The potential on the surface of a hollow spherical shell of radius is held at V(a, )…

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

The potential on the surface of a hollow spherical shell of radius

a

is held at

V(a,\theta) = V_0(5\cos^3\theta - 3\cos\theta),

with

V_0

a constant. The interior of the shell (

r < a

) contains no charge. ((a)) Before doing any integral, state which values of

\ell

appear in the Legendre series for the potential inside the shell, 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 interior region, giving the reason.

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