UNIVERSITY OF SOUTHAMPTON
PHYS2001W1
SEMESTER 2 EXAMINATION 2010/11
ELECTROMAGNETISM
Duration: 120 MINS
VERY IMPORTANT NOTE
Section A answers MUST BE in a separate blue answer book. If any blue
answer booklets contain work for both Section A and B questions - the
latter set of answers WILL NOT BE MARKED.
Answer all questions in Section A and two and only two questions in
Section B.
Section A carries 1/3 of the total marks for the exam paper and you should aim
to spend about 40 mins on it. Section B carries 2/3 of the total marks for the
exam paper and you should aim to spend about 80 mins on it.
A Sheet of Physical Constants will be provided with this examination paper.
An outline marking scheme is shown in brackets to the right of each question.
Only university approved calculators may be used.
Second derivative relation:
∇ × (∇ × A) = ∇(∇.A) − ∇2 A
Number of
c University of Southampton
Copyright 2010 Pages 7
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PHYS2001W1
Section A
A1. Suppose a static electric field E(x, y, z) has components E x = x, Ey = y, Ez = z.
Using Gauss’s law in differential form calculate the charge density ρ.
[2]
Show that a static magnetic field B(x, y, z) with Bx = x, By = y, Bz = z would be
impossible.
[2]
A2. Consider a static magnetic field B(x, y, z) with Bx = y, By = z, Bz = x.
Using Ampère’s law in differential form calculate the current density J .
[3]
Show that a static electromagnetic field whose electric field E(x, y, z) has
components E x = y, Ey = z, Ez = x would be impossible.
[2]
A3. A long straight solenoid of radius a aligned along the x-axis is driven by an
alternating current so that the magnetic field inside is B(t) = B0 x̂ sin(ωt), where
t is time. Using Faraday’s law in differential form calculate the curl of the induced
electric field E(t) in the vacuum inside the solenoid.
[1]
Using Faraday’s law in integral form calculate the electromotive force E induced
around a circular loop of radius a/2 placed inside the solenoid and co-axial with
it.
[2]
A4. Calculate the magnitude of the electric field in the air gap of a parallel plate
capacitor of capacitance C with plates of area A separated by a distance d with
electric charge on the plates given by ±Q(t) where Q(t) = Q0 sin(ωt) where t is
time.
[2]
If the plates are aligned perpendicular to the x-axis, calculate the curl of the
induced magnetic field B(t) between the plates.
[2]
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PHYS2001W1
A5. Write down the real electric and magnetic fields for a cosine plane wave of
electric amplitude E0 , angular frequency ω, wave number k and phase angle
zero that is travelling in the positive z direction and polarised in the x direction.
[2]
Calculate the Poynting vector S and the electromagnetic energy density uem of
the electromagnetic wave.
[2]
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PHYS2001W1
Section B
B1.
(a) Starting from the integral form of Gauss’s law derive the differential form.
[Note Gauss’s divergence theorem
RR
S closed E.dA =
RRR
V
∇.E dV ].
[5]
(b) Consider a charged sphere of radius R centred at the origin with the
spherically symmetric charge density ρ(r) = ρ0 (r4 /R4 ) where ρ0 is a
constant and r is the radial coordinate.
Find the charge dQ0 contained in a spherical shell of radius r0 < R and
infinitesimal thickness dr0 .
[3]
Hence find the charge Q(r) contained inside the sphere as a function of
r < R.
[2]
Using Gauss’s law in integral form, determine the magnitude of the radial
electric field Er (r) inside the sphere as a function of r < R.
Using the fact that ∇.E(r) =
1 ∂ 2
r2 ∂r (r E r (r))
[2]
in this case, verify Gauss’s law in
differential form at a general point inside the sphere.
[2]
(c) A positive point charge Q is added to a point P0 with position vector R0 with
components (1, 1, 1) inside a sphere with the same charge density as in part
(b) apart from the charge Q. Calculate the vector electric field E(r) at a point
P with position vector r with components (1, 2, 1) inside the sphere in terms
of the unit vectors x̂, ŷ, ẑ and Q and Er (considered in part (b)).
