A proton , a deuteron and `alpha )`- particle having the same kinetic energy are moving in circular trajectories in a constant magnetic field . If ` r_(p), r_(d) , and r_(alpha)` denote respectively the radii of the trajectories of these particles , then

Options

A. `r_(alpha)=r_(p) lt r_(d)`
B. `r_(alpha) gt r_(d) lt r_(p)`
C. `r_(alpha)=r_(d) gt r_(p)`
D. `r_(p)=r_(d)=r_(alpha)`

Answer

A

Explanation

Given that `K_(p)=K_(d)=K_(alpha)=K` (say) We know that
`m_(p)=m, m_(d)=2m` and `m_(alpha)=4m` and `q_(p)=e, q_(d)=e` and
`q_(alpha)=2e`
Further ` r=(sqrt(2mK))/(qB) implies r_(p)=(sqrt(2mk))/(eB)`,
` r_(d)=(sqrt(2(2m)K))/(eB)=sqrt(2)r_(p)`
and `r_(alpha)=(sqrt(2(4m)K))/((2e)B)=r_(p)`. Hence `r_(alpha)=r_(p) lt r_(d)`