PHY113: Classical Electrodynamics (2026–27 – Semester I)
Assignment 5
1. Cavities in a polarised material. A large piece of material is uniformly polarised, P = Pẑ, and the electric
field in it is Emat . A small empty cavity is cut at its centre. Assume that the polarisation in the remaining
material stays equal to P.
P
P
(i) Slender cavity
P
(ii) Spherical cavity
(iii) Thin cavity
Figure 1. The cavity axis (or disc normal) is parallel to P.
(a) Imagine filling the cavity with a plug having the same shape and polarisation. Use superposition to
relate Emat , the field Ecav in the cavity, and the field Eplug of the plug alone.
(b) Find Ecav for the three shapes in Figure 1.
(c) Which shapes give Emat and D/ε0 ? Why does the field inside an empty cavity depend on its shape?
Hint. For the three polarised plugs, Eplug = −NP/ε0 , with N → 0, N = 31 , and N → 1, respectively.
1
2. Charge near an earthed sphere. A point charge q is a distance d > R from the centre of an earthed
conducting sphere of radius R. Taking the outward direction as positive, the force when the charge is at r is
F(r) = −
1
q2 Rr
.
4πε0 (r2 − R2 )2
earthed sphere
q′
q
O
R
d
Figure 2. Charge q and its image for an earthed sphere.
(a) Find the work W done by an external agent as q is brought slowly from infinity to r = d. Take U(∞) = 0.
(b) At r = d, the image charge is q′ = −qR/d at R2 /d. Calculate qVimage and compare it with W .
(c) Put d = R + a and take R → ∞. Recover the result for an earthed plane.
(d) Why is qVimage not the energy of the physical system?
Hint. Use u = r2 − R2 in part (a).
2
3. A dielectric sphere in a uniform field. A dielectric sphere of radius R and relative permittivity εr is in
vacuum. The applied field is E0 ẑ. Assume that the polarisation Pẑ and the field Ein ẑ inside the sphere are
uniform.
E0 ẑ
P
R
εr
Figure 3. A dielectric sphere in a uniform applied field.
(a) Write Ein as the sum of the applied field and the field produced by the bound charge.
(b) Write the linear relation between P and Ein .
(c) Solve for Ein and P. Hence find the dipole moment p.
(d) Compare your result with the boundary-condition method.
(e) Why is the assumption of uniform P justified?
Hint. Use χe = εr − 1 and the interior field of a uniformly polarised sphere.
3