Solution of Laplaces Equation in Spherical coordinates by

Solution of Laplace’s Equation in Spherical coordinates by Separation of Variables Laplace’s equation In spherical coordinates it takes the form 2

Solution of Laplace’s Equation in Spherical coordinates by Separation of Variables We shall consider potentials which has azimuthal symmetry, so that V is independent of the angle Laplace’s equation 3

Solution of Laplace’s Equation in Spherical coordinates by Separation of Variables 4

Solution of Laplace’s Equation in Spherical coordinates by Separation of Variables This part depends on r only This part depends on only Each part is a constant 5

Solution of Laplace’s Equation in Spherical coordinates by Separation of Variables 6

Solution of Laplace’s Equation in Spherical coordinates by Separation of Variables 7

Solution of Laplace’s Equation in Spherical coordinates by Separation of Variables The only acceptable solution here is when is integer. That why we took this fancy form for the constant. The solutions are Legendre polynomials in the variable cos. I shall discus Legendre polynomials soon. 8

Solution of Laplace’s Equation in Spherical coordinates by Separation of Variables There is no need to include the constant c here because it is absorbed into A and B. 9

Solution of Laplace’s Equation in Spherical coordinates by Separation of Variables Separation of variables an infinite set of solutions, one for each . The general solution is the linear combination of solutions. 10