There are three concentric thin spherical shells A, B and C of radii R, 2R, and 3R. Shells A and Care given charges q and 2q and shell B is earthed. Then which of the given is correct ?

Options

A. charge on inner surface of shell C is `4/3`q
B. charge on outer surface of shell B is `-4/3` q
C. charge on outer surface of shell C is ` (2)/(3) q`
D. all the above

Answer

To solve the problem, we need to analyze the charges on the three concentric spherical shells A, B, and C, given that shell B is earthed. Here’s a step-by-step solution:

### Step 1: Understand the Setup
We have three concentric spherical shells:
- Shell A with radius R and charge \( q \)
- Shell B with radius \( 2R \) which is earthed (potential = 0)
- Shell C with radius \( 3R \) and charge \( 2q \)

### Step 2: Analyze Shell B (Earthed)
Since shell B is earthed, the potential on its surface must be zero. This means that the total charge on shell B must balance the charges from shells A and C.

### Step 3: Determine Charge on Shell B
Let \( Q' \) be the charge on the outer surface of shell B. The inner surface of shell B will have a charge of \( -q \) (to balance the charge from shell A) and \( -2q \) (to balance the charge from shell C). Therefore, the total charge on the inner surface of shell B is:
\[ Q_{inner, B} = -q - 2q = -3q \]
Since the total charge on shell B must be zero (as it is earthed), we have:
\[ Q' + (-3q) = 0 \implies Q' = 3q \]

### Step 4: Determine Charge on Shell C
The charge on the inner surface of shell C must balance the charge on the outer surface of shell B. Since the outer surface of shell B has a charge of \( 3q \), the inner surface of shell C will have a charge of \( +3q \).

The total charge on shell C is \( 2q \), so the charge on the outer surface of shell C will be:
\[ Q_{outer, C} = 2q - 3q = -q \]

### Step 5: Final Charges on Each Shell
- **Shell A**:
- Inner surface: \( 0 \)
- Outer surface: \( q \)

- **Shell B**:
- Inner surface: \( -3q \)
- Outer surface: \( 3q \)

- **Shell C**:
- Inner surface: \( +3q \)
- Outer surface: \( -q \)

### Conclusion
Now we can summarize the charges:
1. Charge on inner surface of shell C: \( +3q \)
2. Charge on outer surface of shell B: \( 3q \)
3. Charge on outer surface of shell C: \( -q \)

Since all calculations are consistent with the problem statement, the correct answer is that all the options provided are correct.