Match radiation power to the kinematic regime

For a slowly moving point charge, the transverse radiation field has magnitude proportional to charge times transverse acceleration divided by speed squared and distance. Squaring it, multiplying by the vacuum wave factor, and integrating the sine-squared pattern yields Larmor power. This is an instantaneous result for the source's retarded acceleration, not necessarily a cycle average. A sinusoid needs its own averaging; a constant acceleration magnitude does not automatically introduce a factor one half.

At prescribed acceleration, power is quadratic in charge magnitude and acceleration and independent of charge sign. This does not imply that changing mass has no effect in a specified force field. The acceleration itself depends on mass and the chosen force law. Likewise, the formula computes radiated power from a given trajectory. If radiation loss significantly changes the trajectory, one must determine that motion consistently rather than use an indefinitely maintained orbit without an energy source.

The relativistic extension contains a Lorentz factor to the sixth power multiplying acceleration squared minus the squared cross product of beta and acceleration. Split acceleration into components parallel and perpendicular to the velocity. The perpendicular coefficient loses two Lorentz powers because one minus beta squared equals the reciprocal of gamma squared. Thus equal laboratory acceleration magnitudes radiate different powers according to their orientation at relativistic speed, even for the same charge and speed.

The distinction between equal acceleration and equal force is consequential. A parallel force produces coordinate acceleration inversely proportional to gamma cubed times mass, whereas a perpendicular force gives a different response. Comparing particles at fixed field, fixed radius, fixed energy, or fixed acceleration therefore leads to different scalings. State what is held fixed before interpreting a gamma enhancement. In the limit of small beta, both direction-dependent coefficients approach one and the Larmor formula is recovered.

Squaring the exact radiation field gives a sixth power of kappa in the denominator of radius squared times radial Poynting flux. That is received energy per observer time. The corresponding emitted energy per source time requires multiplication by dt/dtᵣ, equal to kappa. The angular emission formula consequently has a fifth power. Integrating the wrong sixth-power expression over directions does not give the Liénard emitted power for a moving charge.

This conversion assumes a fixed remote observer direction for each ray during the infinitesimal comparison. It expresses the compression or stretching of arrival intervals, not energy creation in transit. A practical audit begins by identifying the source time, the detection time, and the acceleration convention. The quadratic lab can test acceleration-amplitude scaling in a normalized slow-motion example, while the gamma and kappa factors must be evaluated analytically; its controls contain no relativistic trajectory or retarded-time calculation.