Separate bound fields from outgoing radiation

Differentiate the Liénard–Wiechert potentials with the chain rule for retarded time. The result divides naturally into a term proportional to inverse distance squared and one proportional to inverse distance times the time derivative of beta. The first is often called the velocity field because it survives uniform motion. The second is the acceleration field. These names refer to terms in the prescribed-source solution, rather than two independently adjustable electromagnetic fields around a single physical trajectory.

The denominator kappa is one minus n dotted into beta. Its powers encode changes in emission time and geometry under differentiation. The acceleration term contains a nested cross product, which is perpendicular to n. Replacing the derivative of beta by ordinary acceleration requires dividing acceleration by c. Missing that conversion changes the field by a factor of light speed. The exact formulas are valid outside a smooth subluminal point-charge worldline in free space, before any far-zone approximation.

For outgoing radiation the magnetic field equals n cross the electric radiation field divided by c. Hence the radial Poynting vector has magnitude epsilon zero times c times the squared radiation electric field. A sphere supplies an area proportional to distance squared, exactly compensating the inverse-square intensity. Radiation is thus diagnosed through a finite asymptotic flux, not merely a field that is visible at a chosen detector or a nonzero local cross product of fields.

By contrast, the velocity field squared contributes intensity scaling as inverse distance to the fourth power. Its integrated large-sphere contribution falls as inverse distance squared. The mixed velocity–acceleration contribution falls as inverse distance after integration. When comparing the same outgoing event at successively larger spheres, observation time must also advance by travel time. Holding reception time fixed can instead compare different stages of a source history and obscure this scaling argument.

An observer near the charge can see substantial energy move between the source and its accompanying field. This stored energy can return later. A uniformly moving charge also has both electric and magnetic fields and generally a nonzero Poynting vector, but no acceleration term. The presence of magnetic field, temporal variation at a detector, or instantaneous outward flow through part of a nearby surface therefore does not independently establish radiative loss. Surface geometry and the complete energy balance matter.

In the slow-motion limit, kappa approaches one and the acceleration field becomes a transverse projection of acceleration divided by c squared and distance. Along the acceleration axis that leading field vanishes; sideways it is largest. The vector lab exposes this projection without representing a charge field. Predict the transverse component first, square it on paper to infer the relative angular intensity, and keep the distinction between field amplitude and intensity explicit throughout the comparison.