Three resistance `P, Q, R` each of `2 Omega` and an unknown resistance `S` from the four amrs of a Wheatstone's bridge circuit. When a resistance of `6 Omega` is connected in parallel to `S` the bridge gets balanced. What is the value of `S` ?

Options

A. `2 Omega`
B. `3 Omega`
C. `6 Omega`
D. `1 Omega`

Explanation

To find the unknown resistance \( S \) in the Wheatstone bridge circuit, we can follow these steps:

### Step 1: Understand the Wheatstone Bridge Configuration
In a Wheatstone bridge, we have four resistances arranged in a diamond shape. The resistances are \( P, Q, R, \) and \( S \). In this case, \( P = 2 \, \Omega \), \( Q = 2 \, \Omega \), and \( R = 2 \, \Omega \). The resistance \( S \) is unknown.

### Step 2: Apply the Balancing Condition
When the bridge is balanced, the following condition holds:
\[ \frac{P}{Q} = \frac{R}{S_{\text{eq}}} \]
Where \( S_{\text{eq}} \) is the equivalent resistance of \( S \) when a \( 6 \, \Omega \) resistor is connected in parallel with it.

### Step 3: Calculate the Equivalent Resistance \( S_{\text{eq}} \)
The equivalent resistance \( S_{\text{eq}} \) of \( S \) and \( 6 \, \Omega \) in parallel is given by:
\[ S_{\text{eq}} = \frac{S \cdot 6}{S + 6} \]

### Step 4: Substitute Known Values into the Balancing Condition
Substituting the known values into the balancing condition:
\[ \frac{2}{2} = \frac{2}{S_{\text{eq}}} \]
This simplifies to:
\[ 1 = \frac{2}{S_{\text{eq}}} \]
Thus, we have:
\[ S_{\text{eq}} = 2 \, \Omega \]

### Step 5: Set Up the Equation
Now, we can set up the equation using the expression for \( S_{\text{eq}} \):
\[ \frac{S \cdot 6}{S + 6} = 2 \]

### Step 6: Cross Multiply to Solve for \( S \)
Cross multiplying gives:
\[ S \cdot 6 = 2(S + 6) \]
Expanding the right side:
\[ 6S = 2S + 12 \]

### Step 7: Rearrange the Equation
Rearranging the equation:
\[ 6S - 2S = 12 \]
This simplifies to:
\[ 4S = 12 \]

### Step 8: Solve for \( S \)
Dividing both sides by 4:
\[ S = 3 \, \Omega \]

### Conclusion
The value of the unknown resistance \( S \) is \( 3 \, \Omega \).

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