The Liénard-Wiechert potential for a particle of charge $q$, assumed to be moving non-relativistically along the trajectory $y^{\mu}(\tau), \tau$ being the proper time along the trajectory,

$$A^{\mu}(x, t)=\left.\frac{\mu_{0} q}{4 \pi} \frac{d y^{\mu} / d \tau}{(x-y(\tau))_{\nu} d y^{\nu} / d \tau}\right|_{\tau=\tau_{0}} .$$

Explain how to calculate $\tau_{0}$ given $x^{\mu}=(x, t)$ and $y^{\mu}=\left(y, t^{\prime}\right)$.

Derive Larmor's formula for the rate at which electromagnetic energy is radiated from a particle of charge $q$ undergoing an acceleration $a$.

Suppose that one considers the classical non-relativistic hydrogen atom with an electron of mass $m$ and charge $-e$ orbiting a fixed proton of charge $+e$, in a circular orbit of radius $r_{0}$. What is the total energy of the electron? As the electron is accelerated towards the proton it will radiate, thereby losing energy and causing the orbit to decay. Derive a formula for the lifetime of the orbit.

Part II