Someone argues: "A hollow

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

The Boundary-Value Problem, Ch Separation of variables in spherical geometry

Problem

Someone argues: "A hollow, charge-free spherical cavity of radius

a

has boundary data

V(a,\theta) = V_0\cos\theta

on its wall. The general solution has both

r^\ell

and

r^{-\ell-1}

terms for every region, so inside the cavity the potential must be

V(r,\theta) = \Big(A_1r + \frac{B_1}{r^2}\Big)\cos\theta,

with both

A_1

and

B_1

fixed by the single condition at

r=a

together with some extra condition at the centre." ((a)) Find the error in this reasoning. ((b)) Give the correct potential inside the cavity for

r < a

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