Question

An electron and a proton having mass \( m_e \) and \( m_p \) respectively, initially at rest, move through the same distance \( s \) in a uniform electric field \( E \). If the time taken by them to cover that distance is \( t_e \) and \( t_p \) respectively, then \( \frac{t_e}{t_p} \) is equal to:

• A. \( \sqrt{\frac{m_p}{m_e}} \)
• B. \( \sqrt{\frac{m_e}{m_p}} \)
• C. \( \frac{m_e}{m_p} \)
• D. \( \frac{m_p}{m_e} \)

Hint:

The time taken by charged particles to cover a given distance in an electric field depends on their mass. The ratio of times can be found using the ratio of their masses.

Answer 1

The Correct Option is B

We know that the force on the charged particle due to the electric field is given by: \[ F = qE \] where \( q \) is the charge on the particle and \( E \) is the electric fiel(D) From Newton's second law, the acceleration of the particle is: \[ a = \frac{F}{m} = \frac{qE}{m} \] Now, the time \( t \) taken to cover a distance \( s \) under constant acceleration can be found using the equation: \[ s = \frac{1}{2} a t^2 \] Substituting for acceleration: \[ s = \frac{1}{2} \frac{qE}{m} t^2 \] Solving for \( t \): \[ t = \sqrt{\frac{2ms}{qE}} \] For the electron (\( e \)) and proton (\( p \)), we have: \[ t_e = \sqrt{\frac{2 m_e s}{e E}} \quad \text{and} \quad t_p = \sqrt{\frac{2 m_p s}{e E}} \] Now, the ratio of the times is: \[ \frac{t_e}{t_p} = \frac{\sqrt{\frac{2 m_e s}{e E}}}{\sqrt{\frac{2 m_p s}{e E}}} = \sqrt{\frac{m_e}{m_p}} \] Thus, the correct ratio is \( \frac{t_e}{t_p} = \sqrt{\frac{m_e}{m_p}} \).