An electron of mass `m_(e )` initially at rest moves through a certain distance in a uniform electric field in time `t_(1)`. A proton of mass `m_(p)` also initially at rest takes time `t_(2)` to move through an equal distance in this uniform electric field.Neglecting the effect of gravity, the ratio of `t_(2)//t_(1)` is nearly equal to

Options

A. 1
B. `sqrt(M_(p)/M_(e))`
C. `sqrt(M_(e)/M_(p))`
D. `1836`

Answer

To solve the problem, we will analyze the motion of an electron and a proton in a uniform electric field and derive the relationship between the time taken by each particle to travel the same distance.

### Step-by-Step Solution:

1. **Understanding the Forces**:
- The force acting on a charged particle in an electric field \( E \) is given by \( F = qE \), where \( q \) is the charge of the particle.
- For an electron, \( q = -e \) (where \( e \) is the elementary charge), and for a proton, \( q = +e \).

2. **Calculating Acceleration**:
- The acceleration \( a \) of a particle is given by Newton's second law \( F = ma \).
- For the electron:
\[ a_e = \frac{F_e}{m_e} = \frac{q_e E}{m_e} = \frac{(-e)E}{m_e} \]
- For the proton:
\[ a_p = \frac{F_p}{m_p} = \frac{q_p E}{m_p} = \frac{eE}{m_p} \]

3. **Using the Equations of Motion**:
- Both the electron and proton start from rest and travel the same distance \( d \) in times \( t_1 \) and \( t_2 \), respectively.
- The equation of motion for distance is:
\[ d = ut + \frac{1}{2} a t^2 \]
- Since both start from rest, \( u = 0 \), so:
\[ d = \frac{1}{2} a_e t_1^2 \quad \text{(for the electron)} \]
\[ d = \frac{1}{2} a_p t_2^2 \quad \text{(for the proton)} \]

4. **Setting the Distances Equal**:
- Since both particles travel the same distance \( d \):
\[ \frac{1}{2} a_e t_1^2 = \frac{1}{2} a_p t_2^2 \]
- Canceling \( \frac{1}{2} \) gives:
\[ a_e t_1^2 = a_p t_2^2 \]

5. **Substituting the Accelerations**:
- Substitute the expressions for \( a_e \) and \( a_p \):
\[ \left(\frac{-eE}{m_e}\right) t_1^2 = \left(\frac{eE}{m_p}\right) t_2^2 \]
- Cancel \( eE \) (since both are positive and non-zero):
\[ -\frac{t_1^2}{m_e} = \frac{t_2^2}{m_p} \]

6. **Rearranging for the Time Ratio**:
- Rearranging gives:
\[ \frac{t_2^2}{t_1^2} = \frac{m_p}{m_e} \]
- Taking the square root:
\[ \frac{t_2}{t_1} = \sqrt{\frac{m_p}{m_e}} \]

### Final Result:
The ratio of the time taken by the proton to the time taken by the electron is:
\[ \frac{t_2}{t_1} = \sqrt{\frac{m_p}{m_e}} \]