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

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Question

(C) The radius $r$ of a charged particle moving in a uniform magnetic field $B$ after being accelerated by a potential difference $\Delta V$ is given

Vedclass · Accepted Answer

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

Vedclass · Answer

(C) The radius $r$ of a charged particle moving in a uniform magnetic field $B$ after being accelerated by a potential difference $\Delta V$ is given by $r = \frac{mv}{qB}$.Since $q\Delta V = \frac{1}{2}mv^2$,we have $v = \sqrt{\frac{2q\Delta V}{m}}$.Substituting $v$ into the radius formula: $r = \frac{m}{qB} \sqrt{\frac{2q\Delta V}{m}} = \frac{1}{B} \sqrt{\frac{2m\Delta V}{q}}$.Since $B$ and $\Delta V$ are constant,$r \propto \sqrt{\frac{m}{q}}$.For $H^+$: $m_1 = 1, q_1 = 1 \Rightarrow \sqrt{\frac{1}{1}} = 1$.For $He^+$: $m_2 = 4, q_2 = 1 \Rightarrow \sqrt{\frac{4}{1}} = 2$.For $O^{++}$: $m_3 = 16, q_3 = 2 \Rightarrow \sqrt{\frac{16}{2}} = \sqrt{8} = 2\sqrt{2}$.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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