In a metre bridge, the length of the wire is 100 cm. At what position will the balance point be obtained if the two resistances are in the ratio 2 : 3?

Answer

To solve the problem of finding the balance point on a metre bridge when two resistances are in the ratio of 2:3, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have a metre bridge (100 cm long) and two resistances P and Q in the ratio 2:3. We need to find the position (length from one end) where the bridge balances.

2. **Assigning Values**:
Let the resistance P be 2x and the resistance Q be 3x, where x is a common factor.

3. **Using the Balance Condition**:
According to the principle of the metre bridge, the ratio of the lengths on either side of the bridge is equal to the ratio of the resistances:
\[ \frac{P}{Q} = \frac{L_1}{L_2} \]
where \(L_1\) is the length from one end to the balance point, and \(L_2\) is the remaining length. Since the total length of the wire is 100 cm, we have:
\[ L_2 = 100 - L_1 \]

4. **Setting Up the Equation**:
Substituting the values of P and Q into the ratio gives:
\[ \frac{2x}{3x} = \frac{L_1}{100 - L_1} \]
Simplifying this, we get:
\[ \frac{2}{3} = \frac{L_1}{100 - L_1} \]

5. **Cross-Multiplying**:
Cross-multiplying to eliminate the fraction:
\[ 2(100 - L_1) = 3L_1 \]
Expanding this, we have:
\[ 200 - 2L_1 = 3L_1 \]

6. **Rearranging the Equation**:
Bringing all terms involving \(L_1\) to one side:
\[ 200 = 3L_1 + 2L_1 \]
This simplifies to:
\[ 200 = 5L_1 \]

7. **Solving for \(L_1\)**:
Dividing both sides by 5 gives:
\[ L_1 = \frac{200}{5} = 40 \text{ cm} \]

8. **Conclusion**:
Therefore, the balance point will be obtained at a length of 40 cm from one end of the metre bridge.

### Final Answer:
The balance point will be at 40 cm.

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