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. `M_(e)/M_(p)`

Answer

To solve the problem, we need to analyze the motion of the electron and the proton in a uniform electric field. Let's break it down step by step.

### Step 1: Understanding the Forces
Both the electron and the proton experience an electric force when placed in a uniform electric field \( E \). The force \( F \) acting on each particle can be expressed as:
- For the electron:
\[ F_e = eE \]
- For the proton:
\[ F_p = eE \]
where \( e \) is the charge of the electron and proton (equal in magnitude).

### Step 2: Calculating Accelerations
The acceleration \( a \) of each particle can be found using Newton's second law, \( F = ma \):
- For the electron:
\[ a_e = \frac{F_e}{m_e} = \frac{eE}{m_e} \]
- For the proton:
\[ a_p = \frac{F_p}{m_p} = \frac{eE}{m_p} \]

### Step 3: Using the Kinematic Equation
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:
\[ s = \frac{1}{2} a_e t_1^2 \implies s = \frac{1}{2} \left(\frac{eE}{m_e}\right) t_1^2 \]
For the proton:
\[ s = \frac{1}{2} a_p t_2^2 \implies s = \frac{1}{2} \left(\frac{eE}{m_p}\right) t_2^2 \]

### Step 4: Setting the Equations Equal
Since both expressions for \( s \) are equal, we can set them equal to each other:
\[ \frac{1}{2} \left(\frac{eE}{m_e}\right) t_1^2 = \frac{1}{2} \left(\frac{eE}{m_p}\right) t_2^2 \]
The \( \frac{1}{2} \) and \( eE \) terms cancel out:
\[ \frac{t_1^2}{m_e} = \frac{t_2^2}{m_p} \]

### Step 5: Rearranging for the Ratio
Rearranging the equation gives:
\[ \frac{t_2^2}{t_1^2} = \frac{m_p}{m_e} \]
Taking the square root of both sides yields:
\[ \frac{t_2}{t_1} = \sqrt{\frac{m_p}{m_e}} \]

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