Physics · Ch 1 — Electrostatics
Distribution of charges in a conductor
Distribution of charges in a conductor
Consider two conducting spheres A and B, of radii r1 and r2, connected to each other by a long thin conducting wire, with the separation between the spheres taken to be much larger than either radius. Once connected, the two spheres, together with the wire, form a single conductor, so (from section 1.7.1) the whole assembly must reach one common potential: V1 = V2, i.e. k q1/r1 = k q2/r2, giving q1/q2 = r1/r2 -- the charge on each sphere divides in direct proportion to its own radius. However, the surface charge density (charge per unit area) on each sphere is sigma = q/(4 pi r^2), so sigma1/sigma2 = (q1/q2)(r2/r1)^2 = (r1/r2)(r2/r1)^2 = r2/r1 -- the surface charge density is inversely proportional to the radius. This means the smaller sphere ends up with the higher surface charge density, even though it carries the smaller total charge. The same conclusion applies, more generally, to any single conductor of irregular shape, which can be thought of as being made up of many differently-curved regions, each locally behaving like part of a sphere of some local radius of curvature: since the whole conductor is at one uniform potential, the more sharply curved (sma …
What this figure shows. Two conducting spheres A and B, of different radii r1 and r2, are drawn joined by a long thin conducting wire, with the distance between the spheres taken to be much larger than either sphere's own radius, so that each sphere's own field can be treated independently of the other's presence. Once connected by the wire, charge flows between the two spheres until they reach one common potential, and the labels q1 and q2 mark the resulting shares of the total charge that end up sitting on each sphere -- generally unequal shares, in direct proportion to each sphere's own radius, as the accompa …