H^+,He^+ and O^{++} are projected in a unifor | vedclass

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(C) The radius $R$ of a charged particle moving in a uniform magnetic field $B$ is given by $R = \frac{mv}{qB} = \frac{p}{qB}$.Since the particle is a

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$1 : 2 : 2\sqrt{2}$

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(C) The radius $R$ of a charged particle moving in a uniform magnetic field $B$ is given by $R = \frac{mv}{qB} = \frac{p}{qB}$.Since the particle is accelerated through a potential $V$,its kinetic energy is $K = qV = \frac{p^2}{2m}$,so $p = \sqrt{2mqV}$.Substituting this into the radius formula: $R = \frac{\sqrt{2mqV}}{qB} = \frac{1}{B} \sqrt{\frac{2mV}{q}}$.Since $B$ and $V$ are constant for all particles,$R \propto \sqrt{\frac{m}{q}}$.For $H^+$: $m = 1, q = 1 \implies R_1 \propto \sqrt{\frac{1}{1}} = 1$.For $He^+$: $m = 4, q = 1 \implies R_2 \propto \sqrt{\frac{4}{1}} = 2$.For $O^{++}$: $m = 16, q = 2 \implies R_3 \propto \sqrt{\frac{16}{2}} = \sqrt{8} = 2\sqrt{2}$.Thus,the ratio is $1 : 2 : 2\sqrt{2}$.

$H^+$,$He^+$ and $O^{++}$ are projected in a uniform transverse magnetic field with equal accelerating potential. If their masses are $1 \, a.m.u.$,$4 \, a.m.u.$ and $16 \, a.m.u.$ respectively,find the ratio of their radii.

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