Conductive Sphere on a Charged Conductive Plane
Birney Robert Fish · 1967
The e lectrostatic problem of a conductive sphere on a charged conductive plane has been so lved using theoretica l and experimenta l methods. From e lementary e lec trostatics i t is seen that there w i l l be a un iform e lectric fie ld E0 vo l ts/meter norma l to a sem i -infin ite plane bearing <70 cou lombs/meter 2• Based on an image charge approach, it was determ ined that a sphere of rad ius R meters w i l l acqu ire a tota l equ i l ibr ium charge q = � 'IT3R2a0 cou lombs a t contact w ith the p lane, and the uniform fie ld w i l l be perturbed, resu l t ing in a field strength E = 4.5 E0 vo l ts/meter at the top of the sphere. An approxi motion, va l id to w ith in less than about I per cent, for the repu l sive force act ing on the sphere was obta ined by extrapolating to contact, series so lut ions of the Laplace equation for fin i te separation between sphere and plane: the force at contact is est imated to be F = 1 . 537 x lo -10k R2E� newtons; where z . k is the d ie lectric constant of the surrounding medium. The charge d istr ibut ion on the sphere was determined experimenta l ly by p lating copper on a sphere plane cathode and, w ith in about 10 per cent, is g iven by the empirica l expression a(9) = (0.8696 cos2 ;+ 3.6304 cos 4 �) <70; where 9 is the ang le of co latitude in spherica l po lar coordinates referred to the axis of contact and concentric w i th the sphere.