Electromagnetism (2nd year): Problems
Problem sheet 0: (Summer vacation 2023)
At the start of the second year you will receive the second part of the Electromagnetism course.
This vacation work contains a set of problems that will enable you to revise the material covered
in the first year Electromagnetism course.
Some of the problems below are taken from:
Introduction to Electrodynamics, David J. Griffiths, 4th Edition
Electricity and Magnetism, Edward M. Purcell and David J. Morin, 3rd Edition.
(Problems first assembled by Prof. Caroline Terquem)
Electrostatics
0.1 Field and potential from charged ring
A thin ring of radius a carries a charge q uniformly distributed. Consider the ring to lie
in the x–y plane with its centre at the origin.
a) Find the electric field E at a point P on the z–axis.
b) Find the electric potential V at P .
c) A charge −q with mass m is released from rest far away along the axis. Calculate
its speed when it passes through the centre of the ring. (Assume that the ring is
fixed in place).
0.2 Field from charged disc
The ring in the previous problem is replaced by a thin disc of radius a carrying a charge
q uniformly distributed. Consider the disc to lie in the x–y plane with its centre at the
origin.
a) Find the electric field E at a point P on the z–axis.
b) Check that the values of E at z = 0 and in the limit z a are consistent with
expectations.
0.3 Hydrogen atom
According to quantum mechanics, the hydrogen atom in its ground state can be described
by a point charge +q (charge of the proton) surrounded by an electron cloud with a
charge density ρ(r) = −Ce−2r/a0 . Here a0 is the Bohr radius, 0.53 × 10−10 m, and C is
a constant.
a) Given that the total charge of the atom is zero, calculate C.
b) Calculate the electric field at a distance r from the nucleus.
1
c) Calculate the electric potential, V (r), at a distance r from the nucleus. We give:
−αr0
e−αr
Z
1 e 0
+ 1 dr = − .
αr0 r0 αr
0.4 Energy of a charged sphere
We consider a solid sphere of radius a and charge Q uniformly distributed.
a) Calculate the electric field E(r) and the electric potential V (r) at a distance r from
the centre of the sphere.
Find the energy U stored in the sphere three different ways:
b) Use the potential energy of the charge distribution due to the potential V (r):
Z
1
U= ρV dτ,
2 V
where ρ is the charge density and the integral is over the volume V of the sphere.
c) Use the energy stored in the field produced by the charge distribution:
0 E 2
Z
U= dτ,
space 2
where the integral is over all space.
d) Calculate the work necessary to assemble the sphere by bringing successively thin
charged layers at the surface.
0.5 Conductors
A metal sphere of radius R1 , carrying charge q, is surrounded by a thick concentric metal
shell of inner and outer radii R2 and R3 . The shell carries no net charge.
a) Find the surface charge densities at R1 , R2 and R3 .
b) Find the potential at the centre, choosing V = 0 at infinity.
c) Now the outer surface is grounded. Explain how that modifies the charge distribu-
tion. How do the answers to questions (a) and (b) change?
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Magnetostatics
0.6 Force on a loop
A long thin wire carries a current
I1 in the positive z–direction along
the axis of a cylindrical co-ordinate
system. A thin, rectangular loop of
wire lies in a plane containing the
axis, as represented on the figure.
The loop carries a current I2 .
a) Find the magnetic field due
to the long thin wire as a
function of distance r from
the axis.
b) Find the vector force on each
side of the loop which results
from this magnetic field.
c) Find the resultant force on
the loop.
0.7 Magnetic field in off–centre hole
A cylindrical rod carries a uniform current den-
sity J. A cylindrical cavity with an arbitrary
radius is hollowed out from the rod at an arbi-
trary location. The axes of the rod and cavity
are parallel. A cross section is shown on the
figure. The points O and O0 are on the axes of
the rod and cavity, respectively, and we note
a = OO0 .
a) Show that the field inside a solid cylinder can be written as B = (µ0 J/2)ẑ×r,
where ẑ is the unit vector along the axis and r is the position vector measured
perpendicularly to the axis.
b) Show that the magnetic field inside the cylindrical cavity is uniform (in both mag-
nitude and direction).
3
0.8 Magnetic field at the centre of a sphere
A spherical shell with radius a and uniform surface charge density σ spins with angular
frequency ω around a diameter. Find the magnetic field at the centre.
0.9 Motion of a charged particle in a magnetic field
A long thin wire carries a current I in the positive z–direction along the axis of a
cylindrical co-ordinate system. A particle of charge q and mass m moves in the magnetic
field produced by this wire. We will neglect the gravitational force acting on the particle
as it is very small compared to the magnetic force.
a) Is the kinetic energy of the particle a constant of motion?
b) Find the force F on the particle, in cylindrical coordinates.
c) Obtain the equation of motion, F = mdv/dt, in cylindrical coordinates for the
particle.
d) Suppose the velocity in the z–direction is constant. Describe the motion.
Electromagnetic induction
0.10 Growing current in a solenoid
An infinite solenoid has radius a and n turns per
unit length. The current grows linearly with time,
according to I(t) = kt, k > 0. The solenoid is
looped by a circular wire of radius r, coaxial with
it. We recall that the magnetic field due to the
current in the solenoid is B = µ0 nI inside the
solenoid and zero outside.
a) Without doing any calculation, explain
which way the current induced in the loop
flows.
b) Use the Hintegral form of Faraday’s law,
which is E · dl = −dΦ/dt, to find the elec-
tric field in the loop for both r < a and
r > a. Check that the orientation of E
agrees with the answer to question (a).
c) Verify that your result satisfies the local
form of the law, ∇×E = −∂B/∂t.
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Maxwell’s equations
0.11 Energy flow into a capacitor
A capacitor has circular plates with ra-
dius a and is being charged by a constant
current I. The separation of the plates
is w a. Assume that the current flows
out over the plates through thin wires
that connect to the centre of the plates,
and in such a way that the surface charge
density σ is uniform, at any given time,
and is zero at t = 0.
a) Find the electric field between the plates as a function of t.
b) Consider the circle of radius r < a shown on the figure (and centered on the axis of
the capacitor). Using the integral form of Maxwell’s equation ∇×B = 0 µ0 ∂E/∂t
over the surface delimited by the circle, find the magnetic field at a distance r from
the axis of the capacitor.
c) Find the energy density u and the Poynting vector S in the gap. Check that the
relation:
∂u
= −∇ · S,
∂t
is satisfied.
d) Consider a cylinder of radius b < a and length w inside the gap. Determine the
total energy in the cylinder, as a function of time. Calculate the total power flowing
into the cylinder, by integrating the Poynting vector S over the appropriate surface.
Check that the power input is equal to the rate of increase of energy in the cylinder.
e) When b = a, and assuming that we can still neglect edge effects in that case, check
that the total power flowing into the capacitor is:
d 1
QV ,
dt 2
where V is the voltage across the capacitor (since QV /2 is the energy stored in the
electric field in the capacitor).
