A particle of mass $m$ experiences, at the point with position vector $\mathbf{r}$, a force $\mathbf{F}$ given by

$$\mathbf{F}=-k \mathbf{r}-e \dot{\mathbf{r}} \times \mathbf{B},$$

where $k$ and $e$ are positive constants and $\mathbf{B}$ is a constant, uniform, vector field.

(i) Show that $m \dot{\mathbf{r}} \cdot \dot{\mathbf{r}}+k \mathbf{r} \cdot \mathbf{r}$ is constant. Give a physical interpretation of each term and a physical explanation of the fact that $\mathbf{B}$ does not arise in this expression.

(ii) Show that $m(\dot{\mathbf{r}} \times \mathbf{r}) \cdot \mathbf{B}+\frac{1}{2} e(\mathbf{r} \times \mathbf{B}) \cdot(\mathbf{r} \times \mathbf{B})$ is constant.

(iii) Given that the particle was initially at rest at $\mathbf{r}_{0}$, derive an expression for $\mathbf{r} \cdot \mathbf{B}$ at time $t$.