A spring-block system undergoes vertical oscillations above a large horizontal metal sheet with uniform positive charge. The time period of the oscillations is T. If the block is given a charge Q, its time period of oscillation will be

Options

A. T
B. `gt T`
C. `lt T`
D. `gt T` if Q is positive and `lt T` if Q is negative

Answer

To solve the problem, we need to analyze how the time period of oscillation of a spring-block system is affected when the block is given a charge \( Q \) while it is oscillating above a large horizontal metal sheet with a uniform positive charge.

### Step-by-Step Solution:

1. **Understanding the System**:
- We have a spring-block system oscillating vertically above a charged metal sheet. The sheet has a uniform positive charge.

2. **Initial Time Period**:
- The time period of oscillation for a spring-block system is given by the formula:
\[ T = 2\pi \sqrt{\frac{m}{k}} \]
where \( m \) is the mass of the block and \( k \) is the spring constant.

3. **Effect of Charge on the Block**:
- When the block is given a charge \( Q \), it experiences an electric force due to the electric field created by the positively charged metal sheet. The electric field \( E \) due to an infinite sheet of charge is given by:
\[ E = \frac{\sigma}{2\epsilon_0} \]
where \( \sigma \) is the surface charge density and \( \epsilon_0 \) is the permittivity of free space.

4. **Force on the Block**:
- The electric force \( F_e \) acting on the block due to the electric field is:
\[ F_e = Q \cdot E \]
- This force will add to the gravitational force acting on the block.

5. **Effective Gravitational Force**:
- The effective gravitational force \( F_{eff} \) acting on the block when it has charge \( Q \) becomes:
\[ F_{eff} = mg + F_e = mg + Q \cdot E \]

6. **Impact on Time Period**:
- The time period \( T \) of the oscillation depends only on the mass \( m \) and the spring constant \( k \), and is independent of the gravitational force acting on the block. Therefore, even though the effective gravitational force changes due to the electric force, the time period remains:
\[ T = 2\pi \sqrt{\frac{m}{k}} \]

7. **Conclusion**:
- Since the time period \( T \) is independent of the gravitational force (and thus the electric force), the time period of oscillation after the block is given a charge \( Q \) remains the same as the initial time period \( T \).

### Final Answer:
The time period of oscillation remains the same, i.e., \( T \).