Three concentric conducting sphereical shells carry charges `+4Q` on the inner shell `-2Q` on the middle shel and `+6Q` on the outer shell. The charge on the inner surface of the outer shell is

Options

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

Explanation

To find the charge on the inner surface of the outer shell, we can follow these steps:

### Step 1: Understand the Configuration
We have three concentric spherical shells:
- Inner shell with charge \( +4Q \)
- Middle shell with charge \( -2Q \)
- Outer shell with charge \( +6Q \)

### Step 2: Determine the Effect of the Inner Shell
The inner shell with charge \( +4Q \) will induce a charge on the inner surface of the middle shell. According to electrostatic induction:
- The inner surface of the middle shell will have a charge of \( -4Q \) (to balance the \( +4Q \) from the inner shell).

### Step 3: Determine the Charge on the Outer Surface of the Middle Shell
Since the middle shell has a total charge of \( -2Q \) and the inner surface has \( -4Q \):
- The charge on the outer surface of the middle shell can be calculated as:
\[ \text{Charge on outer surface of middle shell} = \text{Total charge} + \text{Charge on inner surface} = -2Q + 4Q = +2Q \]

### Step 4: Determine the Effect of the Middle Shell on the Outer Shell
The outer shell has a total charge of \( +6Q \). The charge on the inner surface of the outer shell will be affected by the charge on the outer surface of the middle shell:
- The inner surface of the outer shell will have a charge of \( -2Q \) (to balance the \( +2Q \) from the middle shell).

### Step 5: Calculate the Total Charge on the Inner Surface of the Outer Shell
Now we can summarize the charges:
- The inner surface of the outer shell has a charge of \( -2Q \) (due to the \( +2Q \) from the middle shell).
- The total charge on the inner surface of the outer shell is:
\[ \text{Charge on inner surface of outer shell} = -4Q + 2Q = -2Q \]

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

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