The Electromagnetic Enhancement for a Prolate Metal Spheroid

The aspect ratio a/b defines the nature of the spheroid such that the particle becomes a sphere for a/fe = 1, an oblate spheroid for a/fe 1, and a prolate spheroid for a/b 1. 

Electrostatic Boundary Problem for a Prolate Metal Spheroid 

The electrostatic problem is again to solve the Laplace equation but now we must consider the spheroidal geometry. If we let the particle be a prolate spheroid, it is convenient to solve the Laplace equation in prolate spheroidal coordinates 17 and (p. This is an orthogonal coordinate system where and 7 are orthogonal spheroidal coordinates and (p is an azimuthal coordinate. At the surface of the spheroidal particle, = fo = = 1/[1 ( / ) ] and 

In these expressions, A and B are coefficients to be determined. This determination can be accomplished by requiring that d be continuous across the surface, i.e., 

It is of interest to calculate the surface-averaged squared field, which is defined by 

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