Solution of Laplace’s Equation in Spherical coordinates by Separation of Variables إعداد د. جمال بن حمزة مدني قسم الفيزياء – جامعة الملك عبد العزيز جدة 1

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

Laplace’s equation 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  3

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

This part depends on r only This part depends on  only 5

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

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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 variable variables an infinite set of solutions, one for each. The general solution is the linear combination of solutions. 10

Solution of Laplace’s Equation in Spherical coordinates by Separation of Variables azimuthal symmetry 11

Legendre Polynomials These are the solution of the differential equation 12

Legendre Polynomials We can produce any of P by using Rodrigues formula 13

Legendre Polynomials Example : calculate P 3 14

Legendre Polynomials Example : calculate P 3 15

Legendre Polynomials complete orthogonal 16

Legendre Polynomials orthogonal 17

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Let 22

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تمرين مساعدة اكتب x 2 كمجموع من P 0 و P 2 نهاية المحاضرة 26