A thin spherical shell of radius encloses a charge-free region

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Partial differential equations and boundary-value problems · Partial differential equations and boundary-value problems

Mathematical Methods for Physics, Ch Separation of variables in Cartesian, cylindrical and spherical geometry

Problem

A thin spherical shell of radius a=0.050 ma=0.050\,\mathrm m encloses a charge-free region. External circuitry holds the shell's surface at the axially symmetric potential φ(a,θ)=100 V+(40 V)cos⁡θ,\varphi(a,\theta)=100\,\mathrm V+(40\,\mathrm V)\cos\theta, where θ\theta is measured from the +z+z axis. There is no charge anywhere inside the shell. ((a)) Write the general solution of Laplace's equation inside the sphere that stays finite at r=0r=0, using P0(cos⁡θ)=1P_0(\cos\theta)=1 and P1(cos⁡θ)=cos⁡θP_1(\cos\theta)=\cos\theta, and match it to the surface data to find φ(r,θ)\varphi(r,\theta) for r≤ar\le a. ((b)) Show that the interior solution is a uniform field superposed on a constant, and give the field's magnitude and direction. ((c)) State the potential at the centre of the shell.

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