Starting from the spherical-coordinate Laplacian for an azimuthally symmetric…

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

The Boundary-Value Problem, Ch Separation of variables in spherical geometry

Problem

Starting from the spherical-coordinate Laplacian for an azimuthally symmetric potential V(r,θ)V(r,\theta) (no ϕ\phi dependence), ∇2V=1r2∂∂r(r2∂V∂r)+1r2sin⁡θ∂∂θ(sin⁡θ∂V∂θ)=0,\nabla^2V = \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial V}{\partial r}\right) + \frac{1}{r^2\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial V}{\partial\theta}\right) = 0, ((a)) substitute V(r,θ)=R(r)Θ(θ)V(r,\theta) = R(r)\Theta(\theta) and separate the equation into a radial and an angular ordinary differential equation, calling the separation constant ℓ(ℓ+1)\ell(\ell+1); ((b)) substitute R(r)=rnR(r) = r^n into the radial equation and find the two allowed values of nn in terms of ℓ\ell; ((c)) verify that V=Crcos⁡θV = Cr\cos\theta (with CC a constant) satisfies both separated equations, and state the corresponding value of ℓ\ell.

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