Electric Potential Due to a System of Conducting Spheres

📝 Abstract

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.

💡 Analysis

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.

📄 Content

1 Electric Potential Due to a System of Conducting Spheres

Philip T. Metzger NASA Granular Mechanics and Regolith Operations Laboratory Mail Code: KT-D3 Kennedy Space Center, FL 32899 Philip.T.Metzger@nasa.gov

John E. Lane ASRC Aerospace, MS: ASRC-24 Kennedy Space Center, FL 32899 John.E.Lane@ksc.nasa.gov

Abstract: 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.

2 INTRODUCTION

The first year graduate physics student is likely to be introduced to the problem of calculating the electric field surrounding a conducting sphere in the presence of a point charge during the first few weeks of a standard course in electromagnetism. When this problem is extended to include a cluster of spheres, things get interesting, as well as more difficult. One application of this problem may be found in nuclear physics. Nuclear physicists sometimes need to sum the probability of nuclei breaking into every possible configuration of clusters, each cluster being modeled by a set of charged spheres, and this entails a large number of configurations, each of which must be solved individually. The probability of the nuclei breaking requires a calculation of the stored energy of the electric charges, which depends upon their actual distribution on the spheres, and it is computationally expensive for such a large number of configurations.
Therefore, they use the image charge method truncating the series of image charges for computational efficiency, but at the cost of some accuracy.

Another application involves a proposed spacecraft electrostatic radiation shield, made up of a cluster of conducting spheres surrounding the spacecraft. Similarly, a lunar radiation shield study incorporated conducting spheres of various sizes and potentials. In order to simulate the benefits of a radiation shield configuration, the electric field is needed at every location around the spacecraft in order to calculate the trajectories of the charged particles that constitute the cosmic radiation in space. The electric field throughout space depends upon the actual distribution of charge on the spheres, and it is computationally expensive to (first) solve for the actual distribution of charge, and (second), integrate the contributions to the electric field in each location of space resulting from all the portions of the surfaces of the charged spheres. For computational efficiency, we use the image charge method truncating the series of image charges so that only a finite set of point charges contribute to the electric field in all locations of space around the spheres, and thus summing these contributions is a simple sum over only a finite set of point charges rather than an integral over a set of surfaces.

In many applications it is computationally expensive to solve the exact distribution of charges since the charge distribution on the spheres is not uniform when multiple charged spheres interact with one another.
As spheres move increasingly close to one another, the charges on each sphere are pushed around by the electric fields of adjacent spheres. Since the electric fields from adjacent spheres are also changing, as their own charge distributions are perturbed, the final distribution of charge on the spheres becomes difficult to calculate.

A mathematical technique that can lead to an exact solution in many electrostatic problems is based on conformal mapping in the complex plane. Solving Laplace’s equation by conformal mapping has been primarily restricted (until recently) to two-dimensional problems, or to three-dimensional problems that

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