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Two particles of equal charges after being accelerated through the same potential difference enter in a uniform transverse magnetic field and describe circular paths of radii $R_{1} a n d R_{2}$ . Then the ratio of their respective masses $(M_{1} / M_{2})$ is

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$R_{1} / R_{2}$

${(R_{1} / R_{2})}^{2}$

$(R_{2} / R_{1})$

${(R_{2} / R_{1})}^{2}$

answer is B.

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Detailed Solution

$q ν = \frac{1}{2} M ν^{2} o r ν = \sqrt{\frac{2 q V}{M}} . . . (i)$

$B q ν = \frac{M ν^{2}}{R} o r B q = \frac{M ν}{R} = \frac{M}{R} \sqrt{\frac{2 q V}{M}} = \frac{\sqrt{2 q V M}}{R}$ (Using i)

$o r M = \frac{B^{2} q^{2} R^{2}}{2 q V} = \frac{B^{2} q R^{2}}{2 V}$

$∴ M \propto R^{2}$ ( $∵ B, q a n d V$ are same for the given two particles)

Hence, $\frac{M_{1}}{M_{2}} = {(\frac{R_{1}}{R_{2}})}^{2}$

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