Electric Potential Due to a System of Conducting Spheres

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. 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 [1], made up of a cluster of conducting spheres surrounding the spacecraft. Similarly, a lunar radiation shield [2] 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 *Address correspondence to this author at the NASA Granular Mechanics and Regolith Operations Laboratory, Mail Code: KT-D3, Kennedy Space Center, FL 32899, USA; E-mail: [email protected] 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 threedimensional problems that have rotational symmetry, or to cases where the Separation of Variables method can be applied [3, 4]. These cases do not encompass the problem solved in this paper, charged spheres placed arbitrarily in three-dimensional space, which is not generally reducible to a two-dimensional problem or amenable to the Separation of Variables. However, the techniques of conformal mapping are advancing at a rapid pace, extending the scope of Electric Potential Due to a System of Conducting Spheres The Open Applied Physics Journal, 2009, Volume 2 33 problems that can be solved by this method [5-7]. At least some of these advances involve infinite products and/or infinite series as a part of the solution, and therefore may provide no advantage in computing a numerical solution in electrostatics. However, that waits to be seen, and this paper makes no attempt to evaluate the rapidly advancing methods in conformal mapping. Another technique that is of general usefulness in threedimensional electrostatic problems is the Method of Moments [8, 9]. The Method of Moments can be used to solve Poisson’s equation by finding the unknown charge density on the surface of a conductor when the potential of the conductor is known. Fairly large matrices can result which are inverted to find the surface charge density. A key strategy to this method is find useful basis functions that can be solved analytically, thus reducing the number of matrix elements that need to be solved numerically. The Method of Moments is a powerful method which can be used to solve a large variety of electrostatic, as well as general electromagnetic problems. For the specific problem discussed in this paper, the Method of Images leads to a more direct approach, which is simpler conceptually and requires less complex computer code to implement as compared to the Method of Moments. Also, since the problem addressed by this paper generally involves three-dimensional geometry without rotational symmetry, the Method of Images is a simpler solution method compared to a method based on conformal mapping into the complex plane. IMAGE CHARGE SOLUTIONS The Method of Images [10, 11] is a convenient procedure for finding the electric potential due to a system of conductors and point charges without having to solve a differential equation, where the solution is guaranteed to be a solution of Laplace’s equation in the exterior region. The work involved with this method is to simply match boundary conditions via vector algebra using a finite or infinite series of point charge solutions. However, finding the necessary image point charges is not necessarily trivial. Since image charge solutions generally involve an infinite series of images, except in very simple cases, truncation of the series will always result in a less than perfect solution. Point Charge Near a Conducting Sphere The simplest example problem that will help lead into the general problem of calculating the electric potential due to a system of conducting spheres, is that of a single point charge outside of a conducting sphere of radius a held a constant potential V0. This problem is presented in detail in Jackson [11], but will be repeated below in brief to serve as an introduction into the more difficult cases to be considered next. The point charge has a charge q and is a distance d from the sphere’s center, noting that d > a The potential can be written as that due to one real charge plus two image charges: U(r) = 1 4 0 q0 r + q1 r bex + q r dex = aV0 r + q 4 0 r bex + 1 r dex (1) where the charges are placed on the x-axis for convenience, without loss of generality and where ex denotes the unit vector along the x-axis, as shown in Fig. (1). The first term is that due to an image charge at the center of the sphere, proportional to sphere’s potential. The second term is a single image charge that is induced at r = bex by the proximity of the real charge (the third term), located at r = dex. Note that b < a. The magnitude of the image charge is designated as q. Fig. (1). Point charge q a distance d from a conducting sphere of radius a. The solution for the position and magnitude of the image charge, as described in the references [11-13] is: b = a / d (2a) = a / d (2b) A demonstration of the single conducting sphere and single point charge is shown in Fig. (2a) with a = 1.2, d = 2.5, and V0 = 0, with the image charge parameters arbitrarily set to = -1 and b = 0. In Fig. (2b), and b are set to the correct values, according to Eq. (2). Image Charges for Two Conducting Spheres Fig. (3) shows two conducting spheres at different constant potentials, V1 and V2. The electric field potential can again be solved by use of image charges (see reference [14] for historical insights). The result is an infinite series of image charges inside of each sphere where the magnitude decreases with increasing order. As the order increases, positions of the image charges move closer to the inside surface of the sphere. No image charges appear outside of the spheres, in accordance with Laplace’s equation and the uniqueness theorem. The strategy (as well as similar notation) behind solving this problem is based on Jackson [11]. Because the electric field potential can be decomposed into a sum of fields due to image point charges, the solution of the single sphere with a single external point charge, as described in the previous section, can be applied in an iterative fashion to solve this problem. The zeroth order image charges are located at the center of each sphere: q1(0) = 4 0a1V1 (i = 1) (3a) q2 (0) = 4 0a2V2 (i = 2) (3b) b d q  q a V0