A spherical electrode of radius is made of many narrow metal bands running round the axis

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

A spherical electrode of radius

a

is made of many narrow metal bands running round the

z

axis, insulated from one another, each held by a control circuit at its own fixed potential, so that the potential on the sphere

r=a

V(a,\theta) = V_0\sin^2\theta,

with

V_0

a constant. The region of interest is the whole of space outside the sphere,

r > a

, which is charge-free and extends to infinity where the potential tends to zero. ((a)) Before doing any integral, state which values of

\ell

appear in the Legendre series for the potential outside the sphere, giving the reason from the form of the boundary data. ((b)) Before writing the solution, state which of

r^\ell

and

r^{-\ell-1}

must be kept for the exterior region, giving the reason.

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