Question

A particle of charge ‘ $q$ ’ and mass ‘ $m$ ’ is projected from the origin with velocity $\left(u_{0} \hat{i} + v_{0} \hat{j}\right) \, $ in a gravity free region where uniform electric field $-E_{0}\hat{i}$ and uniform magnetic field $-B_{0}\hat{i}$ exist. Find the condition so that the particle would return to origin at least for once.

• A. $\frac{\mu _{0} B_{0}}{2 \pi E_{0}}$ is an integer
• B. $\sqrt{u_{0}^{2} + v_{0}^{2}} \, \, \frac{B_{0}}{\pi E_{0}}$ is an integer
• C. $\frac{v_{0} B_{0}}{\pi B_{0}}$ is an integer
• D. $\frac{u_{0} B_{0}}{\pi E_{0}}$ is an integer

Answer 1

Answer: (D)

Time for one complete rotation $T_{0}=\frac{2 \pi m}{q B_{0}}$
Time for retracing motion in $x$ direction $T_{1}=\frac{2 u_{0} m}{q E_{0}}$
$\therefore \, \, \frac{T_{1}}{T_{0}}$ must be an integer. So, $\frac{2 u_{0} m}{q E_{0}}\times \frac{q B_{0}}{2 \pi m}=\frac{u_{0} B_{0}}{\pi E_{0}}$