Application of the Dirac delta function to electric charge and multipole distributions

We introduce the concept of equivalent volume charge density for all types of charge and multipole distributions, discrete as well as continuous. The equivalent volume charge density for point charges, point dipoles, and point quadrupoles is expressed in terms of the Dirac delta function and its derivatives. The equivalent volume charge density for surface charge distributions and surface dipole distributions is obtained by introducing a ’’surface delta function.’’ The well-known equivalence between a continuous distribution of dipoles within a specified volume and the volume and surface densities is derived directly by means of the delta function formalism. The same method is also applied to find the equivalent volume and charge distributions for a continuous distribution of quadrupoles confined in a specified volume.