Four resistances `6Omega,6Omega,6Omega` and `18Omega` form a Wheatstone bridge. Find the resistance which connected across the `18Omega` resistance will balance the network.

Answer

To solve the problem of finding the resistance that needs to be connected across the 18Ω resistance in a Wheatstone bridge to balance the network, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Wheatstone Bridge Configuration**:
- The Wheatstone bridge consists of four resistances arranged in a diamond shape. In this case, we have three resistances of 6Ω each and one resistance of 18Ω.

2. **Label the Resistances**:
- Let the resistances be labeled as follows:
- R1 = 6Ω (top)
- R2 = 6Ω (bottom)
- R3 = 6Ω (left)
- R4 = 18Ω (right)

3. **Condition for Balance**:
- For the Wheatstone bridge to be balanced, the ratio of the resistances in one branch must equal the ratio in the other branch:
\[ \frac{R1}{R2} = \frac{R4}{R3 + R} \]
- Here, R is the unknown resistance we need to find, connected across the 18Ω resistance.

4. **Substituting Known Values**:
- Substitute the known values into the balance equation:
\[ \frac{6}{6} = \frac{18}{6 + R} \]
- This simplifies to:
\[ 1 = \frac{18}{6 + R} \]

5. **Cross-Multiplying**:
- Cross-multiply to eliminate the fraction:
\[ 6 + R = 18 \]

6. **Solving for R**:
- Rearranging the equation gives:
\[ R = 18 - 6 \]
\[ R = 12Ω \]

7. **Final Result**:
- The resistance that needs to be connected across the 18Ω resistance to balance the Wheatstone bridge is **12Ω**.