Polarizability of a Dielectric Sphere

The requirement that VxE = 0 is immediately satisfied by seeking a solution in the form of E = -VO, where O (x) is a potential function. The divergence condition, V-E = 0, on the electric field leads to 

Equation (4.19) is Laplace s equation and has fundamental solutions in the form of harmonic functions. These functions are of two types growing harmonics, which are appropriate for bounded, interior regions, and decaying harmonics, which apply to unbounded space. These functions are expansions of the Green s function solution to Laplace s equation, G (x) = 1/(4jc x ), and are 

In general, the solution to Laplace s equation will be a superposition of harmonic functions. For the problem of the sphere, the fields J , and f 2 are 

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