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 the two resistances are in the ratio of 2:3, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Setup**:
A metre bridge consists of a wire of length 100 cm. When two resistances \( P \) and \( Q \) are connected at either end of the bridge, a jockey is used to find the balance point along the wire.

2. **Define the Balance Point**:
Let the balance point be at a distance \( x \) cm from one end of the bridge. Consequently, the remaining length from the other end will be \( 100 - x \) cm.

3. **Use the Principle of the Meter Bridge**:
According to the principle of the metre bridge, the ratio of the resistances is equal to the ratio of the lengths of the wire on either side of the jockey at the balance point:
\[ \frac{P}{Q} = \frac{x}{100 - x} \]

4. **Substitute the Given Ratio**:
We know that the resistances are in the ratio \( \frac{P}{Q} = \frac{2}{3} \). Therefore, we can substitute this into our equation:
\[ \frac{2}{3} = \frac{x}{100 - x} \]

5. **Cross-Multiply to Solve for \( x \)**:
Cross-multiplying gives:
\[ 2(100 - x) = 3x \]
Expanding this:
\[ 200 - 2x = 3x \]

6. **Rearrange the Equation**:
Combine like terms:
\[ 200 = 3x + 2x \]
\[ 200 = 5x \]

7. **Solve for \( x \)**:
Divide both sides by 5:
\[ x = \frac{200}{5} = 40 \text{ cm} \]

### Final Answer:
The balance point will be obtained at \( 40 \) cm from one end of the metre bridge.