A sphere of radius is uniformly polarised with ,

PHYSICS · Subjective · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Conductors, capacitance and boundary-value electrostatics · Electric fields in matter

The Boundary-Value Problem, Ch Polarisation and bound charge, completed

Problem

A sphere of radius

a = 3.0\,\mathrm{cm}

is uniformly polarised with

\mathbf{P} = P\hat{\mathbf{z}}

P = 2.0\times10^{-6}\,\mathrm{C\,m^{-2}}

. There is no free charge anywhere. (a) Starting from the fact that the bound surface charge is

\sigma_b = P\cos\theta

and the bound volume charge vanishes, argue that the interior and exterior potentials must each contain only the

\ell = 1

Legendre term, so that

V_{\rm in} = Ar\cos\theta

and

V_{\rm out} = B\cos\theta/r^2

. (b) Impose continuity of

V

r=a

and the interface condition

\varepsilon_0(\partial V/\partial n)_{\rm out} - \varepsilon_0(\partial V/\partial n)_{\rm in} = -\sigma_b

(with

\hat{\mathbf{n}}

pointing outward from the sphere) to find

A

and

B

. (c) Give the numerical value of

\mathbf{E}_{\rm in}

. (d) Show that

B

corresponds to a point dipole of moment

p = \tfrac43\pi a^3 P

sitting at the centre, give its numerical value, and check that

\mathbf{E}_{\rm in}

times the sphere's volume equals

-p/3\varepsilon_0

Browse the problem library