A spherical electrode of radius is divided into many narrow

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Partial differential equations and boundary-value problems · Special functions and orthogonal polynomials

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

Problem

A spherical electrode of radius a=0.050 ma = 0.050\,\mathrm{m} is divided into many narrow, mutually insulated rings about a fixed axis, and an external control circuit holds them at the surface potential φ(a,θ)=V0+V2P2(cos⁡θ),\varphi(a,\theta) = V_0 + V_2P_2(\cos\theta), where θ\theta is the polar angle from that axis, V0=100 VV_0 = 100\,\mathrm{V}, V2=40 VV_2 = 40\,\mathrm{V}, and P2(cos⁡θ)=12(3cos⁡2θ−1)P_2(\cos\theta) = \tfrac{1}{2}(3\cos^2\theta - 1). There is no charge outside the sphere and the potential vanishes as r→∞r\to\infty. (a) Explain why, of all terms Aℓr−(ℓ+1)Pℓ(cos⁡θ)A_\ell r^{-(\ell+1)}P_\ell(\cos\theta) that solve Laplace's equation and vanish at infinity, only ℓ=0\ell=0 and ℓ=2\ell=2 can appear outside the sphere. (b) Match the general exterior solution to the stated surface potential at r=ar=a and find A0A_0 and A2A_2. (c) Give φ(r,θ)\varphi(r,\theta) for r≥ar\ge a and evaluate it at r=0.20 mr = 0.20\,\mathrm{m}, θ=π/3\theta = \pi/3. (d) Find the radial field Er=−∂φ/∂rE_r = -\partial\varphi/\partial r just outside the sphere at the pole (θ=0\theta=0, r=ar=a).

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