Two particles of equal charge are accelerated by applying the same potential difference and them allowed to enter a uniform magnetic field normally. If the particles keep revolving along circular path of radii `R_(1) and R_(2)`, determine the ratio of their masses.

Explanation

If the charge of each particle be q and the potential difference applied be V, kinetic energy acequired =qV
So, `qV=(1)/(r)m_(1)v_(1)^(2)=(1)/(2)m_(2)v_(2)^(2))`
or `(v_(1))/(v_(2))= sqrt((m_(2))/(m_(1)))`
Again magnetic force requred to revolve along a circular path= centripetal force.
`:.` force the first paricle,
`qv_(1)B=(m_(1)v_(1)^(2))/(R_(1)) or, qB=(m_(1)v_(1))/(R_(1))`
For the second particle, `qB=(m_(2)v_(2))/(R_(2))`
`:. (m_(1)v_(1))/(R_(1))=(m_(2)v_(2))/(R_(2)) or, (m_(1))/(m_(2))*(v_(1))/(v_(2))=(R_(1))/(R_(2))`
or, `(m_(1))/(m_(2))* sqrt((m_(2))/(m_(1)))=(R_(1))/(R_(2)) or sqrt((m_(1))/(m_(2)))=(R_(1))/(R_(2))`
or `(m_(1))/(m_(2))=((R_(1))/(R_(2)))^(2)`