Larmor formula ($P=\frac{q^2{\mu }_0{a}^2}{6\pi c}$)

The nonrelativistic radiated power of an accelerated point charge is $P=q^2{\mu }_0{a}^2/(6\pi c)$ in SI units. Integrating its dipole radiation pattern over solid angle gives this result; relativistic motion requires the Lienard generalization.

For a nonrelativistic point charge, let $C_{ij}=⟨a_i{a}_j⟩$ be the time-averaged second moment of acceleration, without subtracting its mean. The outgoing radiation zone fields satisfy $\mathbf{B}=\mathbf{n}\times \mathbf{E}/c$ and $\mathbf{E}=q\mathbf{n}\times (\mathbf{n}\times \mathbf{a})/(4\pi {ϵ}_0{c}^{2}r)$. Their Poynting vector gives the displayed power per solid angle. Its integral is $⟨P⟩=q^2\mathrm{tr}C/(6\pi {ϵ}_0{c}^3)$, the averaged Larmor formula.

For periodic motion the mean acceleration vanishes and $C$ is also its covariance matrix. More generally the second moment, rather than the centered covariance matrix, is the relevant quantity. The angular formula is valid to leading nonrelativistic order and evaluates source motion at retarded time.

The relativistic generalization of the Larmor formula uses the invariant proper-acceleration magnitude: $P=-{\mu }_0{q}^2{A}^a{A}_a/(6\pi )$ in signature $(+---)$. In coordinate time this is the displayed expression. For perpendicular magnetic circular motion, $\gamma m|a|=|q|vB$, giving $P={\mu }_0{q}^4({\gamma }^2-1)B^2/(6\pi m^2)$. The statement concerns total instantaneous emitted power; directional radiation flux is not itself invariant.

For one particle at a fixed Lorentz factor, the Liénard radiated-power formula gives $P_{\parallel }=C_q{\gamma }^6{a}_{\parallel }^2$ and $P_{\perp }=C_q{\gamma }^4{a}_{\perp }^2$, where $C_q=q^2/(6\pi {ϵ}_0{c}^3)$. Consequently equal coordinate acceleration magnitudes make transverse motion radiate less by ${\gamma }^2$. Equal applied laboratory force magnitudes instead give $a_{\parallel }=F/(m{\gamma }^3)$ and $a_{\perp }=F/(m\gamma )$, so $P_{\perp }/P_{\parallel }={\gamma }^2$. These opposite orderings concern distinct constraints and are consistent. An energy budget without a time, length, force, or acceleration constraint cannot by itself select a unique minimum-loss accelerator design.