The electric or magnetic field of an ideal dipole is known to have a Dirac delta function at the
origin. The usual textbook derivation of this delta function is rather ad hoc and cannot be used to
calculate the delta-function structure for higher multipole moments. Moreover, a naive application
of Gauss’s law to the ideal dipole field appears to give an incorrect expression for the dipole’s
effective charge density. We derive a general result for the delta-function structure at the origin
of an arbitrary ideal multipole field without using any advanced techniques from distribution
theory. We find that the divergence of a singular vector field can contain a derivative of a Dirac
delta function even if the field itself does not contain a delta function. We also argue that a
physical interpretation of the delta function in the dipole field previously given in the literature
is perhaps misleading and may require clarification. Both the explanation of and the resolution to
thi…