A particle of mass m and charge q is placed at rest in a uniform electric field E and then released, the kinetic energy attained by the particle after moving a distance y will be

Options

A. `q^(2)Ey`
B. `qEy`
C. `aE^(2)y`
D. `qEy^(2)`

Explanation

To solve the problem of finding the kinetic energy attained by a charged particle after moving a distance \( y \) in a uniform electric field \( E \), we can follow these steps:

### Step-by-Step Solution

1. **Identify the Forces Acting on the Particle**:
The particle has a charge \( q \) and is placed in a uniform electric field \( E \). The force \( F \) acting on the particle due to the electric field is given by:
\[ F = qE \]

2. **Apply Newton's Second Law**:
According to Newton's second law, the force acting on an object is equal to the mass of the object multiplied by its acceleration \( a \):
\[ F = ma \]
Therefore, we can equate the two expressions for force:
\[ qE = ma \]
From this, we can solve for acceleration \( a \):
\[ a = \frac{qE}{m} \]

3. **Use Kinematic Equation to Find Velocity**:
The particle starts from rest, so its initial velocity \( u = 0 \). We can use the kinematic equation that relates initial velocity, final velocity, acceleration, and distance:
\[ v^2 = u^2 + 2as \]
Substituting \( u = 0 \), \( a = \frac{qE}{m} \), and \( s = y \):
\[ v^2 = 0 + 2\left(\frac{qE}{m}\right)y \]
Thus, we have:
\[ v^2 = \frac{2qEy}{m} \]

4. **Calculate the Kinetic Energy**:
The kinetic energy \( KE \) of the particle when it has reached the velocity \( v \) is given by:
\[ KE = \frac{1}{2}mv^2 \]
Substituting \( v^2 \) from the previous step:
\[ KE = \frac{1}{2}m\left(\frac{2qEy}{m}\right) \]
Simplifying this expression:
\[ KE = \frac{1}{2} \cdot 2qEy = qEy \]

5. **Final Result**:
Therefore, the kinetic energy attained by the particle after moving a distance \( y \) in the electric field \( E \) is:
\[ KE = qEy \]