A particle of mass M and charge q is released from rest in a region of uniform electric field of magnitude E. After a time t, the distance travelled by the charge is S and the kinetic energy attained by the particle is T. Then, the ratio T/S

Options

A. Remains constant with time t
B. Varies linearly with the mass M of the particle
C. Is independent of the charge q
D. Is independent of the magnitude of the electric field E

Explanation

To solve the problem, we need to find the ratio of the kinetic energy \( T \) attained by a particle of mass \( M \) and charge \( q \) after a time \( t \) to the distance \( S \) traveled by the particle in a uniform electric field of magnitude \( E \).

### Step-by-Step Solution:

1. **Determine the Force on the Particle**:
The force \( F \) acting on the charged particle in an electric field \( E \) is given by:
\[ F = qE \]

2. **Apply Newton's Second Law**:
According to Newton's second law, the acceleration \( a \) of the particle can be expressed as:
\[ F = Ma \implies a = \frac{F}{M} = \frac{qE}{M} \]

3. **Calculate the Distance Traveled (S)**:
Since the particle is released from rest, we can use the equation of motion:
\[ S = ut + \frac{1}{2} a t^2 \]
where \( u = 0 \) (initial velocity). Thus, we have:
\[ S = \frac{1}{2} a t^2 = \frac{1}{2} \left(\frac{qE}{M}\right) t^2 \]
Simplifying this gives:
\[ S = \frac{qEt^2}{2M} \]

4. **Calculate the Kinetic Energy (T)**:
The kinetic energy \( T \) of the particle after time \( t \) is given by:
\[ T = \frac{1}{2} M v^2 \]
To find \( v \), we use the equation:
\[ v = u + at = 0 + \left(\frac{qE}{M}\right) t = \frac{qEt}{M} \]
Now substituting \( v \) into the kinetic energy formula:
\[ T = \frac{1}{2} M \left(\frac{qEt}{M}\right)^2 = \frac{1}{2} M \cdot \frac{q^2 E^2 t^2}{M^2} = \frac{q^2 E^2 t^2}{2M} \]

5. **Find the Ratio \( \frac{T}{S} \)**:
Now, we can find the ratio of kinetic energy \( T \) to distance \( S \):
\[ \frac{T}{S} = \frac{\frac{q^2 E^2 t^2}{2M}}{\frac{qEt^2}{2M}} = \frac{q^2 E^2 t^2}{2M} \cdot \frac{2M}{qEt^2} \]
Simplifying this gives:
\[ \frac{T}{S} = \frac{qE}{1} = qE \]

### Final Answer:
The ratio \( \frac{T}{S} \) is:
\[ \frac{T}{S} = qE \]