1. Math
  2. Advanced Math
  3. b3 consider a threedimensional potential u which obeys the laplace...

Question: b3 consider a threedimensional potential u which obeys the laplace...

Question details

B3. Consider a three-dimensional potential U which obeys the Laplace equation V2U 0. In spherical polar coordinates (r, θ, φ), and assuming that the potential does not depend on ф, the Laplace equation can be written as (i) We can solve the Laplace equation using a trial function U(r, θ)-R(r)0(9) Show that Rdr dr and O du where μ cos θ and k is a constant. (ii) You may assume that the differential equation for Θ has valid solutions given by the Legendre polynomials P(u) only when k-(C 1), where e- 0,1,2,3, are positive integers. Show that R oc rA is a valid solution, and determine the two possible values of A in terms of the numbers £. (iii) Suppose that the angular dependence of the potential on the surface of the unit sphere is given by 2 Using your result from (ii), find the potential both inside and outside the sphere (no sources) assuming that R(r 1)1 Hints: Recall the second Legendre polynomial P2(1) (3/2 - 1, and

Solution by an expert tutor
Blurred Solution
This question has been solved
Subscribe to see this solution