A particle of mass `m` and charge `-q` is released from rest in a uniform electric field. If there is no other force on the particle, the dependence of its potential energy on the distance `x` travelled by it is correctly given by (assume initial potential energy to be zero ,graphs are schematic and not drawn to scale)

Options

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Explanation

To solve the problem, we need to analyze the behavior of a charged particle in a uniform electric field and how its potential energy changes as it moves.

### Step-by-Step Solution:

1. **Understanding the System**:
- We have a particle with mass \( m \) and charge \( -q \) in a uniform electric field \( \mathbf{E} \).
- The electric field exerts a force on the particle given by \( \mathbf{F} = q\mathbf{E} \). Since the charge is negative, the force will act in the opposite direction of the electric field.

2. **Force on the Particle**:
- The force acting on the particle is \( \mathbf{F} = -q\mathbf{E} \).
- If we assume the electric field is directed in the positive \( x \)-direction, then the force on the particle will be in the negative \( x \)-direction.

3. **Work Done by the Electric Field**:
- The work done \( W \) by the electric field when the particle moves a distance \( x \) is given by:
\[ W = \mathbf{F} \cdot \mathbf{d} = (-qE)(-x) = qEx \]
- Here, \( \mathbf{d} \) is the displacement vector in the direction of the movement.

4. **Change in Potential Energy**:
- The change in potential energy \( \Delta U \) is related to the work done by the conservative force:
\[ \Delta U = -W \]
- Substituting the expression for work done:
\[ \Delta U = -qEx \]

5. **Initial Potential Energy**:
- We are given that the initial potential energy \( U_i = 0 \). Therefore, the final potential energy \( U_f \) is:
\[ U_f = U_i + \Delta U = 0 - qEx = -qEx \]

6. **Dependence of Potential Energy on Distance**:
- The final expression for potential energy as a function of distance \( x \) is:
\[ U(x) = -qEx \]
- This shows that the potential energy decreases linearly with increasing \( x \) since the coefficient of \( x \) is negative.

7. **Graphical Representation**:
- The graph of potential energy \( U \) versus distance \( x \) will be a straight line with a negative slope, indicating that as \( x \) increases, \( U \) decreases.

### Conclusion:
The dependence of potential energy on the distance \( x \) travelled by the particle is given by:
\[ U(x) = -qEx \]
This indicates a linear decrease in potential energy with distance.