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LAPLACE’S EQUATION IN SPHERICAL COORDINATES

LAPLACE S EQUATION IN SPHERICAL COORDINATES With Applications to Electrodynamics We have seen that Laplace s EQUATION is one of the most significant equations in physics. It is the solution to problems in a wide variety of fields including thermodynamics and electrodynamics. In your careers as physics students and scientists, you will encounter this EQUATION in a variety of contexts. It is important to know how to solve Laplace s EQUATION in various coordinate systems. The coordinate systems you will encounter most frequently are Cartesian, cylindrical and SPHERICAL polar. We investigated Laplace s EQUATION in Cartesian COORDINATES in class and just began investigating its solution in SPHERICAL COORDINATES . Let s expand that discussion here. We begin with Laplace s EQUATION : 02= V (1) We can write the Laplacian in SPHERICAL COORDINATES as: )(sin1)(sinsin1)(122222222 + + = VrVrrVrrrV (2) where is the polar angle measured down from the north pole, and is the azimuthal angle, analogous to longitude in earth measuring COORDINATES .

Legendre polynomials is possible since we have learned that Legendre polynomials are a complete set of orthogonal functions on (-1, 1). Thus, we can expand any function f(x) on ( …

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The coordinate systems you will encounter most frequently are Cartesian, cylindrical and spherical polar. We investigated Laplace’s equation in Cartesian coordinates in class and just began investigating its solution in spherical coordinates. Let’s expand that discussion here. We begin with Laplace’s equation: 2V. ∇ = 0 (1)

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