A small spherical grain of a ferroelectric ceramic

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Electric fields in matter · Electric fields in matter

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

Problem

A small spherical grain of a ferroelectric ceramic, radius

a = 3.0\,\mathrm{mm}

, carries a frozen-in, uniform polarisation

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

with

P = 2.0\times10^{-5}\,\mathrm{C\,m^{-2}}

. The grain sits in vacuum, carries no free charge anywhere, and its polarisation is unaffected by the field it produces. ((a)) Starting from the potential of a continuous distribution of dipoles,

V(\mathbf{r}) = \frac{1}{4\pi\varepsilon_0}\int_{\tau'} \frac{\mathbf{P}(\mathbf{r}')\cdot(\mathbf{r}-\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|^3}\,d\tau',

integrate by parts to show this is the potential of a surface charge

\sigma_b = \mathbf{P}\cdot\hat{\mathbf{n}}

and a volume charge

\rho_b = -\nabla\cdot\mathbf{P}

, and evaluate both for the grain. ((b)) Because

\sigma_b

varies as

\cos\theta

and

\rho_b = 0

, write

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

and

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

. Use continuity of

V

and the jump condition

\varepsilon_0\!\left[\left(\frac{\partial V}{\partial r}\right)_{\rm out} - \left(\frac{\partial V}{\partial r}\right)_{\rm in}\right]_{r=a} = -\sigma_b

(normal pointing from the grain's interior to the vacuum outside) to find

A

and

B

. ((c)) Show that

\mathbf{E}_{\rm in}

is uniform and that

4\pi\varepsilon_0 B

equals the grain's total dipole moment

p = \tfrac43\pi a^3 P

. ((d)) Evaluate

\int_{\rm sphere}\mathbf{E}\,d\tau

directly from

\mathbf{E}_{\rm in}

and the grain's volume, show it equals

-\mathbf{p}/3\varepsilon_0

, and state what this proves about the coefficient of the point term in the field of an ideal dipole. Give numerical values for

\mathbf{E}_{\rm in}

p

, and

\int\mathbf{E}\,d\tau

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