A spherical electrode of radius is divided into insulated segments and driven by…

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

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

Problem

A spherical electrode of radius a=0.10 ma=0.10\,\mathrm m is divided into insulated segments and driven by external circuitry so that its surface is held at the axially symmetric potential φ(a,θ)=50 V+(30 V)cos⁡2θ,\varphi(a,\theta)=50\,\mathrm V+(30\,\mathrm V)\cos^2\theta, where θ\theta is the polar angle from the +z+z axis. There is no charge outside the sphere, and φ→0\varphi\to0 as r→∞r\to\infty. ((a)) Write cos⁡2θ\cos^2\theta as a combination of the Legendre polynomials P0(cos⁡θ)=1P_0(\cos\theta)=1 and P2(cos⁡θ)=12(3cos⁡2θ−1)P_2(\cos\theta)=\tfrac12(3\cos^2\theta-1), and hence express φ(a,θ)\varphi(a,\theta) as C0P0(cos⁡θ)+C2P2(cos⁡θ)C_0P_0(\cos\theta)+C_2P_2(\cos\theta). ((b)) Write the general solution of Laplace's equation outside the sphere that stays finite as r→∞r\to\infty, match it to the surface data, and give φ(r,θ)\varphi(r,\theta) for r≥ar\ge a. ((c)) Evaluate φ\varphi in volts at r=0.20 mr=0.20\,\mathrm m, at the pole (θ=0\theta=0) and at the equator (θ=π/2\theta=\pi/2).

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