Question

A particle of specific charge $0.5\text{C/kg}$ is projected in a transverse magnetic field with velocity making angle $\theta =\frac{\pi }{3}$ at $t=0$ as shown in figure.


The time spent by the particle in the magnetic field is:

• A. $\frac{\pi }{3}\text{sec}$
• B. $\frac{2\pi }{3}\text{sec}$
• C. $\frac{\pi }{12}\text{sec}$
• D. $\frac{\pi }{4}\text{sec}$

Answer 1

The correct option is B $\frac{2\pi }{3}\text{sec}$

The particle will move in a circular path as shown in the figure.


The force acting on particle $F$ interset at centre $\text{O}$.

The arc subtends an angle $(\pi -2\theta )$ at the centre.

Thus, time spent in magnetic field is

$t=\frac{m(\pi -2\theta )}{qB}$

$t=\frac{\pi -2(\frac{\pi }{3})}{(\frac{q}{m})B}=\frac{\pi -2\left(\pi /3\right)}{\frac{1}{2}\times 1}$

$\therefore t=\frac{2\pi }{3}\text{s}$

Therefore, option $\left(b\right)$ is right.

Why this question ? Tip: The spent in magnetic field region is related with angular frequency of particle as, $\omega t=\pi -2\theta \Rightarrow t=\frac{\pi -2\theta }{\omega }$ Where, $\omega =\frac{qB}{m}$