A spherical capacitor in vacuum consists of two concentric conducting shells: an…

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

The Boundary-Value Problem, Ch The field as a differential equation: Poisson, Laplace and the well-posed problem

Problem

A spherical capacitor in vacuum consists of two concentric conducting shells: an inner shell of radius

a = 3.0\,\mathrm{cm}

held at potential

V_0 = 90\,\mathrm{V}

, and an outer shell of radius

b = 9.0\,\mathrm{cm}

held at

V = 0

. The region between the shells,

a \le r \le b

, contains no free charge. A proposed potential there is

V(r) = V_0\,\frac{\dfrac{1}{r}-\dfrac{1}{b}}{\dfrac{1}{a}-\dfrac{1}{b}}.

((a)) Show that

V(r)

satisfies Laplace's equation throughout

a < r < b

. ((b)) Check that

V(r)

takes the stated value on each shell. ((c)) The mean-value property forbids

V

from having an interior maximum or minimum in this charge-free region. Compute

dV/dr

and confirm it is never zero for

a < r < b

, so the extreme values of

V

occur only on the two shells, as the property requires.

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