A particle of mass $m$ and charge $q$ moving in a vacuum through a magnetic field $\mathbf{B}$ and subject to no other forces obeys

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

where $\mathbf{r}(t)$ is the location of the particle.

For $\mathbf{B}=(0,0, B)$ with constant $B$, and using cylindrical polar coordinates $\mathbf{r}=$ $(r, \theta, z)$, or otherwise, determine the motion of the particle in the $z=0$ plane if its initial speed is $u_{0}$ with $\dot{z}=0$. [Hint: Choose the origin so that $\dot{r}=0$ and $\ddot{r}=0$ at $t=0$.]

Due to a leak, a small amount of gas enters the system, causing the particle to experience a drag force $\mathbf{D}=-\mu \dot{\mathbf{r}}$, where $\mu \ll q B$. Write down the new governing equations and show that the speed of the particle decays exponentially. Sketch the path followed by the particle. [Hint: Consider the equations for the velocity in Cartesian coordinates; you need not apply any initial conditions.]