Two large conducting plates are placed parallel to each other with a separation of d between them. An electron starting from rest near one of the plates reaches the other plate in time t. If e is the charge on the electron and m is its mass, then the surface charge density on the inner surface is

Options

A. `(d m e)/(4 pi epsilon_(0)t^(2))`
B. `( d m epsilon_(0)) /(4 pi et^(2))`
C. `(2 d m epsilon_(0))/(et^(2))`
D. None of the above

Explanation

To find the surface charge density on the inner surface of the two large conducting plates, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have two parallel conducting plates separated by a distance \( d \).
- An electron starts from rest near one of the plates and reaches the other plate in time \( t \).

2. **Electric Field Between the Plates**:
- The surface charge density on one plate is \( +\sigma \) and on the other plate is \( -\sigma \).
- The electric field \( E \) between the plates is given by:
\[ E = \frac{\sigma}{\epsilon_0} \]
- Since there are two plates, the net electric field \( E \) is:
\[ E = \frac{\sigma}{\epsilon_0} \]

3. **Force on the Electron**:
- The force \( F \) acting on the electron due to the electric field is:
\[ F = eE = e \left(\frac{\sigma}{\epsilon_0}\right) \]
- The acceleration \( a \) of the electron can be calculated using Newton's second law:
\[ a = \frac{F}{m} = \frac{e\sigma}{m\epsilon_0} \]

4. **Using the Equation of Motion**:
- The electron travels a distance \( d \) in time \( t \) starting from rest. The equation of motion is:
\[ d = ut + \frac{1}{2} a t^2 \]
- Since the initial velocity \( u = 0 \), we have:
\[ d = \frac{1}{2} a t^2 \]
- Substituting the expression for acceleration:
\[ d = \frac{1}{2} \left(\frac{e\sigma}{m\epsilon_0}\right) t^2 \]

5. **Solving for Surface Charge Density \( \sigma \)**:
- Rearranging the equation to solve for \( \sigma \):
\[ d = \frac{e\sigma t^2}{2m\epsilon_0} \]
\[ \sigma = \frac{2dm\epsilon_0}{et^2} \]

### Final Result:
The surface charge density on the inner surface of the plates is:
\[ \sigma = \frac{2dm\epsilon_0}{et^2} \]