Ring symmetry in electric potential calculation extended to discs and cylinders

Abstract

The electric potential—but not the electric field—of any charge distribution on a ring at a point on its axis is equivalent to that of an equal point charge placed anywhere on the ring; this is the well-known ring symmetry. We point out similar symmetries for finding the potential at points on the axis of a charged disc: disc symmetry, and the potential at points along the axes of a charged cylindrical shell and a solid cylinder; and cylinder symmetry. As such, a uniformly charged disc, or parts of it, map onto a line of equal charge parallel to its radius that gives the same potential at the same points on its axis. In the same manner, a uniformly charged thin cylindrical shell, or parts of it, maps onto a line of equal charge parallel to its axis that gives the same potential as the shell for points on its axis. Similarly, a cylinder, or parts of it, with a uniform volume charge density is seen to map onto a sheet carrying the same charge with the potential above one edge of the sheet equal to that at the corresponding points on the axis of the cylinder. The disc (cylinder) symmetry leads also to the fact the potential at points on the axis of a disc (cylinder) with charge distributions that vary with the azimuthal angle is the same as the potential of a uniformly charged disc (cylinder) at these points. These observations and their generalizations can supplement the methods for calculating the electric potential used in introductory electricity and magnetism classes as further useful symmetries that can be applied, whenever relevant, to find the electric potential.