Archive for Laplace’s Equation

A Potential Problem with a Sphere

Posted in Cute Problems with tags , , on March 9, 2016 by telescoper

Busy busy busy again today so I thought I’d post a quick entry to the cute problems folder. I set this as a problem to my second-year Theoretical Physics students recently, which is appropriate because I encountered it when I was a second-year student at Cambridge many moons ago!

Sphere_problem

HINT: You can solve this by finding the general solution for the potential at any point inside the sphere, but that isn’t the smart way to do it!

FURTHER HINT: The question asks for the Electric Field at the origin. What terms in the solution for the potential can contribute to this?

Answers through the comments box please!

OUTLINE SOLUTION: A numerically correct answer has now been posted so I’ll give an outline solution. The potential V inside the sphere is governed by Laplace’s equation, the general solution of which is a series expansion in powers of r and Legendre polynomials, i.e. rn Pn(θ). The coefficients of this expansion can be determined for the given boundary conditions (V=V0 at r=a for θ = +1, V=0 for cos θ = -1). However this is a lot more work than necessary. The question asks for the electric field, i.e. the gradient of the potential, and if you look at the form of the potential there is only one term that can possibly contribute to the field at r=0, namely the one involving rP1(cosθ) =rcos θ (which is actually z). Any higher power of r would give a derivative that vanishes at the origin. Hence we just have to determine the coefficient of one term. Using the orthogonality properties of the Legendre polynomials this can easily be seen to be 3V0/4a. The electric field is thus -3V0/4a in the z-direction, i.e. vertically downwards from the top of the sphere.