Question

A proton, a deuteron and an $\alpha -particle$ having the same kinetic energy are moving in circular trajectories in a constant magnetic field. If $r_p,r_d$ and $r_{\alpha }$ denote, respectively, the radii of the trajectories of these particles, then

• A. $r_{\alpha }=r_p<r_d$
• B. $r_{\alpha }>r_d>r_p$
• C. $r_{\alpha }=r_d>r_p$
• D. $r_p=r_d=r_{\alpha }$

Answer 1

The correct option is A $r_{\alpha }=r_p<r_d$

The centripetal force is provided by the magnetic force

$\frac{m{v}^2}{r}=qvB\Rightarrow \frac{mv}{r}=qB$

$\Rightarrow r=\frac{mv}{qB}=\frac{p}{qB}[\because p-mv]$

But $KE=\frac{p^2}{2m}\Rightarrow p=\sqrt{2mKE}$

Here, KE and B are same for the three particles

$\therefore r\propto \frac{\sqrt{m}}{q}$

$\therefore r_p:r_d:r_{\alpha }=\frac{\sqrt{1}}{2}:\frac{\sqrt{2}}{1}:\frac{\sqrt{4}}{2}=1:\sqrt{2}:1$

$\Rightarrow r_{\alpha }=r_p<r_d$.