A proton and an electron moving with same kinetic energy enter uniform magnetic field at right- angle to its direction. The ratio of radii of their trajectories will be:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( 43 : 1 \)
A charge moving perpendicular to a uniform magnetic field moves in a circle; the magnetic force provides the centripetal force.
\( \displaystyle qvB = \frac{mv^{2}}{r} \;\Rightarrow\; r = \frac{mv}{qB} \)
Writing the momentum in terms of the kinetic energy \( K \), \( mv = \sqrt{2mK} \):
\( \displaystyle r = \frac{\sqrt{2mK}}{qB} \)
The proton and the electron have the same \( K \), the same magnitude of charge \( e \) and move in the same \( B \), so \( r \propto \sqrt{m} \).
\( \displaystyle \frac{r_p}{r_e} = \sqrt{\frac{m_p}{m_e}} = \sqrt{\frac{1.67\times10^{-27}\,\mathrm{kg}}{9.1\times10^{-31}\,\mathrm{kg}}} = \sqrt{1835} \approx 42.8 \)
\( r_p : r_e \approx 43 : 1 \)