Three concentric conducting spherical shells carry charges as : `+4Q` on the inner shell, `-2Q` on the middle shell and `-5Q` on the outer shell. The charge on the inner surface of the outer shell is

Options

A. 0
B. `4Q`
C. `-Q`
D. `-2Q`

Answer

To find the charge on the inner surface of the outer shell, we will analyze the charge distribution on the concentric conducting spherical shells step by step.

### Step-by-Step Solution:

1. **Identify the Charges on the Shells**:
- Inner shell (A) has a charge of \( +4Q \).
- Middle shell (B) has a charge of \( -2Q \).
- Outer shell (C) has a charge of \( -5Q \).

2. **Determine Charge Induction on the Inner Surface of the Middle Shell (B)**:
- The inner shell (A) has a charge of \( +4Q \).
- By electrostatic induction, the inner surface of the middle shell (B) will acquire a charge of \( -4Q \) to neutralize the electric field inside the conductor.

3. **Determine Charge on the Outer Surface of the Middle Shell (B)**:
- The total charge on the middle shell (B) is \( -2Q \).
- Since the inner surface has \( -4Q \), the outer surface of the middle shell will have:
\[ \text{Charge on outer surface of B} = \text{Total charge of B} - \text{Charge on inner surface of B} = -2Q - (-4Q) = +2Q \]

4. **Determine Charge Induction on the Inner Surface of the Outer Shell (C)**:
- The outer surface of the middle shell (B) has a charge of \( +2Q \).
- Therefore, the inner surface of the outer shell (C) will acquire a charge of \( -2Q \) to neutralize the electric field due to the \( +2Q \) charge on the outer surface of the middle shell.

5. **Final Charge on the Inner Surface of the Outer Shell**:
- Thus, the charge on the inner surface of the outer shell (C) is \( -2Q \).

### Final Answer:
The charge on the inner surface of the outer shell is \( -2Q \).

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