A proton, a deuteron ion and an `alpha`-particle of equal kinetic energy perform circular motion normal to a uniform magnetic field B. If the radii of their paths are `r_(p),r_(d)andr_(alpha)` respectively then _____ [Here, `q_(d)=q_(p),m_(d)=2m_(p)`]

Options

A. `r_(alpha)=r_(p)ltr_(d)`
B. `r_(alpha)=r_(d)gtr_(p)`
C. `r_(alpha)gtr_(d)gtr_(p)`
D. `r_(alpha)=r_(d)=r_(p)`

Answer

A

Explanation

Centripetal force required electric charge `(mv^(2))/r=qvB`
`thereforev=(qBr)/m" "...(1)`
Kinetic energy of electric charge = K = `1/2mv^(2)`
= `1/2m((qBr)/m)^(2)`
`thereforeK=(q^(2)B^(2)r^(2))/(2m)`
`thereforer^(2)=(2Km)/(q^(2)B^(2))rArrrpropsqrtm/q`
`thereforer_(alpha):r_(p):r_(d)=sqrt4/2:sqrt1/(1)1:sqrt2/1`
= `1:1:sqrt2`
`thereforer_(alpha)=r_(p)ltr_(d)`