In a Wheatstone's bridge, three resistances P,Q and R connected in the three arms and the fourth arm is formed by two resistances `S_1 and S_2` connected in parallel. The condition for the bridge to be balanced will be

Options

A. `(R(s_(1)+s_(2)))/(s_(1)s_(2))`
B. `(s_(1)s_(2))/(R(s_(1)+s_(2)))`
C. `(R s_(1)s_(2))/((s_(1)+s_(2)))`
D. `((s_(1)+s_(2)))/(R s_(1)s_(2))`

Answer

To find the condition for a Wheatstone bridge to be balanced when three resistances \( P, Q, R \) are connected in three arms and the fourth arm is formed by two resistances \( S_1 \) and \( S_2 \) connected in parallel, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Wheatstone Bridge Configuration**:
- Draw the Wheatstone bridge circuit with resistances \( P, Q, R \) in three arms and \( S_1 \) and \( S_2 \) in parallel in the fourth arm.

2. **Condition for Balance**:
- The bridge is balanced when the current through the galvanometer \( I_G \) is zero. This means that the potential difference across the galvanometer is zero.

3. **Using the Balance Condition**:
- For the bridge to be balanced, the ratio of the resistances in one arm must equal the ratio of the resistances in the other arm:
\[ \frac{P}{Q} = \frac{R}{S_{eq}} \]
where \( S_{eq} \) is the equivalent resistance of \( S_1 \) and \( S_2 \) in parallel.

4. **Calculating the Equivalent Resistance**:
- The equivalent resistance \( S_{eq} \) of two resistances \( S_1 \) and \( S_2 \) in parallel is given by:
\[ S_{eq} = \frac{S_1 \cdot S_2}{S_1 + S_2} \]

5. **Substituting \( S_{eq} \) into the Balance Condition**:
- Substitute \( S_{eq} \) into the balance condition:
\[ \frac{P}{Q} = \frac{R}{\frac{S_1 \cdot S_2}{S_1 + S_2}} \]

6. **Cross-Multiplying**:
- Cross-multiply to eliminate the fraction:
\[ P \cdot \left(\frac{S_1 \cdot S_2}{S_1 + S_2}\right) = Q \cdot R \]

7. **Rearranging the Equation**:
- Rearranging gives:
\[ P \cdot (S_1 + S_2) = Q \cdot R \cdot \frac{S_1 \cdot S_2}{S_1 \cdot S_2} \]

8. **Final Formulation**:
- This leads to the final balanced condition:
\[ P \cdot (S_1 + S_2) = Q \cdot R \implies \frac{P}{Q} = \frac{R \cdot (S_1 + S_2)}{S_1 \cdot S_2} \]

### Final Answer:
The condition for the Wheatstone bridge to be balanced is:
\[ \frac{P}{Q} = \frac{R \cdot (S_1 + S_2)}{S_1 \cdot S_2} \]