Paper 4, Section II, A

Dynamics and Relativity | Part IA, 2018

Write down the Lorentz force law for a charge qq travelling at velocity v\mathbf{v} in an electric field E\mathbf{E} and magnetic field B\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 hh from a wall. The ball has charge q>0q>0 and at time tt, it is a distance z(t)z(t) from the wall. A constant electric field of magnitude EE 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 γ<1\gamma<1. Show that the total distance travelled by the ball before it comes to rest is

L=hq1(γ)q2(γ)L=h \frac{q_{1}(\gamma)}{q_{2}(\gamma)}

where q1q_{1} and q2q_{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 D=αvv\mathbf{D}=-\alpha|\mathbf{v}| \mathbf{v}, where v\mathbf{v} is the instantaneous velocity of the ball and α>0\alpha>0. Solve the system before the first bounce, finding an explicit solution for the distance z(t)z(t) between the ball and the wall as a function of time of the form

z(t)=hAf(Bt)z(t)=h-A f(B t)

where ff is a function and AA and BB are dimensional constants, all of which you should find explicitly.

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