A proton , a deutron 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_(a)=r_(p) lt r_(d)`
B. `r_(a) gt r_(d) gt r_(p)`
C. `r_(a)=r_(d)gt gt r_(p)`
D. `r_(p)=r_(d)=r_(a)`

Answer

A

Explanation

`(a)` The centripetal force is provided by the magnetic force.
`(mv^(2))/(r)=qvBimplies(mv)/(r)=qBimpliesr=(mv)/(qB)=(q)/(qB)`
`[ :' p=mv]`
But `K.E.=(p^(2))/(2m)impliesp=sqrt(2mK.E.)`
`:. r=(sqrt(2mK.E.))/(qB)`
Here `K.E.` and B is same for the three particles
`:.rprop(sqrt(m))/(q)" "," "r_(p):r_(d):r_(alpha)=(sqrt(1))/(1):(sqrt(2))/(1):(sqrt(4))/(2)`
`implies r_(alpha)=r_(p) lt r_(d)`
Hence `(a)` is the correct correct option