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ScanSolve AI Answer
Here are Poisson's and Laplace's equations in spherical coordinates: The Laplacian operator ^2 in spherical coordinates (r, , ) is given by: ^2 V = (1)/(r^2) ()/( r) ( r^2 ( V)/( r) ) + (1)/(r^2 ) ()/( ) ( ( V)/( ) ) + (1)/(r^2 ^2 ) (^2 V)/( ^2) 1. Poisson's Equation in Spherical Coordinates Poisson's equation is ^2 V = -()/(_0), where V is the electric potential, is the volume charge density, and _0 is the permittivity of free space. (1)/(r^2) ()/( r) ( r^2 ( V)/( r) ) + (1)/(r^2 ) ()/( ) ( ( V)/( ) ) + (1)/(r^2 ^2 ) (^2 V)/( ^2) = -()/(_0) 2. Laplace's Equation in Spherical Coordinates Laplace's equation is ^2 V = 0, which is a special case of Poisson's equation where the charge density = 0. (1)/(r^2) ()/( r) ( r^2 ( V)/( r) ) + (1)/(r^2 ) ()/( ) ( ( V)/( ) ) + (1)/(r^2 ^2 ) (^2 V)/( ^2) = 0 What's next?