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A proton and a deuteron having the same ...

A proton and a deuteron having the same kinetic energies enter a region of uniform magnetic field perpendicularly, Deuteron's mass is twice that of proton. Calculate the ratio of the radii of their circular paths.

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1. The trajectory of a charged particle entering perpendicular to magnetic field is circular. Cetripetal force required for circular motion of particle is provided by magnetic force qvB.
`therefore(mv^(2))/r=qvB`
`therefore(mv)/r=qB=p/r`
2. For a motion of proton, `(m_(p)v_(p))/r_(p)=Bq=p_(p)/r_(p)" "...(1)`
For a motion of deuteron `(m_(d)v_(d))/r_(d)=B_(q)=p_(d)/r_(d)" "...(2)`
From equation (1) and (2),
`thereforep_(p)/r_(p)=p_(d)/r_(d)`
`thereforer_(d)/r_(p)=p_(d)/p_(p)" "...(3)`
but, `E=p^(2)/(2m)" "thereforep=sqrt(2mE)`
`thereforep_(p)=sqrt(2m_(p)E_(p))andp_(d)=sqrt(2mE_(d))`
From equation (3),
`thereforer_(d)/r_(p)=sqrt((2m_(d)E_(d))/(2m_(p)E_(p)))`
But kinetic energy of both are equal so
`E_(d)=E_(p)andm_(d)=2m_(p)`
`thereforer_(d)/r_(p)=sqrt((2m_(p))/m_(p))=sqrt2`
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