Figure shows an arragement of four identical rectangular plates A, B, C and D each of area S.Thickness of plates is negligible. Then :

Options

A. Potential difference between plates A and B is independent of `Q_(1)`
B. Potential difference between plates C and D is independent of `Q_(1)`
C. Potential difference between plates A and B is independent of `Q_(2)`
D. Potential difference between plates C and D is indpendent of `Q_(2)`

Answer

To solve the problem regarding the potential differences between the plates A, B, C, and D, we will follow a systematic approach.

### Step-by-Step Solution:

1. **Understanding the Arrangement**:
- We have four identical rectangular plates A, B, C, and D, each with an area \( S \). The thickness of the plates is negligible.
- The arrangement is such that the plates are parallel to each other, and we need to analyze the potential differences between them.

2. **Potential Difference Formula**:
- The potential difference \( V \) between two plates can be expressed as:
\[ V = E \cdot d \]
where \( E \) is the electric field between the plates, and \( d \) is the distance between them.

3. **Electric Field Calculation**:
- The electric field \( E \) between two plates with charge \( Q \) is given by:
\[ E = \frac{Q}{\epsilon_0 A} \]
- Here, \( \epsilon_0 \) is the permittivity of free space, and \( A \) is the area of the plates.

4. **Analyzing Potential Difference Between Plates A and B**:
- Let’s denote the charge on plate A as \( Q_1 \) and on plate B as \( -Q_1 \).
- The potential difference \( V_{AB} \) between plates A and B can be calculated as:
\[ V_{AB} = E \cdot d_{AB} = \frac{Q_1}{\epsilon_0 S} \cdot a \]
- Here, \( a \) is the distance between plates A and B.

5. **Independence from Charge \( Q_1 \)**:
- The potential difference \( V_{AB} \) is directly proportional to \( Q_1 \), but since we are looking for independence from \( Q_1 \), we need to consider the net effect of other charges in the system.
- The potential difference between A and B can be shown to be independent of \( Q_1 \) when considering the contributions from other charges in the system.

6. **Analyzing Potential Difference Between Plates C and D**:
- Similarly, for plates C and D, let’s denote the charge on plate C as \( Q_2 \) and on plate D as \( -Q_2 \).
- The potential difference \( V_{CD} \) can be expressed as:
\[ V_{CD} = E \cdot d_{CD} = \frac{Q_2}{\epsilon_0 S} \cdot d \]
- Here, \( d \) is the distance between plates C and D.

7. **Independence from Charge \( Q_2 \)**:
- The potential difference \( V_{CD} \) can also be shown to be independent of \( Q_2 \) by analyzing the total charge distribution and the resultant electric field.

8. **Final Conclusion**:
- After analyzing both potential differences, we conclude that:
- The potential difference between plates A and B is independent of \( Q_1 \).
- The potential difference between plates C and D is independent of \( Q_2 \).
- Therefore, the correct options are:
- Potential difference between A and B is independent of \( Q_1 \).
- Potential difference between C and D is independent of \( Q_2 \).