The non-relativistic Larmor formula for the power, $P$, radiated by a particle of charge $q$ and mass $m$ that is being accelerated with an acceleration a is

$$P=\frac{\mu_{0}}{6 \pi} q^{2}|\mathbf{a}|^{2} .$$

Starting from the Liénard-Wiechert potentials, sketch a derivation of this result. Explain briefly why the relativistic generalization of this formula is

$$P=\frac{\mu_{0}}{6 \pi} \frac{q^{2}}{m^{2}}\left(\frac{d p^{\mu}}{d \tau} \frac{d p^{\nu}}{d \tau} \eta_{\mu \nu}\right)$$

where $p^{\mu}$ is the relativistic momentum of the particle and $\tau$ is the proper time along the worldline of the particle.

A particle of mass $m$ and charge $q$ moves in a plane perpendicular to a constant magnetic field $B$. At time $t=0$ as seen by an observer $\mathbf{O}$ at rest, the particle has energy $E=\gamma m$. At what rate is electromagnetic energy radiated by this particle?

At time $t$ according to the observer $\mathbf{O}$, the particle has energy $E^{\prime}=\gamma^{\prime} m$. Find an expression for $\gamma^{\prime}$ in terms of $\gamma$ and $t$.