A particle of mass $m$ and charge $q$ moves with trajectory $\mathbf{r}(t)$ in a constant magnetic field $\mathbf{B}=B \hat{\mathbf{z}}$. Write down the Lorentz force on the particle and use Newton's Second Law to deduce that

$$\dot{\mathbf{r}}-\omega \mathbf{r} \times \hat{\mathbf{z}}=\mathbf{c}$$

where $\mathbf{c}$ is a constant vector and $\omega$ is to be determined. Find $\mathbf{c}$ and hence $\mathbf{r}(t)$ for the initial conditions

$$\mathbf{r}(0)=a \hat{\mathbf{x}} \quad \text { and } \quad \dot{\mathbf{r}}(0)=u \hat{\mathbf{y}}+v \hat{\mathbf{z}}$$

where $a, u$ and $v$ are constants. Sketch the particle's trajectory in the case $a \omega+u=0$.

[Unit vectors $\hat{\mathbf{x}}, \hat{\mathbf{y}}, \hat{\mathbf{z}}$ correspond to a set of Cartesian coordinates. ]