A particle of mass and charge is released from rest in a uniform electric field. If there is no other force on the particle, the dependence of its kinetic energy on the distance travelled by it is correctly given by (graphs are schematic and not drawn to scale)

Options

A.
B.
C.
D.

Explanation

To solve the problem of how the kinetic energy of a charged particle changes as it travels through a uniform electric field, we can follow these steps:

### Step 1: Understand the problem
We have a particle of mass \( m \) and charge \( q \) that is released from rest in a uniform electric field \( E \). We need to find the relationship between its kinetic energy \( K \) and the distance \( x \) it travels.

**Hint:** Remember that the particle starts from rest, which means its initial velocity is zero.

### Step 2: Determine the force acting on the particle
The only force acting on the particle due to the electric field is given by:
\[ F = qE \]

**Hint:** The force on a charged particle in an electric field is the product of the charge and the electric field strength.

### Step 3: Calculate the acceleration of the particle
Using Newton's second law, we can find the acceleration \( a \) of the particle:
\[ a = \frac{F}{m} = \frac{qE}{m} \]

**Hint:** Acceleration is the force divided by mass.

### Step 4: Use kinematics to find the final velocity
Since the particle starts from rest, we can use the kinematic equation:
\[ v^2 = u^2 + 2as \]
Here, \( u = 0 \) (initial velocity), \( a = \frac{qE}{m} \), and \( s = x \) (distance traveled). Thus, we have:
\[ v^2 = 0 + 2 \left(\frac{qE}{m}\right) x \]
This simplifies to:
\[ v^2 = \frac{2qEx}{m} \]

**Hint:** This equation relates the final velocity to the distance traveled.

### Step 5: Find the kinetic energy
The kinetic energy \( K \) of the particle is given by:
\[ K = \frac{1}{2} mv^2 \]
Substituting the expression for \( v^2 \):
\[ K = \frac{1}{2} m \left(\frac{2qEx}{m}\right) \]
This simplifies to:
\[ K = qEx \]

**Hint:** The kinetic energy is directly related to the distance traveled in the electric field.

### Step 6: Analyze the relationship
From the equation \( K = qEx \), we see that the kinetic energy \( K \) is directly proportional to the distance \( x \). This indicates a linear relationship.

**Hint:** A linear relationship means that if you plot \( K \) against \( x \), you will get a straight line that passes through the origin.

### Step 7: Choose the correct graph
Given the options (A, B, C, D), the graph that represents a straight line passing through the origin is the correct answer.

**Final Answer:** The correct graph is option B.