CONSIDER A SPHERICAL BATTERY

We introduce a simplified model of a ''spherical battery'' embedded within an infinite conducting medium, wherein the source of emf is taken to be a uniform vertical force per unit charge F, of nonelectrical nature. This model can be solved analytically for the surface charge distribution, the electric field, the electric current density, the magnetic field, the Poynting vector, and the angular momentum density. Through the net current and net power loss, a ''lumped'' resistance can be defined, and from the net current and the lumped resistance a lumped emf can be defined. This example explicitly illustrates that the surface charge on a battery causes it, at a distance, to behave like an electric dipole. It also provides an explicit example where the nonelectrical ''field'' of the battery acts only within the battery, whereas the electrostatic field of the surface charge acts both within and outside of the battery. A spherical inclusion within a material with otherwise uniform current flow is also analyzed to obtain the electric field, current density, and magnetic field. Additional geometries with current flow and surface charge, both with and without batteries, for which there exist analytic but sometimes complex solutions, are discussed. Following a discussion of voltaic cells, it is pointed out that in real voltaic cells the emf occurs at the electrode-electrolyte interface rather than throughout the volume. From that we propose a more realistic but less tractable model for a spherical battery, in which the emf acts only within two hemispherical shells.