Paper 4, Section II, A

Write down the Lorentz force law for a charge $q$ travelling at velocity $\mathbf{v}$ in an electric field $\mathbf{E}$ and magnetic field $\mathbf{B}$.

In a space station which is in an inertial frame, an experiment is performed in vacuo where a ball is released from rest a distance $h$ from a wall. The ball has charge $q>0$ and at time $t$, it is a distance $z(t)$ from the wall. A constant electric field of magnitude $E$ points toward the wall in a perpendicular direction, but there is no magnetic field. Find the speed of the ball on its first impact.

Every time the ball bounces, its speed is reduced by a factor $\gamma<1$. Show that the total distance travelled by the ball before it comes to rest is

$L=h \frac{q_{1}(\gamma)}{q_{2}(\gamma)}$

where $q_{1}$ and $q_{2}$ are quadratic functions which you should find explicitly.

A gas leak fills the apparatus with an atmosphere and the experiment is repeated. The ball now experiences an additional drag force $\mathbf{D}=-\alpha|\mathbf{v}| \mathbf{v}$, where $\mathbf{v}$ is the instantaneous velocity of the ball and $\alpha>0$. Solve the system before the first bounce, finding an explicit solution for the distance $z(t)$ between the ball and the wall as a function of time of the form

$z(t)=h-A f(B t)$

where $f$ is a function and $A$ and $B$ are dimensional constants, all of which you should find explicitly.

*Typos? Please submit corrections to this page on GitHub.*