A particle of mass $m$ and charge $q$ moves through a magnetic field $\mathbf{B}$. There is no electric field or external force so that the particle obeys

$$m \ddot{\mathbf{r}}=q \dot{\mathbf{r}} \times \mathbf{B},$$

where $\mathbf{r}$ is the location of the particle. Prove that the kinetic energy of the particle is preserved.

Consider an axisymmetric magnetic field described by $\mathbf{B}=(0,0, B(r))$ in cylindrical polar coordinates $\mathbf{r}=(r, \theta, z)$. Determine the angular velocity of a circular orbit centred on $\mathbf{r}=\mathbf{0}$.

For a general orbit when $B(r)=B_{0} / r$, show that the angular momentum about the $z$-axis varies as $L=L_{0}-q B_{0}\left(r-r_{0}\right)$, where $L_{0}$ is the angular momentum at radius $r_{0}$. Determine and sketch the relationship between $\dot{r}^{2}$ and $r$. [Hint: Use conservation of energy.] What is the escape velocity for the particle?