Question

A charged particle of charge $+Q$ and mass $M$ is projected perpendicular to a uniform magnetic field(thickness $d$) as shown. It undergoes a deviation of ${60}^{\circ }$ while moving through the magnetic field. Then,

• A. If it had negative charge, it would deflect downwards with same angle of deviation
• B. If its mass would have been $2M,$ it would undergo a deviation of ${30}^{\circ }$
• C. If its mass would have been $3M,$ it would undergo a deviation of ${180}^{\circ }$
• D. If its mass would have been $\frac{M}{2},$ it would undergo a deviation of ${180}^{\circ }.$

Answer 1

Answer: (A), (D)

If its mass would have been $\frac{M}{2},$ it would undergo a deviation of ${180}^{\circ }.$



When the charge entered perpendicularly, then
$\frac{m{v}^2}{R}=qvB$
$R=\frac{mv}{qB}$
$R\propto m$
(A) From figure
$\sin \theta =\frac{d}{R}$
$\sin {60}^{\circ }=\frac{d}{mv}\times qB$
(B) If mass is doubled, radius is doubled
$\sin \theta =\frac{\sin {60}^{\circ }}{2}=\frac{\sqrt{3}}{4}$
(C) If mass is $3M$, radius is $3$ times
$\sin {\theta }^{\prime }=\frac{\sin {60}^{\circ }}{3}=\frac{1}{2\sqrt{3}}$
(D) If mass is halved, then the radius is
$R=\frac{mv}{2qB}$
$\Rightarrow \frac{mv}{qB}=\frac{2d}{\sqrt{3}}$
$R=\frac{2d}{\sqrt{3}}\times \frac{1}{2}=\frac{d}{\sqrt{3}}$

As $R<d$, particle will deflect ${180}^{\circ }$