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_∝ respectively in a uniform magnetic field B. The relation between r_e, r_p, r_∝ is:

r_e < r_∝ < r_p

r_e > r_p = r_∝

r_e < r_p = r_∝

r_e < r_p < r_∝

Solution

r_e < r_p = r_∝

The radius of circular path in a magnetic field is given as,

$r = \frac{\sqrt{2 K m}}{q B} F o r e l e c t r o n, r_{e} = \frac{\sqrt{2 K m_{e}}}{e B} . . . . (i) F o r P r o t o n, r_{p} = \frac{\sqrt{2 K m_{p}}}{e B} . . . . (i i) F o r α p a r t i c l e, r_{α} = \frac{\sqrt{2 K m_{a}}}{q_{α} B} = \frac{\sqrt{2 K 4 m_{p}}}{2 e B} = \frac{\sqrt{2 K m_{p}}}{e B} . . . (i i i) ∴ r_{e} < r_{p} = r_{α} (∵ m_{e} < m_{p})$