Continuous Distributions of Charge

Coulomb's law and the superposition principle, as stated so far, apply directly only to a finite collection of point charges. A real charged rod, plate, ring, disc or sphere, however, is made up of an enormous number of individual electrons and ions -- far too many to add up one at a time -- spread essentially continuously over the body's length, surface, or volume. For such bodies it is both more practical and more accurate to describe the charge using a charge density: the amount of charge present per unit length, per unit area, or per unit volume, at each point of the body.

Three charge densities are defined, matching the three ways charge can be physically distributed:

Linear charge density $\lambda$ (used for a charged wire, rod, or ring), the charge per unit length:

Surface charge density $\sigma$ (used for a charged plate, sheet, disc, or the surface of a charged sphere), the charge per unit area:

Volume charge density $\rho$ (used for charge distributed throughout a solid three-dimensional body), the charge per unit volume:

If the charge is spread uniformly, the corresponding density is simply the total charge divided by the total length, area or volume (e.g. $\sigma=Q/A$ for a uniformly charged plate of total charge $Q$ and area $A$); if the distribution is non-uniform, the density itself varies from point to point. …