Equations describing the complete series of image charges for a system of conducting spheres are presented.
The method of image charges, originally described by J. C. Maxwell in 1873, has been and continues to be a useful
method for solving many three dimensional electrostatic problems. Here we demonstrate that as expected when the series
is truncated to any finite order N, the electric field resulting from the truncated series becomes qualitatively more similar
to the correct field as N increases. A method of charge normalization is developed which provides significant
improvement for truncated low order solutions. The formulation of the normalization technique and its solution via a
matrix inversion has similarities to the method of moments, which is a numerical solution of Poisson’s equation, using an
integral equation for the unknown charge density with a known boundary potential. The last section of this paper presents
a gradient search method to optimize a set of L point charges for M spheres. This method may use the image charge series
to initialize the gradient search. We demonstrate quantitatively how the metric can be optimized by adjusting the locations
and amounts of charge for the set of points, and that an optimized set of charges generally performs better than truncated
normalized image charges, at the expense of gradient search iteration time.