A particle of mass .m. and charge q is placed at rest in a uniform electric field E and then released. The K.E. attained by the particle after moving a distance y is

Options

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

Answer

To find the kinetic energy attained by a particle of mass \( m \) and charge \( q \) 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 experiences an electrostatic force due to the electric field. The force \( F \) acting on the particle can be expressed as:
\[ F = qE \]
where \( q \) is the charge of the particle and \( E \) is the strength of the electric field.

2. **Determine the Acceleration of the Particle:**
Using Newton's second law, we can find the acceleration \( a \) of the particle:
\[ F = ma \implies a = \frac{F}{m} = \frac{qE}{m} \]

3. **Use Kinematic Equations to Relate Distance, Acceleration, and Velocity:**
Since the particle starts from rest, we can use the kinematic equation:
\[ v^2 = u^2 + 2as \]
where \( u \) is the initial velocity (which is 0), \( a \) is the acceleration, and \( s \) is the distance traveled. Substituting the known values:
\[ v^2 = 0 + 2 \left(\frac{qE}{m}\right) y \]
Simplifying this gives:
\[ v^2 = \frac{2qEy}{m} \]

4. **Calculate the Kinetic Energy:**
The kinetic energy \( KE \) of the particle is given by:
\[ KE = \frac{1}{2} mv^2 \]
Substituting the expression for \( v^2 \):
\[ KE = \frac{1}{2} m \left(\frac{2qEy}{m}\right) \]
The mass \( m \) cancels out:
\[ KE = qEy \]

### Final Result:
Thus, the kinetic energy attained by the particle after moving a distance \( y \) is:
\[ \boxed{qEy} \]