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 of how the potential energy \( U \) of a particle with mass \( m \) and charge \( -q \) changes as it moves through a uniform electric field, we can follow these steps:

### Step-by-Step Solution

1. **Understand the Electric Field**:
The electric field \( E \) is uniform, meaning it has a constant magnitude and direction throughout the region. The force \( F \) acting on the charged particle is given by:
\[ F = qE \]
Since the charge is negative, the force will act in the opposite direction to the electric field.

2. **Potential Energy Definition**:
The potential energy \( U \) of a charge \( q \) in an electric field is given by:
\[ U = -qV \]
where \( V \) is the electric potential at a distance \( x \).

3. **Electric Potential in a Uniform Electric Field**:
The electric potential \( V \) at a distance \( x \) in a uniform electric field can be expressed as:
\[ V = -Ex \]
Here, \( E \) is the magnitude of the electric field, and the negative sign indicates that the potential decreases in the direction of the electric field.

4. **Substituting for Potential Energy**:
Substituting the expression for \( V \) into the potential energy equation gives:
\[ U = -q(-Ex) = qEx \]
This indicates that the potential energy \( U \) is directly proportional to the distance \( x \) travelled by the particle.

5. **Initial Conditions**:
We are given that the initial potential energy is zero when the particle is at rest. Therefore, at \( x = 0 \), \( U = 0 \).

6. **Graphical Representation**:
Since \( U \) is proportional to \( x \), the graph of potential energy \( U \) versus distance \( x \) will be a straight line passing through the origin. The slope of this line will be \( qE \).

7. **Conclusion**:
The dependence of potential energy on the distance travelled by the particle is linear, and the correct representation of this relationship is a straight line graph starting from the origin.

### Final Expression:
Thus, the potential energy \( U \) as a function of distance \( x \) is given by:
\[ U(x) = qEx \]