A solid spherical conductor of radius R has a spherical cavity of radius at its centre. A charge is kept at the centre. The charge at the inner surface, outer surface are respectively.

Options

A. `+Q,-Q`
B. `-Q,+Q`
C. `Q, - Q`
D. `+Q,0`

Explanation

To solve the problem, we need to analyze the situation involving a solid spherical conductor with a cavity and a charge placed at the center. We will use Gauss's law to determine the charges induced on the inner and outer surfaces of the conductor.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a solid spherical conductor of radius \( R \).
- There is a spherical cavity of radius \( A \) at the center of the conductor.
- A charge \( +Q \) is placed at the center of the cavity.

2. **Applying Gauss's Law Inside the Cavity**:
- Inside the cavity, the electric field \( E \) is zero because the conductor shields the cavity from external electric fields.
- According to Gauss's law, the electric flux \( \Phi \) through a closed surface is given by:
\[ \Phi = E \cdot A \]
- Since \( E = 0 \), the electric flux \( \Phi \) is also zero. Thus, the charge enclosed by any Gaussian surface inside the cavity must also be zero.

3. **Determining Charge on the Inner Surface**:
- Since there is a charge \( +Q \) at the center, to maintain the zero electric field inside the cavity, the inner surface of the conductor must have an induced charge of \( -Q \).
- This is because the total enclosed charge must equal zero, so:
\[ Q_{\text{enclosed}} = +Q + Q_{\text{inner surface}} = 0 \implies Q_{\text{inner surface}} = -Q \]

4. **Determining Charge on the Outer Surface**:
- The conductor as a whole must remain electrically neutral. If the inner surface has a charge of \( -Q \), the total charge on the conductor must balance this.
- Therefore, the outer surface must have a charge of \( +Q \) to ensure that the total charge of the conductor remains zero.
- Thus, the charge on the outer surface is:
\[ Q_{\text{outer surface}} = +Q \]

5. **Final Answer**:
- The charge at the inner surface is \( -Q \) and the charge at the outer surface is \( +Q \).

### Summary of Charges:
- Charge at the inner surface: \( -Q \)
- Charge at the outer surface: \( +Q \)