Question

Two ions having equal masses, but charges in the ratio 1:2 are projected perpendicular to a uniform magnetic field with speeds in the ratio 2:3. The ratio of the radii of the circular paths along which the two ions move is:

• A. 4 : 3
• B. 2 : 3
• C. 3 : 2
• D. 1 : 4

Answer 1

Answer: (A)

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} $