Electrostatic Potential
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Another illustration of the use of Legendre polynomials is provided by the problem of a neutral conducting sphere (radius r_0) placed in a (previously) uniform electric field (see attachment). The problem is to find the new, perturbed, electrostatic potential. If we call the electrostatic potential v, it satisfies [see the attachment for equation] Laplaces equation.
We select spherical polar coordinates because of the spherical shape of the conductor. (This will simplify the application of the boundary condition at the surface of the conductor.) Separating variables and glancing at Table 9.2, we can write the unknown potential V (r, omega) in the region outside the sphere as a linear combination of solutions:
[see the attachment for the equation].
How do we get the new electrostatic potential?
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