Question

2. Potential Inside a Sphere We are interested in the electric potential inside a spherical shell that is radius a and centered on the origin. There are no charges inside the she, so the potential satisfies the Laplace equation, However, there is an external voltage applied to the surface of the shell which holds the potential on the surface to a value which depends on θ: As a result, the potential Ф(r,0) -by symmetry, it does not depend on ф-is non-zero inside the spherical she, despite the lack of electric charges. A. Write the solution using separtion of variables as Φ(r,0) R(r)F(9), and plug this into the Laplace equation in spherical coordinates. You should get a tangled up differential equation in terms of R(r) and F(0) (and their first and second derivatives) B. Show that a solution with: R(r) = Arn (where A is an unknown coefficient and n is an integer) leads to an equation for F(cs) which is the Legendre equation C. Using this fact, the general solution in terms of Legendre polynomials, cluding an infinite number of undetermined coefficients (corresponding to all of the infinite number of possible values of n), is given by, Apply the boundary conditions to fix the undetermined coefficients An in this general solution HINT: This last step is much easier to do if you remember the properties of the Legendre polynomials, including their orthogonality!

Answer 1

[A] Delta^2 phi = 0: phi(r, theta) In spherical polar coordinates- d^2 phi/partial differential r^2 + 2/r partial differential phi/partial differential r + 1/r^2 partial differential^2 phi/partial differential theta^2 + cot theta/r^2 partial differential phi/ partial differential theta = 0 Let phi(r, theta) = R(r) F(theta) F d^2 R/dr^2 + 2F/r dR/dr + R/r^2 d^2 F/d theta^2 + cot theta R/r^2 dF/d theta = 0 rightwardsdoublearrow 1/R d^2 R/dr^2 + 2/rR dR/dr + 1/r^2 F d^2 F/d theta^2 + cot theta/r^2 F dF/d theta = 0 rightwardsdoublearrow 1/R d^2 R/dr^2 + 2/R dR/dr = -1/r^2 F d^2 F/d theta^2 - cot theta/r^2 F dF/d theta rightwardsdoublearrow r^2(1/R d^2 R/dr^2 + 2/rR dR/dr) = -1/F d^2 F/d theta^2 - cot theta/F dF/d theta Since LHS is a function of 'r' only and RHS is a function of 'theta' alone so they can be equated to some constant. Therefore r^2 (1/R d^2 R/dr^2 + 2/rR dR/dr) = -1/F d^2 F/d theta^2 - cot theta/F dF/d theta = k constant)