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. `(m_(p)//m_(e ))^(1//2)`
C. `(m_(e )//m_(p))^(1//2)`
D. 1836

Explanation

To solve the problem, we need to analyze the motion of both the electron and the proton in a uniform electric field. Here’s a step-by-step solution:

### Step 1: Understanding the Forces
Both the electron and the proton experience forces due to the electric field. The force \( F \) acting on a charged particle in an electric field \( E \) is given by:
\[ F = qE \]
where \( q \) is the charge of the particle.

### Step 2: Calculate the Acceleration
The acceleration \( a \) of a particle can be calculated using Newton's second law:
\[ F = ma \]
Thus, the acceleration can be expressed as:
\[ a = \frac{F}{m} = \frac{qE}{m} \]

For the electron (mass \( m_e \)):
\[ a_e = \frac{qE}{m_e} \]

For the proton (mass \( m_p \)):
\[ a_p = \frac{qE}{m_p} \]

### Step 3: Kinematic Equation for Distance
Since both particles start from rest and travel the same distance \( s \), we can use the kinematic equation:
\[ s = \frac{1}{2} a t^2 \]
For the electron:
\[ s = \frac{1}{2} a_e t_1^2 = \frac{1}{2} \left(\frac{qE}{m_e}\right) t_1^2 \]

For the proton:
\[ s = \frac{1}{2} a_p t_2^2 = \frac{1}{2} \left(\frac{qE}{m_p}\right) t_2^2 \]

### Step 4: Set the Distances Equal
Since both particles travel the same distance \( s \):
\[ \frac{1}{2} \left(\frac{qE}{m_e}\right) t_1^2 = \frac{1}{2} \left(\frac{qE}{m_p}\right) t_2^2 \]

### Step 5: Simplify the Equation
We can cancel \( \frac{1}{2} \) and \( qE \) from both sides:
\[ \frac{t_1^2}{m_e} = \frac{t_2^2}{m_p} \]

### Step 6: Rearranging for the Time Ratio
Rearranging gives:
\[ \frac{t_1^2}{t_2^2} = \frac{m_e}{m_p} \]

Taking the square root of both sides:
\[ \frac{t_1}{t_2} = \sqrt{\frac{m_e}{m_p}} \]

### Step 7: Finding the Ratio \( \frac{t_2}{t_1} \)
To find \( \frac{t_2}{t_1} \):
\[ \frac{t_2}{t_1} = \sqrt{\frac{m_p}{m_e}} \]

### Conclusion
Thus, the ratio \( \frac{t_2}{t_1} \) is nearly equal to:
\[ \frac{t_2}{t_1} \approx \sqrt{\frac{m_p}{m_e}} \]