Proton , deuton and alpha particle of same kinetic energy are moving in circular trajectories in a constant magnetic field. The radii of proton , deuteron and alpha particle are respectively `r_(p), r_(d) and r_(alpha)`. Which one of the following relation is correct?

Options

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

Answer

c

Explanation

The radius of circle described in magnetic field is given by `r = (mv)/(qB)` .
We need to write the radius in terms of kinetic energy.
`K = 1/2 mv^(2) rArr m^(2) v^(2) = 2m K`
Hence we can write the radius of circle as follows :
`r = sqrt(2 mK)/(qB) rArr r propto sqrt(m/q^(2))`
If m is mass of proton and e is charge on proton them
`r_(p) propto sqrt(m/e^(2))`
Mass of deuterium becomes 2m but charge remains the same.
` r_(d) propto sqrt((2m)/e^(2))`
Mass of alpha becomes 4m and charge is 2e.
`r_(alpha) propto sqrt((4m)/((2e)^(2)) ) rArr r_(alpha) propto sqrt(m/e^(2))`
Hence, we can see that option (c ) is correct.