A charge Q is placed at the centre of an uncharged, hollow metallic sphere of radius a. (a) Find the surface charge density on the inner surface and on the outer surface. (b) If a charge q is put on the sphere, what would be the surface charge densities on the inner and the outer surface? (c) Find the electric field inside the sphere at a distance x from the centre in the situations (a) and (b).

Explanation

To solve the problem step by step, we will address each part of the question systematically.

### Part (a): Surface Charge Density on Inner and Outer Surfaces

1. **Understanding the Setup**: We have a hollow metallic sphere of radius \( a \) with a charge \( Q \) placed at its center. The sphere is initially uncharged.

2. **Induced Charges**: Since the charge \( Q \) is placed at the center, it will induce a charge of \( -Q \) on the inner surface of the sphere. This is because the electric field inside a conductor in electrostatic equilibrium must be zero.

3. **Charge on the Outer Surface**: The total charge of the sphere must remain zero (since it was initially uncharged). Therefore, the outer surface must have a charge of \( +Q \) to balance the \( -Q \) on the inner surface.

4. **Calculating Surface Charge Densities**:
- **Inner Surface Charge Density (\( \sigma_{\text{inner}} \))**:
\[ \sigma_{\text{inner}} = \frac{-Q}{A_{\text{inner}}} = \frac{-Q}{4\pi a^2} \]
- **Outer Surface Charge Density (\( \sigma_{\text{outer}} \))**:
\[ \sigma_{\text{outer}} = \frac{Q}{A_{\text{outer}}} = \frac{Q}{4\pi a^2} \]

### Part (b): Surface Charge Densities with Charge \( q \) on the Sphere

1. **Adding Charge \( q \)**: When a charge \( q \) is placed on the outer surface of the sphere, the total charge on the outer surface becomes \( Q + q \).

2. **Inner Surface Charge Density**: The inner surface charge density remains the same as before because the electric field inside the hollow sphere is still zero. Thus:
\[ \sigma_{\text{inner}} = \frac{-Q}{4\pi a^2} \]

3. **Outer Surface Charge Density**:
\[ \sigma_{\text{outer}} = \frac{Q + q}{4\pi a^2} \]

### Part (c): Electric Field Inside the Sphere at Distance \( x \)

1. **Electric Field in Situation (a)**:
- Inside the hollow metallic sphere, the electric field is zero because of the properties of conductors in electrostatic equilibrium.
\[ E_{\text{inside}} = 0 \quad \text{(for situation (a))} \]

2. **Electric Field in Situation (b)**:
- Similarly, since the charge \( Q \) is at the center and the electric field inside a conductor is always zero, the electric field remains zero.
\[ E_{\text{inside}} = 0 \quad \text{(for situation (b))} \]

### Summary of Answers:
- **(a)**:
- Inner Surface Charge Density: \( \sigma_{\text{inner}} = \frac{-Q}{4\pi a^2} \)
- Outer Surface Charge Density: \( \sigma_{\text{outer}} = \frac{Q}{4\pi a^2} \)

- **(b)**:
- Inner Surface Charge Density: \( \sigma_{\text{inner}} = \frac{-Q}{4\pi a^2} \)
- Outer Surface Charge Density: \( \sigma_{\text{outer}} = \frac{Q + q}{4\pi a^2} \)

- **(c)**:
- Electric Field Inside Sphere: \( E = 0 \) (for both situations (a) and (b))