Question

A proton and an $\alpha -$ particle enter the same magnetic field which is perpendicular to their velocity. If both particles have same kinetic energy, then the ratio of their radii of the circular path is

• A. $1:1$
• B. $1:2$
• C. $2:1$
• D. $1:4$

Answer 1

The correct option is A $1:1$

Radius of circular path for a charged particle having charge $q$ and mass $m$ with kinetic energy $K$ in a uniform magnetic field $B$ is given by,

$r=\frac{\sqrt{2mK}}{qB}$

For constant $B$ and $K$,

$r=\frac{\sqrt{m}}{q}$

$\therefore \frac{r_{proton}}{r_{\alpha }}=\sqrt{\frac{m_p}{m_{\alpha }}}\times \frac{q_{\alpha }}{q_p}$

As we know that
$m_{\alpha }=4m_p;q_{\alpha }=2q_p$

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

Therefore, option $\left(a\right)$ is correct.