ion of any electrostatic problem involving two spheres, need for the solution of a differential equation either by saying that the gradient of the field is too smal ver the diameter of the small sphere or by noting that the image charges are small and highly localized on both APPENDIX spheres. When the small sphere is charged, however, and ught near the uncharged large sphere the resulting force Consider two similar charges, o, a distance d apart. If very large. Essentially, the small sphere sees its negative we surround one charge with a conducting sphere of radius image close by and is strongly attrached to it. Or, in a differ- a and require that the sphere be an equipotential surface, ent description, the potential of the small sphere is large we find that an image charge of magnitude and the field gradient over the diameter of the large sphere q1 (A1) is also large must be placed a distance a'/d from the center of the When the two spheres are of different radii a surprising sphere to produce a spherical equipotential at the radius a2 esult occurs. If a,> 1.24a2 there exists a center-to-center he original charge go at the center of the sphere must be spacing where the force between equally charged spheres increased to go 1q1 to maintain the total charge equal to becomes attractive.Of course, in the laboratory we seldom go. A second conducting sphere of radius a around the deal with equal charges on dissimilar spheres. It is more second charge go will be an equipotential if two new image common to charge the spheres to same potential charges are placed inside, corresponding to the images of In Fig 4 is shown the position of the null point, at which 1 and of o' lg l. Each image charge is smaller than its the force between equally charged spheres changes sign object charge and of opposite sign. By repeated iteration of from repulsion to attraction. Distances in Fig. 4 are ex- this scheme, a task well suited to a personal computer,we pressed as d/(a,+ a2)and the abscissa is the"eccentric- found that a string of charges of alternating sign and rapid ly decreasing magnitude can replace the charged spheres These results should not surprise us. If the larger sphere and produce the appropriate field. Figure 5 shows the loca were very much larger than the smaller one its charge tions of the image charges when d=2.1a would create only a small repulsive force on the small The computer program calculates all of the anand sphere. The small sphere, on the other hand would be at tracted to its own oppositely charged image by a much from larger force. Our calculations show that the onset of this D,=d-xn+1 effect occurs at about a,= 1. 24a2 D I=l/Dn, The program we have written is convenient for the solu- -gn/Dn The resulting charges are in the right proportions but no longer sum up to @o so their sum S is calculated and a new o= 1/S is chosen, The process is repeated until IS-1<10-. To calculate the total force between the spheres we simply sum all of the forces between the sets of charges, a task that the computer does very quickly. In ractice, when the distance between the spheres d is greater than 4a, as few as 10 charges suffice for six-digit accuracy On the other hand, when the spheres touch, d= 2a, and more than 100 charges on each side are needed to compute 2*419 q。+1428 46" Fig. 4. Unequal spheres, equally charged; center-to-center distance where force changes sign. Fig. 5. Distribution of image charges when D=2. 1a, repulsive case 1198 Am J Phys., Vol 58, No 12, December 1990 Jack a. soule