A) 4 : 3
B) 2 : 3
C) 3 : 2
D) 1 : 4
Correct Answer: A
Solution :
Given: The ion have equal masses \[{{m}_{1}}={{m}_{2}}\] Ratio of charges \[\frac{{{q}_{1}}}{q{{ & }_{2}}}=\frac{1}{2}\] Ratio of speeds\[\frac{{{v}_{1}}}{{{v}_{2}}}=\frac{2}{3}\] The necessary centripetal force for circular path is provided by magnetic force \[F=\frac{m{{v}^{2}}}{r}=qvB\Rightarrow r=\frac{mv}{qB}\] Hence \[\frac{{{r}_{1}}}{{{r}_{2}}}=\frac{{{m}_{1}}{{v}_{1}}}{{{q}_{1}}B}\times \frac{{{q}_{2}}B}{{{m}_{2}}{{v}_{2}}}=\frac{{{m}_{1}}{{v}_{1}}{{q}_{2}}}{{{m}_{2}}{{v}_{2}}{{q}_{1}}}\] \[\frac{{{r}_{1}}}{{{r}_{2}}}=\frac{{{m}_{1}}\times 2\times 2}{{{m}_{1}}\times 3\times 1}=\frac{4}{3}\] \[\frac{{{r}_{1}}}{{{r}_{3}}}=\frac{4}{3}\]
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