PHYS20141

ONE HOUR THIRTY MINUTES

A list of constants is enclosed.

UNIVERSITY OF MANCHESTER

Electromagnetism

32nd January 2022, 2.00 p.m. - 3.30 p.m.

Answer ALL parts of question 1 and TWO other questions

Electronic calculators may be used, provided that they cannot store text.

The numbers are given as a guide to the relative weights of the different parts of
each question.

c University of Manchester, 2022 1 of 6 P.T.O

 PHYS20141

Formula Sheet

For vector V,
∇2 V = ∇(∇ · V) − ∇ × ∇ × V.

In cylindrical polar coordinates, for scalar Φ and vector V = (vr , vθ , vz ),

∂Φ 1 ∂Φ ∂Φ
∇Φ =r̂ + θ̂ + ẑ,
∂r r ∂θ ∂z
1 ∂ 1 ∂vθ ∂vz
∇·V = (rvr ) + + ,
r ∂r r ∂θ ∂z
     
1 ∂vz ∂vθ ∂vr ∂vz 1 ∂ ∂vr
∇×V = − r̂ + − θ̂ + (rvθ ) − ẑ.
r ∂θ ∂z ∂z ∂r r ∂r ∂θ

In spherical polar coordinates, for scalar Φ and vector V = (vr , vθ , vφ ),

∂Φ 1 ∂Φ 1 ∂Φ
∇Φ = r̂ + θ̂ + φ̂,
∂r r ∂θ r sin θ ∂φ
1 ∂ 2 1 ∂ 1 ∂vφ
∇·V = (r v r ) + (sin θ vθ ) + ,
r2 ∂r r sin θ ∂θ r sin θ ∂φ
     
1 ∂ ∂vθ 1 1 ∂vr ∂rvφ 1 ∂rvθ ∂vr
∇×V = (sin θ vφ ) − r̂ + − θ̂ + − φ̂.
r sin θ ∂θ ∂φ r sinθ ∂φ ∂r r ∂r ∂θ

The Lorenz gauge condition, relating vector potential, A, and scalar potential, φ, is given
by
1 ∂φ
∇·A=− 2 ,
c ∂t
and when time-independent this defines the Coulomb gauge,

∇ · A = 0.

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 PHYS20141

1. a) An electric field vector in free space is given by

E(r, t) = E0 ŷ cos (kx − ωt) ,

where k = (2π/3) m−1 and E0 = 300 Vm−1 .

(i) Show that this vector, subject to an appropriate dispersion relationship, is com-
patible with Maxwell’s requirements for an electromagnetic wave.
(ii) Calculate the angular frequency ω of the wave.
(iii) State the direction of propagation of the wave.
(iv) Evaluate the magnitude of the magnetic vector associated with this wave.
(v) Evaluate the maximum magnitude and direction of the Poynting vector.
[9 marks]

b) A stationary electron is located a distance d from the surface of a large vertical

metal plate held at 0 V. For what value of d will the force on the electron give rise
to an instantaneous horizontal acceleration of 1 ms−2 ?
[3 marks]

c) Express the electric field in terms of the vector potential, A, and scalar potential,
φ. If A satisfies the Coulomb gauge show that φ satisfies Poisson’s equation.
[3 marks]

d) Write down an expression for the potential energy of a magnetic dipole, formed by
a current I flowing in a circular loop of radius a, if the loop is placed with its axis
at an angle θ to a uniform magnetic field B.
[3 marks]

e) A solenoid consists of 200 turns of wire wound on a cylinder of magnetic material,

µr = 1.1, of length 0.25 m and diameter 5 mm. Calculate the magnetic energy
stored in the cylinder when a current of 2 A is passed through the wire.
[7 marks]

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2. a) Explain, why Ampère, exploring magnetic effects using electrically conductive ma-
terials, could not experimentally detect the correction to his work later applied by
Maxwell.
[4 marks]

b) Use Maxwell’s equations to show that, in a non-magnetic, neutral electrical conduc-

tor, the electric field, E, satisfies
∂E
∇ 2 E ' µ0 σ ,
∂t
where σ is the electrical conductivity.
[8 marks]

c) Assuming an exact equality, show that plane wave solutions to the equation in part
b), of the form
E = E0 ei(kz−ωt) ,
where E0 is a constant vector, have a complex wave number k that may be written
as
k = a(1 + i),
and evaluate the term a.
[5 marks]

d) Two lengths of conductors are made from,

i) a single solid copper wire of radius 5 mm, and,
ii) a bundle of 25 solid copper wires each of radius 1 mm.
For each conductor, estimate the resistance per unit length at a frequency of 1 MHz.
You may assume a conductivity of σ = 6.7 × 107 Ω−1 m−1 and that the skin depth,
δ, is given by r
2
δ= .
µ0 σω

[8 marks]

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3. a) Show, using the appropriate Maxwell equations, that charge is conserved in electro-
magnetism.
[5 marks]

b) i) In a region of space where the current density vector j = 0 and µr = 1 show that
magnetic field, B, satisfies
1 ∂ 2B
∇2 B = 2 2 .
c ∂t

[3 marks]

ii) If the magnetic field inp

this region is axially symmetric and independent of z then
B = g(r, t) θ̂, where r = x2 + y 2 . Show, for this field, that

∂ 2 Bθ 1 ∂Bθ Bθ 1 ∂ 2 Bθ
+ − 2 = 2 ,
∂r2 r ∂r r c ∂t2
where Bθ is the θ-component of the magnetic field.
[6 marks]

iii) If the magnetic field, for r ≤ s, is given by

µ0 f r
B= θ̂,
2πs2
where f and s are independent of position, show that the rate of change of the
magnetic field in the region r ≤ s is constant.
[4 marks]

iv) Using Ampère’s law, provide a physical interpretation of f and s and write down
an expression for the electric field in the region r ≤ s in terms of f and s.
[7 marks]

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4. a) Define the magnetic vector potential, A, and state its units. Show, that in static
situations, where the Coulomb gauge applies, that this vector satisfies Poisson’s
equation,
∇2 A = −µ0 j,
where j is the current density vector.
[6 marks]

b) (i) What are the boundary conditions that the magnetic field, B, and magnetic
intensity, H, must satisfy at the interface between two magnetic materials?
[4 marks]

(1)
(ii) Two materials form an interface at z=0. The relative permeability is µr = 1.5
(2)
for z > 0 and µr = 4.0 for z < 0. The magnetic field vector is given by

B = (5x̂ + 6ŷ + 8ẑ)T

in the region z < 0. What is the magnetic field vector in the region z > 0?
[6 marks]

c) A long iron rod, with a uniform magnetisation of 5000 Am−1 along its symmetry
axis, is formed into a ring of radius 10 cm. A section of width 1 mm is removed
from the ring as shown in the figure (leaving parallel faces). Calculate the values
of the magnetic energy density in the gap and in the iron. [Ignore any edge effects
and assume that the ring is in vacuum.]
[9 marks]

10 cm

1 mm

END OF EXAMINATION PAPER

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