Electric potential on solid spheres in a plasma

Derivation of the general expression for the potential on a solid sphere immersed in a plasma, showing the dependence of the potential on the radius (a) of the sphere and the radius (s) of the plasma sheath that develops around the sphere. In the limit where the radius a is much larger than the sheath thickness s-a, the well-known result for the potential on an infinite wall in contact with a plasma is recovered from this expression. At the other extreme, where s is much larger than a, the result derived by Spitzer (1941) for the potential on spherical grains in the interstellar plasma is obtained. Since the surface of the sphere forms a sink for the charged particles, there is a net drift of the plasma towards the surface. The effect of this drift on the potential is examined. Finally, for very small metallic spheres, an effect leading to a revision of the potential is discussed. This effect consists in a lowering of the potential barrier for the electrons due to the image force. The various effects limiting the potential on spheres are discussed.