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Dipole radiation

In addition to dipoles in electrostatics, it is also common to consider an electric or magnetic dipole that is oscillating in time. It is an extension, or a more physical next-step, to spherical wave radiation.

In particular, consider a harmonically oscillating electric dipole, with angular frequency ω and a dipole moment p along the ẑ direction of the form

In vacuum, the exact field produced by this oscillating dipole can be derived using the retarded potential formulation as:

For rω / c ≫ 1, the far-field takes the simpler form of a radiating "spherical" wave, but with angular dependence embedded in the cross-product:

The time-averaged Poynting vector

is not distributed isotropically, but concentrated around the directions lying perpendicular to the dipole moment, as a result of the non-spherical electric and magnetic waves. In fact, the spherical harmonic function (sin θ) responsible for such toroidal angular distribution is precisely the l = 1 "p" wave.

The total time-average power radiated by the field can then be derived from the Poynting vector as

Notice that the dependence of the power on the fourth power of the frequency of the radiation is in accordance with the Rayleigh scattering, and the underlying effects why the sky consists of mainly blue colour.

A circular polarized dipole is described as a superposition of two linear dipoles.