An electron is moving towards x-axis. An electric field is along y-direction then path of electron is

Options

A. Circular
B. Elliptical
C. Parabola
D. None of these

Answer

To determine the path of an electron moving towards the x-axis while an electric field is directed along the y-axis, we can analyze the motion step by step.

### Step-by-Step Solution:

1. **Understanding the Initial Conditions**:
- The electron is initially moving along the x-axis with some initial velocity \( v_0 \).
- The electric field \( \vec{E} \) is directed along the y-axis.

2. **Force Acting on the Electron**:
- The force \( \vec{F} \) acting on the electron due to the electric field is given by:
\[ \vec{F} = q \vec{E} \]
- Since the electron has a negative charge (\( q = -e \)), the direction of the force will be opposite to that of the electric field. Thus, if the electric field is in the positive y-direction, the force on the electron will be in the negative y-direction.

3. **Calculating the Acceleration**:
- The acceleration \( \vec{a} \) of the electron can be calculated using Newton's second law:
\[ \vec{a} = \frac{\vec{F}}{m} = \frac{-e \vec{E}}{m} \]
- This indicates that the electron will experience a downward acceleration (in the negative y-direction).

4. **Motion in the y-Direction**:
- The initial velocity in the y-direction is zero (\( u_y = 0 \)).
- Using the second equation of motion for the y-direction:
\[ y = u_y t + \frac{1}{2} a_y t^2 \]
- Substituting \( u_y = 0 \) and \( a_y = -\frac{eE}{m} \):
\[ y = 0 + \frac{1}{2} \left(-\frac{eE}{m}\right) t^2 = -\frac{1}{2} \frac{eE}{m} t^2 \]

5. **Motion in the x-Direction**:
- The motion in the x-direction is uniform since there is no force acting in that direction:
\[ x = v_0 t \]
- Rearranging gives:
\[ t = \frac{x}{v_0} \]

6. **Substituting Time into the y-Equation**:
- Substitute \( t = \frac{x}{v_0} \) into the equation for \( y \):
\[ y = -\frac{1}{2} \frac{eE}{m} \left(\frac{x}{v_0}\right)^2 \]
- Simplifying gives:
\[ y = -\frac{eE}{2mv_0^2} x^2 \]

7. **Identifying the Path**:
- The equation \( y = -\frac{eE}{2mv_0^2} x^2 \) is a quadratic equation in \( x \) and represents a parabola that opens downwards.

### Conclusion:
The path of the electron is parabolic due to the influence of the electric field acting in the y-direction while the electron moves in the x-direction.