Assertion : A proton and an alpha particle having the same kinetic energy are moving in circular paths in a uniform magnetic field. The radii of their circular paths will be equal.
Reason : Any two charged particles having equal kinetic energies and entering a region of uniform magnetic field $\overrightarrow B $ in a direction perpendicular to $\overrightarrow B $, will describe circular trajectories of equal radii.
AIIMS 2009, Medium
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The radius of the circular path is given by
$r=\frac{m v}{q B}=\frac{\sqrt{2 m K}}{q B} ;$ where $K=\frac{1}{2} m v^{2}$
since $K$ and $B$ are the same for the two particles, $r \propto \frac{\sqrt{m}}{q} .$ Now, the charge of an alpha particle is twice that of a proton and its mass is four times the mass of a proton, $\sqrt{m} / q$ will be the same for both particles. Hence, $r$ will be the same for both particles.
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