Question

A proton, a deuteron and an $\alpha -$particle having the same kinetic energy enters a region of uniform magnetic field, moving at right angles to the magnetic field $B$. What is the ratio of the radii of their circular paths?

• A. $1:\sqrt{2}:1$
• B. $1:\sqrt{2}:\sqrt{2}$
• C. $\sqrt{2}:1:1$
• D. $\sqrt{2}:\sqrt{2}:1$

Answer 1

The correct option is A $1:\sqrt{2}:1$

The radius of a charged particle in a uniform magnetic field is given by,

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

For proton, $r_p=\frac{\sqrt{2m_{p}K}}{eB}$

For deuteron, $r_d=\frac{\sqrt{2\left(2m_p\right)K}}{eB}$

For $\alpha -$particle, $r_{\alpha }=\frac{\sqrt{2\left(4m_p\right)K}}{\left(2e\right)B}$

Therefore, ratio of their radii is,

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

<!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> Hence, option $\left(A\right)$ is the correct answer.