What is the potential at the center of a sphere with given boundary conditions?

AI Thread Summary
To find the potential at the center of a sphere with a given boundary condition of r=2m and u(2,theta,psi)=5sin(theta)sin(0.25psi), Laplace's equation must be solved in spherical coordinates. The angular component should be matched with the boundary condition, utilizing spherical harmonics for expansion. After establishing the angular solutions, the radial ordinary differential equation (ODE) can be solved while ensuring non-singular solutions at r=0. This approach will yield the potential at the sphere's center. Proper application of these mathematical principles is essential for accurate results.
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Homework Statement


Given potential r=2m on the surface of the sphere which meets the Laplasian equation (triangle)u=0 and is u(2,theta,psi)=5sin(theta)sin(0.25psi). I need to find the potential of sphere center. Can anyone help me?


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The Attempt at a Solution


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Write Laplace's equation in spherical coordinates, plug in your boundary condition and solve.

Note that the angular component has solutions which you can expand in terms of http://en.wikipedia.org/wiki/Spherical_harmonics" . You should first match the angular component with the boundary condition. Then it is a simple matter of solving the radial ODE for the boundary conditions... noting you must have non-singular solutions only at r=0.
 
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