Question

An electron falls from rest through a vertical distance h in a uniform and vertically upward directed electric field E. The direction of electric field is now reversed, keeping its magnitude the same. A proton is allowed to fall from rest in it through the same vertical distance h. The time of fall of the electron, in comparison to the time of fall of the proton is

• A. smaller
• B. 10 times greater
• C. 5 times greater
• D. equal

Answer 1

The Correct Option is A

To answer this question, we need to analyze the motion of an electron and a proton falling through an electric field.

**Concepts involved:**

• **Electrostatic force:** It acts on a charged particle in an electric field, given by F = qE, where q is the charge and E is the electric field strength.
• **Motion of a charged particle in an electric field:** The acceleration a experienced by a particle with mass m and charge q in an electric field E is a = \frac{qE}{m}.
• **Time of fall:** If a particle falls from rest, its time of fall through a height h under constant acceleration a is given by t = \sqrt{\frac{2h}{a}}.

Now, let's solve the problem step-by-step:

1. For the **electron**:
   • Charge of electron, q = -e and mass m_e.
   • Initial acceleration when field is upward: a_e = \frac{eE}{m_e}.
   • The time of fall t_e = \sqrt{\frac{2h}{a_e}} = \sqrt{\frac{2hm_e}{eE}}.
2. For the **proton** after reversing the field direction:
   • Charge of proton, q = +e and mass m_p.
   • Initial acceleration: a_p = \frac{eE}{m_p}.
   • The time of fall t_p = \sqrt{\frac{2h}{a_p}} = \sqrt{\frac{2hm_p}{eE}}.

**Comparative Analysis:**

To find the ratio of the times of fall:

\frac{t_e}{t_p} = \frac{\sqrt{\frac{2hm_e}{eE}}}{\sqrt{\frac{2hm_p}{eE}}} = \sqrt{\frac{m_e}{m_p}}

Since the mass of a proton is much greater than the mass of an electron (m_p \approx 1836 \times m_e), we find:

\frac{t_e}{t_p} = \sqrt{\frac{1}{1836}} \approx \frac{1}{\sqrt{1836}}

This ratio indicates that t_e is much smaller than t_p, meaning the time of fall for the electron is smaller compared to the proton.

Thus, the correct answer is that the time of fall of the electron is smaller than that of the proton.