A particle of charge $q$ and mass $m$ moves non-relativistically with 4 -velocity $u^{a}(t)$ along a trajectory $x^{a}(t)$. Its electromagnetic field is determined by the Liénard-Wiechert potential

$$A^{a}\left(\mathbf{x}^{\prime}, t^{\prime}\right)=\frac{q}{4 \pi \epsilon_{0}} \frac{u^{a}(t)}{u_{b}(t)\left(x^{\prime}-x(t)\right)^{b}}$$

where $t^{\prime}=t+\left|\mathbf{x}-\mathbf{x}^{\prime}\right|$ and $\mathbf{x}$ denotes the spatial part of the 4 -vector $x^{a}$.

Derive a formula for the Poynting vector at very large distances from the particle. Hence deduce Larmor's formula for the rate of loss of energy due to electromagnetic radiation by the particle.

A particle moves in the $(x, y)$ plane in a constant magnetic field $\mathbf{B}=(0,0, B)$. Initially it has kinetic energy $E_{0}$; derive a formula for the kinetic energy of this particle as a function of time.