PY3103 - PROBLEM SET 3
Due Monday 24 October, 2016
1) Consider two magnetic media with permeability µ1 and µ2 , separated
by a straight interface. The magnetic field in medium 1 makes an angle
θ1 to the normal to the interface and the corresponding magnetic field in
medium 2 makes an angle θ2 to the normal to the interface. By matching
boundary conditions, show that these angles are given by tan θ2 / tan θ1 =
µ2 /µ1 , assuming there is no free current at the interface.
2) Show that the electric field can be written in terms of the magnetic vector
potential as
∂A
− ∇V
∂t
where V is the electric (scalar) potential.
E = −
3) A pair of parallel wires separated by a distance x carries equal currents
I in opposite directions, with I increasing at the rate dI/dt. A rectangular
loop lies in the plane of the two wires, with length l parallel to the wires
and width w perpendicular to the wires. The closer side of the loop is a
distance a from the nearer of the two wires. Find the emf in the loop. (b)
Find the emf in the loop from the change in the vector potential using the
equation given above.
4) Two parallel rectangular loops lying in the same plane have lengths l1
and l2 and widths w1 and w2 , with l2 ≪ l1 . The loops lie with their long
sides parallel and do not overlap, and the distance between the near sides
is s. Show that the mutual inductance between the loops is
M =
µo l 2
s + w2
ln 2π
s 1 + w2
s+w1
5) The long co-axial line shown with radii a, b and c carries a current I
in opposite directions in the inner and outer conductors. The currents are
distributed uniformly in each region. (a) Find the magnetic energies per
unit length of the line in regions 1, 2, 3 and 4. (b) Find the self-inductance
per unit length L of the line.