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 $T_1=\frac{2u_{0}m}{q{E}_0}$
$\therefore \frac{T_1}{T_0}$ must be an integer. So, $\frac{2u_{0}m}{q{E}_0}\times \frac{q{B}_0}{2\pi m}=\frac{u_0{B}_0}{\pi E_0}$