(a) Travelling in the negative x direction and polarized in the z direction; [4]
(b) Travelling in the direction from the origin to the point (1, 1, 1), with polarization parallel to the x z plane.
[4]
⃗⃗ and 𝒏
In each case, sketch the wave and give the explicit Cartesian components of 𝑘
̂.
Consider the reflection and transmittance of electromagnetic waves at oblique incidence for the air/diamond
interface. The index of refraction of diamond is 2.42.
(a) Calculate the reflected and transmitted amplitudes at normal incidence
[4]
(b) Determine the Brewster’s angle
[2]
(c) Calculate the “crossover” angle, at which the reflected and transmitted amplitudes are equal.
[2]
(d) Using the values obtained in (a), (b) and (c) above, sketch a graph of the transmitted and reflected amplitudes
as functions of the incidence angle for the air/diamond interface.
[4]
3.2
Show that the TE00 cannot occur in a rectangular wave guide.
[8]
𝜔
(Note that in this case = 𝑘.)
𝑐
4.1
The potentials of a point charge moving at constant velocity are given by
1
𝑞𝑐
𝑉(𝑟⃗, 𝑡) =
4𝜋𝜖𝑜 √(𝑐 2 𝑡 − 𝑟⃗ ∙ 𝑣⃗)2 + (𝑐 2 − 𝑣 2 )(𝑟 2 − 𝑐 2 𝑡 2 )
𝜇𝑜
𝑞𝑐𝑣⃗
𝐴⃗(𝑟⃗, 𝑡) =
4𝜋 √(𝑐 2 𝑡 − 𝑟⃗ ∙ 𝑣⃗)2 + (𝑐 2 − 𝑣 2 )(𝑟 2 − 𝑐 2 𝑡 2 )
Show that they satisfy the Lorentz gauge condition.
[12]
4.2 Consider an electron that decelerated at a constant rate a from some initial velocity 𝑣𝑜 down to zero.
(a) What fraction of its initial kinetic energy is lost to radiation? (The rest of the energy is absorbed by whatever mechanism
keeps the acceleration constant.) Assume 𝑣𝑜 ≪ 𝑐 so that the Lamor formula can be used.
[4]
(b) Suppose the initial velocity is thermal (around 105 m/s) and the distance the electron goes is 30 Å. What can you
conclude about radiation losses for the electrons in an ordinary conductor?
[4]
5 Figure 1 shows a model for electric quadrupole radiation in which two oppositely oriented oscillating electric dipoles are
separated by a distance d. Note that the dipoles are not located at the origin.
Using the potentials
𝑝 𝜔 cos 𝜃
𝑉(𝑟⃗, 𝜃, 𝑡) = − 𝑜 (
) sin[𝜔(𝑡 − 𝑟/𝑐)]
4𝜋𝜖𝑜 𝑐
𝑟
and keeping only the terms of first order in d:
(a)
Find the scalar and vector potentials.
(b)
Find the electric and magnetic fields.
(c)
Find the Poynting vector.
(d)
Find the power radiated.
17 | P a g e
𝜇 𝑝 𝜔
𝐴⃗(𝑟⃗, 𝜃, 𝑡) = − 𝑜 𝑜 sin[𝜔(𝑡 − 𝑟/𝑐)] 𝑧̂
4𝜋𝑟
[8]
[5]
[3]
[4]