Continue beyond the electric dipole

Start from the retarded vector-potential integral for a localized conserved source. At a remote observer, approximate the inverse distance by inverse observer radius while retaining the source-position correction to the retarded time. Expanding that correction produces powers of source size divided by reduced wavelength. The leading integral of current is tied to the electric dipole derivative. The next current moment splits into antisymmetric circulation and symmetric shape change, giving magnetic dipole and electric quadrupole terms.

All radiative multipoles have inverse-radius far fields. A quadrupole is not identified by an inverse-radius-to-the-fourth far radiation field; that familiar power belongs to static multipole fields. Suppression of higher radiative multipoles comes from extra powers of the small source phase and the structure of the moments. Relative estimates fail when a nominally leading moment vanishes by symmetry. In that case a formally smaller multipole can supply the entire leading radiation.

The magnetic moment of a localized current is one half the integral of position crossed with current density. A small planar loop gives current times oriented area. The factor one half in the volume expression is essential; using it again on the loop formula halves the answer incorrectly. A steady loop has magnetic field but no magnetic dipole radiation. An oscillating moment radiates with the same sine-squared angular pattern as an electric dipole, with electric and magnetic polarization roles exchanged.

The total magnetic dipole power depends on the squared second time derivative of magnetic moment. In SI units its coefficient contains epsilon zero and speed to the fifth power, or equivalently mu zero divided by speed cubed. Comparing magnetic and electric dipoles requires comparing magnetic moment divided by c with electric moment. Their raw numerical moments have different units. This conversion also explains why a compact ordinary current loop often radiates weakly compared with a comparable separated-charge antenna.

Choose a traceless quadrupole convention explicitly because published coefficients differ with definitions. Here the moment weights three times the product of two coordinates minus radius squared times the identity tensor. Its units are coulomb square metres. Reflection symmetry can make the electric dipole zero while leaving this even moment nonzero. To radiate as a quadrupole the shape moment must vary with a nonzero third derivative, rather than merely have a large static value.

For example, equal positive charges at opposite positions on an axis, balanced by twice the negative charge at the origin, have zero net charge and zero electric dipole at every separation. Changing separation changes their quadrupole. Their leading electric dipole radiation cancels coherently; adding the separate charge powers would miss that cancellation. A wave-addition lab can demonstrate cancellation of the leading amplitudes but does not compute the residual quadrupole term, its polarization, or its angular pattern.