[6]
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B2.
PHYS2001W1
(a) The Biot-Savart law below gives the differential magnetic field at a point
described by a displacement vector r due to a current element at another
point described by a displacement vector R0 :
dB(r) =
µ0 IdL × r̂0
.
4π r02
Indicate on a diagram the vectors r, R0 and r0 . Express r0 , r0 and rˆ0 in terms
of r and R0 .
[5]
(b) Draw a diagram in the (x, y) plane to illustrate the application of the BiotSavart law to the case of a differential magnetic field dB(r) at a point on the
x-axis described by the vector r = a x̂ due to an element of wire at a point
R0 = yŷ running along the y-axis and carrying a current I in the negative
y-direction. Label the angle θ which satisfies tan θ = y/a.
[3]
Using the Biot-Savart law show that:
dB(r) =
µ0 aIdyẑ
.
4π (a2 + y2 )3/2
[3]
Hence show that
dB(r) =
µ0 I cos θdθẑ
.
4π
a
[3]
(c) Using the above results, with the aid of a diagram and symmetry arguments,
calculate the magnetic field at the centre of a square loop of wire whose
corners have (x, y) coordinates (0, a), (0, −a), (2a, a), (2a, −a), and which
carries a current I anticlockwise about the z axis.
[6]
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B3.
PHYS2001W1
(a) Using the time-dependent Maxwell’s equations for the divergence of the
magnetic field and the curl of the electric field, show that the magnetic and
electric fields can be written in terms of the magnetic vector potential A and
the scalar potential V as:
B = ∇ × A,
E = −∇V −
∂A
.
∂t
[4]
(b) Show that the electric and magnetic fields are invariant under the gauge
transformations of the potentials,
A → A + ∇φ, V → V −
∂φ
∂t
for any arbitrary function φ = φ(r, t).
[4]
(c) Using the time-dependent Maxwell’s equations for the divergence of the
electric field and the curl of the magnetic field, show that the equations
satisfied by V and A in the presence of a charge density ρ and a current
density J are:
∂
1
−∇2 V − (∇.A) = ρ
∂t
0
∂2 A 2
∂V
µ0 0 2 −∇ A + ∇ ∇.A + µ0 0
= µ0 J.
∂t
∂t
[4]
Hence find the equations satisfied by V and A in Lorentz gauge.
[3]
Find the equations satisfied by V and A in Coulomb gauge.
[3]
Calculate the charge density for the potentials V = −q/(0 x) and A = Ct in
Coulomb gauge, where q and C are constant.
[2]
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B4.
PHYS2001W1
(a) Show that Maxwell’s equations in an infinite, linear, isotropic, homogeneous,
dielectric medium, in the absence of external charge and current, imply that
the electric and magnetic fields E and B satisfy wave equations.
[6]
Express the speed of the wave v in terms of the permittivity and
permeability µ of the medium, and show how for many materials the
refractive index n is approximately related to the dielectric constant.
[2]
(b) Show that the complex wave functions
Ẽ(x, y, z, t) = E0 ei(k.r−ωt+δ)
B̃(x, y, z, t) = B0 ei(k.r−ωt+δ)
describe electromagnetic waves moving along k with speed v = ω/k.
[4]
Show, for the electric field component, that the waves must be transverse.
[2]
(c) Explain the physical interpretation of the conventionally defined fields D and
H using a capacitor with a dielectric and a solenoid with an iron core as
examples, relating them to the corresponding vacuum fields E0 and B0 in
each case.
[2]
Write down Maxwell’s equations in a finite, non-linear, non-isotropic, nonhomogeneous medium in the presence of an external charge density ρ and
an external current density J .
[2]
Give the relations between D and E and between H and B for the case of
an infinite, linear, isotropic, homogeneous, dielectric medium as in part (a).
END OF PAPER
[2]