A proton, at deuteron and an `alpha`-particle are projected perpendicular to the direction of a uniform magnetic field with same kinetic energy. The ratio of the radii of the circular paths described by them is

Options

A. `1:sqrt(2):sqrt(2)`
B. `1:sqrt(2):1`
C. `sqrt(2):sqrt(2):1`
D. `sqrt(2):1:1`

Answer

B

Explanation

When particle is moving perpendicular to a magnetic field then it moves in a circular path of radius, `r=mv // qB`
`:'` kinetic energy of all the three particles is same.
`:. (1)/(2)m_(p)v_(p)^(2)=(1)/(2)m_(d)v_(d)^(2)=(1)/(2)m_(alpha)v_(alpha)^(2)`
As we know `m_(p)=1 u`, `m_(d)=2 u`, `m_(alpha)=4 u`
`:. v_(p):v_(d):v_(alpha)=2:sqrt2:1`
or `v_(d)=(1)/(sqrt2)v_(p)` and `v_(alpha)=(1)/(2)v_(p)`
`r_(p):r_(d):r_(alpha)=(m_(p)v_(p))/(q_(p)B):(m_(d)v_(d))/(q_(d)B):(m_(alpha)v_(d))/(q_(alpha)B)`
`=(1 xx v_(p))/(1 xx B):(2 xx 1 xx v_(p))/(1 xx sqrt2 xx B):(4 xx 1 xx v_(p))/(2 xx 2 xx B)=1:sqrt2:1`