A non-relativistic particle of rest mass $m$ and charge $q$ is moving slowly with velocity $\mathbf{v}(t)$. The power $d P / d \Omega$ radiated per unit solid angle in the direction of a unit vector $\mathbf{n}$ is

$$\frac{d P}{d \Omega}=\frac{\mu_{0}}{16 \pi^{2}}|\mathbf{n} \times q \dot{\mathbf{v}}|^{2} .$$

Obtain Larmor's formula

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

The particle has energy $\mathcal{E}$ and, starting from afar, makes a head-on collision with a fixed central force described by a potential $V(r)$, where $V(r)>\mathcal{E}$ for $r<r_{0}$ and $V(r)<\mathcal{E}$ for $r>r_{0}$. Let $W$ be the total energy radiated by the particle. Given that $W \ll \mathcal{E}$, show that

$$W \approx \frac{\mu_{0} q^{2}}{3 \pi m^{2}} \sqrt{\frac{m}{2}} \int_{r_{0}}^{\infty}\left(\frac{d V}{d r}\right)^{2} \frac{d r}{\sqrt{V\left(r_{0}\right)-V(r)}}$$