A physical dipole consists of point charges and at and respectively

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 physical dipole consists of point charges

+q

and

-q

z=+d/2

and

z=-d/2

respectively, with

q = 4.0\times10^{-9}\,\mathrm{C}

and

d = 1.0\,\mathrm{mm}

. Let

a = 4.0\,\mathrm{cm}

(so

a \gg d/2

) and consider an imaginary sphere of radius

a

centred at the origin, enclosing both charges. ((a)) Using the result that the field of a single point charge

q'

at position

\mathbf{r}'

inside a ball of radius

a

, integrated over the ball's volume, equals

-q'\mathbf{r}'/3\varepsilon_0

regardless of

a

, find

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

for this dipole in terms of

p=qd

, and give its numerical value. ((b)) Divide by the ball's volume to get the average field

\langle\mathbf{E}\rangle

inside the sphere, and evaluate it numerically. ((c)) A uniformly polarised sphere of the same radius

a

, carrying total dipole moment

p

spread uniformly through its volume as

P = p/V

, has interior field

-P/3\varepsilon_0

(a result derived from matching

V

and its normal derivative at

r=a

). Show algebraically that this equals

\langle\mathbf{E}\rangle

found in part (b).