Capacitance of Two Overlapping Conducting Spheres

Objectives: We calculate the capacitance of two conducting spheres, which are partially overlapping. Methods: Two sequences of image charges are needed to make the surfaces of the conductors equipotential by the method of images. For some special contact angles, the number of image charges is finite and they are located inside the unphysical region (that is, the conducting spheres). Findings: We obtain the closed-form expressions for the charges and positions of the image charges for some special contact angles from which any physical quantities including the capacitance are calculated. Application: The result can be applicable to estimating the capacitances of some biological cells and nanoparticles.


Introduction
Capacitors are one of passive elements used in electric and electronic circuits. Their capacitances depend only on the geometry of conductors and are usually calculated for parallel-plate, cylindrical, and spherical capacitors [1]. However, in dealing with parallelplate and cylindrical capacitors with finite sizes, their edge effects are neglected. Spherical capacitor consisting of concentric conducting spheres is special in that its size is finite from the start and its exact capacitance can be easily calculated. Recently, capacitors consisting of a pair of conducting spheres whose centers do not coincide have been discussed where one surface is located inside another surface with different centers or one is located outside another [2][3][4].
Here, we consider the situation where two conductors with spherical surfaces S a and S b of radii a and b, respectively, are partially overlapping. We want to calculate the capacitance of the combined conductor using the method of images in the next section.

Two Overlapping Conducting Spheres
The center of one conducting sphere is chosen as the origin of the coordinate system (see Figure 1). The other conducting sphere has its center at x = s on the positive x-axis and its surface crosses the positive x-axis at x = s-b. The separation between the centers of the two spheres is s with a b s a b − < ≤ + . The problem is to make their electric potential held at a constant value V. The method of images [5] is to simulate the boundary condition with suitably placed point charges with finite magnitudes inside the unphysical region surrounded by S a and S b . An image charge q 1 at the origin x 1 = 0 makes the surface S a equipotential. Then, S b is not equipotential any more so that an image charge 1 q of q 1 with respect to S b is required to make the potential on S b due to charges q 1 and 1 q vanishing. Now 1 q is not located at the center of S a and breaks the equipotential condition of S a . Thus, an image charge q 2 of 1 q with respect to S a is introduced to make S a equipotential. This process continues indefinitely in order to make the surfaces S a and S b each equipotential, requiring two infinite sequences 1 2 , , q q … and 1 2 , , q q … of image charges. All the image charges lie on the x-axis. In Figure 1, solid dots represent charges with the same sign as q 1 , while open dots represent those with the opposite sign to q 1 .
The image of the charge q n at x = x n with respect to S b is given by Similarly, the image of the charge n q at n x x = with respect to S a is given by  From equation (1) with n replaced with n + 1 gives where equations (3) and (4) have been used. Equation (5) Eliminating n x by combining equations (4) and (2) gives If equation (6) is substituted into equation (7), then we obtain a 2nd-order difference equation for 1/ n q : Then, n x are determined by equation (4) ) sin sin ( 1 ] . sin N N Hence, a finite number of image charges are used to produce the potential in the physical region outside the conductors: n q for 1, 2, , n N = … , and n q for 1,2, , 1 n N = … − , which are all located in the unphysical region.
For each n the potentials on S b due to charges n q and n q cancel out by construction, and similarly, the potentials on S a due to charges 1 n q + and n q do. Since due to equation (21), the potentials on S a and S b are given, respectively, by meaning that the surfaces of the conductors are equipotential. When the potential of the conducting spheres is V, we have The electrostatic potential at any point in the physical region is given by the image charges. The surface charge density and the total charge on each surface can be calculated from the normal derivative of the potential at the surface. More simply, the total charge on S a and S b is, by Gauss's law, given by   Hence, the capacitance of the overlapping conducting spheres is  

Concluding Remarks
Smythe [6] considered overlapping conducting spheres for the special case 2 N = . In this study, we generalized his result for any 2, ≥ where / N θ π = is the contact angle between the two spheres. The closed-form expressions for the charges and positions of the