Normalize electric dipole radiation

Define the electric dipole moment as the integral of position times charge density. Its units are coulomb metres. Continuity connects its first time derivative to the volume integral of current for a localized source with no boundary current escaping the chosen region. The radiation field depends on the second time derivative of the moment. A constant dipole has an electrostatic field but no electric dipole radiation; constant first derivative likewise gives no leading radiation term.

Two independent scale comparisons are needed. The source size must be small compared with the reduced wavelength, so source elements have nearly equal retarded phase. The observer must also be in the radiation zone, far compared with the source size and reduced wavelength. A distant observer does not by itself make an electrically large antenna a dipole. Higher multipoles and angular phase structure remain important for a large source even if its individual charges move slowly.

Suppose the second derivative of the moment lies along a fixed axis. Its projection onto the plane perpendicular to the observation direction has magnitude proportional to sine of the polar angle. Squaring the transverse field and multiplying by epsilon zero times c and radius squared gives power per solid angle. This quantity has units of watts per steradian. It is not power per unit polar angle, because a band of fixed polar-angle width contains a different solid angle near the axis and equator.

Integrate over azimuth and include the sine from the solid-angle measure. The integral of sine squared over the sphere is eight pi over three. This factor turns the angular coefficient into the total dipole power. The pattern's axial zeros concern the radiation term, not the full near electric field. A detector located on the dipole axis can still sense a changing induction field nearby without contradicting the far-zone radiation null.

For a moment with cosine time dependence, the second derivative contributes angular frequency squared and a minus sign. The sign does not survive the square, while the squared cosine averages to one half. Average power therefore grows as frequency to the fourth power at fixed moment amplitude. This statement holds only while source size remains small compared with wavelength and the prescribed moment remains physically realizable. Increasing frequency indefinitely eventually violates the approximation.

Fields from coherent sources must be added before squaring. Two equal dipoles with identical orientation and phase can radiate four times one dipole's intensity in a direction where their path phases agree. Opposite phases can cancel the leading dipole moment and reveal higher multipoles. Adding independently computed powers would miss that interference. Random relative phases, averaged over suitable times or ensembles, are a different physical preparation and may justify an incoherent sum.