A uniform electrif field of intensit E is in the Y-negative direction. An electron of mass m and charge e is fired through the origin with initial velocity u in the X-positive direction. The displacement undergone of the electron during a time interval t is

Options

A. ut
B. `2Et^(2)//2m`
C. `sqrt(ut+(eEt^(2)//2m)^(2))`
D. `sqrt(ut-(eEt^(2)//2m)^(2))`

Explanation

To solve the problem, we need to analyze the motion of the electron under the influence of the electric field. The electric field is directed in the negative y-direction, and the electron is fired with an initial velocity in the positive x-direction. We will calculate the displacement of the electron in both the x and y directions and then find the total displacement.

### Step-by-Step Solution:

1. **Identify the Forces Acting on the Electron**:
- The electron has a charge \( e \) and is subjected to an electric field \( E \) in the negative y-direction.
- The force \( F \) on the electron due to the electric field is given by:
\[ F = eE \]
- This force causes an acceleration \( a_y \) in the y-direction, calculated using Newton's second law:
\[ F = ma \implies a_y = \frac{F}{m} = \frac{eE}{m} \]

2. **Analyze Motion in the X-Direction**:
- The initial velocity in the x-direction is \( u \).
- Since there is no force acting in the x-direction, the acceleration \( a_x = 0 \).
- The displacement in the x-direction \( S_x \) after time \( t \) is given by:
\[ S_x = u t + \frac{1}{2} a_x t^2 = u t + \frac{1}{2} \cdot 0 \cdot t^2 = u t \]

3. **Analyze Motion in the Y-Direction**:
- The initial velocity in the y-direction \( u_y = 0 \) (since it is fired horizontally).
- The displacement in the y-direction \( S_y \) after time \( t \) is given by:
\[ S_y = u_y t + \frac{1}{2} a_y t^2 = 0 + \frac{1}{2} \left(\frac{eE}{m}\right) t^2 = \frac{eE}{2m} t^2 \]

4. **Calculate the Total Displacement**:
- The total displacement \( S \) of the electron can be found using the Pythagorean theorem since the displacements in the x and y directions are perpendicular:
\[ S = \sqrt{S_x^2 + S_y^2} = \sqrt{(u t)^2 + \left(\frac{eE}{2m} t^2\right)^2} \]
- Substituting the expressions for \( S_x \) and \( S_y \):
\[ S = \sqrt{(u t)^2 + \left(\frac{eE}{2m} t^2\right)^2} \]

### Final Answer:
The displacement undergone by the electron during the time interval \( t \) is:
\[ S = \sqrt{(u t)^2 + \left(\frac{eE}{2m} t^2\right)^2} \]