An electron, a proton and an alpha particle having the same kinetic energy are moving in circular orbits of radii `r_(e),r_(p),r_(alpha)` respectively in a uniform magnetic field B. The relation between `r_(e),r_(p),r_(alpha)` is:

Options

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

Answer

D

Explanation

Linear momentum in terms of kinetic energy can be written as follows:
`1/2 mv^(2) = K rArr m^(2) v^(2) = 2mK rArr mv = sqrt(2 mK) `
Radius of the circular path of the charged particle inside magnetic field is given by
`r = (mv)/(qB) = sqrt(2mK)/(qB) `
`rArr " " r_(e) = sqrt(2 m_(e)K)/(eB) `
`rArr " " r_(p) = sqrt(2m_(p)K)/(eB) `
` rArr " " r_(alpha) = sqrt(2(4 m_(p))K)/((2e)B) `
Comparing the above three radii we can infer that
` r_(e) lt r_(p) = r_(alpha)`
Hence option (d) is correct.