A proton and an alpha particle having the same kinetic energy are… - Vidyadip

Question

A proton and an alpha particle having the same kinetic energy are, in turn, passed through a region of the uniform magnetic field, acting normal to the plane of the paper and travel in circular paths. Deduce the ratio of the radii of the circular paths described by them.

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Answer

Derivation of ratio of the radii of the circular paths $r =\frac{m v}{q B}$
$\begin{array}{l}\text { But } \frac{p^2}{2 m}=k \Rightarrow p=\sqrt{2 m k}= mv \\
\Rightarrow \quad \frac{r_\alpha}{r_p}=\frac{\sqrt{2 m_\alpha k_\alpha} / q_a B}{\sqrt{2 m_p k_p} / q_p B}\end{array}$
$\begin{array}{l}\text { But } k _\alpha= k _{ p } \\
\frac{r_\alpha}{r_p}=\frac{q_p}{q_\alpha} \sqrt{\frac{m_\alpha}{m_p}}\end{array}$
Since, charge of alpha particle $q _\alpha=2 q _{ p }$
mass of alpha particle $m _\alpha=4 m_{ p }$
$\Rightarrow \quad \frac{r_\alpha}{r_p}=\frac{q_p}{2 q_p} \sqrt{\frac{4 m_p}{m_p}}=1: 1$

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