Question

A particle of mass m and charge +q enters a region of magnetic field with a velocity v, as shown in fig.
Time which the particle stay inside the magnetic field is $t=\frac{T}{X\pi }(\pi -2\theta )$. FInd X?

Answer 1

The length of the arc traced by the particle: $l=R(\pi -2\theta )$
Time spent in the field, $t=\frac{l}{v}=\frac{R(\pi -2\theta )}{v}$ and $R=\frac{mv}{Bq}$ which gives $t=\frac{m}{Bq}(\pi -2\theta )$
As time period: $T=\frac{2\pi m}{Bq}$

Hence, $t=\frac{T}{2\pi }(\pi -2\theta )$
We can generalize this result. If $\varphi$ is the angle subtended by the arc traced by the charged particle in the magnetic field. The time spent is, $t=T\left(\frac{\varphi }{2\pi }\right)$