The radii of two concentric spherical conducting shells are `r_(1) and r_(2)(gt r_(1))`. The change on the oute shell is q. The charge on the inner shell which is connected to the earth is

Options

A. `q((r_(2))/(r_(1)))`
B. `q^(2)((r_(1))/(r_(2)))`
C. `-q(r_(1)//r_(2))`
D. `q^(2)((r_(2))/(r_(1)))`

Answer

To solve the problem, we need to determine the charge on the inner spherical conducting shell, which is connected to the earth, given that the outer shell has a charge \( q \).

### Step-by-Step Solution:

1. **Understanding the System**:
- We have two concentric spherical conducting shells.
- The inner shell has a radius \( r_1 \) and the outer shell has a radius \( r_2 \) where \( r_2 > r_1 \).
- The outer shell carries a charge \( q \).
- The inner shell is connected to the earth.

2. **Potential of the Inner Shell**:
- When a conductor is connected to the earth, its potential becomes zero because the earth can provide or absorb an infinite amount of charge.
- Therefore, the potential \( V_1 \) of the inner shell must be zero.

3. **Calculating the Potential**:
- The potential \( V_1 \) at the surface of the inner shell due to its own charge \( Q_1 \) and the charge \( q \) on the outer shell can be expressed as:
\[ V_1 = k \frac{Q_1}{r_1} + k \frac{q}{r_2} \]
- Here, \( k \) is the electrostatic constant.

4. **Setting the Potential to Zero**:
- Since the potential \( V_1 \) is zero (because the inner shell is grounded), we can set up the equation:
\[ k \frac{Q_1}{r_1} + k \frac{q}{r_2} = 0 \]

5. **Simplifying the Equation**:
- We can factor out \( k \) (since \( k \) is a constant and does not equal zero):
\[ \frac{Q_1}{r_1} + \frac{q}{r_2} = 0 \]

6. **Solving for \( Q_1 \)**:
- Rearranging the equation gives:
\[ \frac{Q_1}{r_1} = -\frac{q}{r_2} \]
- Multiplying both sides by \( r_1 \):
\[ Q_1 = -\frac{q \cdot r_1}{r_2} \]

7. **Conclusion**:
- The charge on the inner shell, which is connected to the earth, is:
\[ Q_1 = -\frac{q \cdot r_1}{r_2} \]

### Final Answer:
The charge on the inner shell is \( Q_1 = -\frac{q \cdot r_1}{r_2} \).

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