A spherical capacitor consists of a solid metal ball of radius inside a concentric…

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Electrostatic fields and potential · Conductors, capacitance and boundary-value electrostatics

Electricity, Magnetism and Electromagnetism, Ch Conductors, screening and capacitance

Problem

A spherical capacitor consists of a solid metal ball of radius

a

inside a concentric metal shell whose inner radius is

b

, with

b>a

. The gap between them is evacuated. The ball carries charge

+Q

and the shell carries

-Q

on its inner surface, and both conductors are in electrostatic equilibrium. ((a)) Starting from Gauss's law on a concentric spherical surface of radius

r

with

a<r<b

, and then integrating the field along a radial path, derive the capacitance

C

of this pair of conductors in terms of

\varepsilon_0

a

and

b

. ((b)) Show that as

b\to\infty

your result gives the capacitance of an isolated sphere of radius

a

, and write that capacitance down. ((c)) Evaluate

C

for

a=8.0\,\mathrm{cm}

and

b=10.0\,\mathrm{cm}

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