A particle of mass $m$ and charge $q>0$ moves in a time-dependent magnetic field $\mathbf{B}=\left(0,0, B_{z}(t)\right)$.

Write down the equations of motion governing the particle's $x, y$ and $z$ coordinates.

Show that the speed of the particle in the $(x, y)$ plane, $V=\sqrt{\dot{x}^{2}+\dot{y}^{2}}$, is a constant.

Show that the general solution of the equations of motion is

$$\begin{aligned} &x(t)=x_{0}+V \int_{0}^{t} d t^{\prime} \cos \left(-\int_{0}^{t^{\prime}} d t^{\prime \prime} q \frac{B_{z}\left(t^{\prime \prime}\right)}{m}+\phi\right) \\ &y(t)=y_{0}+V \int_{0}^{t} d t^{\prime} \sin \left(-\int_{0}^{t^{\prime}} d t^{\prime \prime} q \frac{B_{z}\left(t^{\prime \prime}\right)}{m}+\phi\right) \\ &z(t)=z_{0}+v_{z} t \end{aligned}$$

and interpret each of the six constants of integration, $x_{0}, y_{0}, z_{0}, v_{z}, V$ and $\phi$. [Hint: Solve the equations for the particle's velocity in cylindrical polars.]

Let $B_{z}(t)=\beta t$, where $\beta$ is a positive constant. Assuming that $x_{0}=y_{0}=z_{0}=$ $v_{z}=\phi=0$ and $V=1$, calculate the position of the particle in the limit $t \rightarrow \infty$ (you may assume this limit exists). [Hint: You may use the results $\int_{0}^{\infty} d x \cos \left(x^{2}\right)=\int_{0}^{\infty} d x \sin \left(x^{2}\right)=$ $\sqrt{\pi / 8} .]$