Two particles of masses `M_(1)andM_(2)` and having the equal electric charge are accelerated through equal potential difference and then move inside a uniform magnetic field, normal to it. If the radii of their circular paths are `R_(1)andR_(2)` respectively, find the ratio of their masses.

Explanation

1. Two particles having mass `M_(1)andM_(2)` and equal electric charge move under potential v (suppose)
2. For a particle having mass `M_(1)=qV=1/2M_(1)v_(1)^(2)`
For a particle having mass `M_(2)=qV=1/2M_(2)v_(2)^(2)`
`therefore1/2M_(1)v_(1)^(2)=1/2M_(2)v_(2)^(2)`
`thereforeM_(1)/M_(2)=v_(2)^(2)/v_(1)^(2)" "...(1)`
3. After entering into uniform magnetic field, both the particles perform uniform circular motion due to centripetal force acting on them.
For a particle having mass `M_(1)=(M_(1)v_(1)^(2))/R_(1)=qv_(1)B`
`therefore(M_(1)v_(1)^(2))/R_(1)=qB`
Similarly for a particle having mass `M_(2)`,
`(M_(2)v_(2)^(2))/R_(2)=qv_(2)B`
`therefore(M_(2)v_(2))/R_(2)=qB`
`therefore(M_(1)v_(1))/R_(1)=(M_(2)v_(2))/R_(2)`
`thereforeM_(1)/M_(2)=v_(2)/v_(1)R_(1)/R_(2)`
From equation (1) `v_(2)/v_(1)=(M_(1)/M_(2))^(1/2)`
`thereforeM_(1)/M_(2)=(M_(1)/M_(2))^(1/2)*R_(1)/R_(2)`
`therefore(M_(1)/M_(2))^(1/2)=R_(1)/R_(2)`
`thereforeM_(1)/M_(2)=(R_(1)/R_(2))^(2)`