When a heater wire is connected to a 230 V main supply power dissipation is `P_(1)`. Now the same wire is cut into three equal pieces and are connected in parallel to the same source, power dissipation in this case is `P_(2)`. The ratio `P_(2)`, `P_(1)`, is

Options

A. `1:3`
B. `3:1`
C. `9:1`
D. `1:9`

Answer

To solve the problem, we need to determine the ratio of power dissipation when a heater wire is connected to a voltage supply in two different configurations: first as a whole wire and then as three equal pieces connected in parallel.

### Step-by-Step Solution:

1. **Identify the Power Dissipation in the First Case (P1)**:
- When the heater wire of resistance \( R \) is connected to a voltage supply \( V \), the power dissipation \( P_1 \) can be calculated using the formula:
\[ P_1 = \frac{V^2}{R} \]
- Here, \( V = 230 \, \text{V} \).

2. **Determine the Resistance of Each Piece After Cutting**:
- When the wire is cut into three equal pieces, the length of each piece becomes \( \frac{L}{3} \).
- The resistance of one piece \( R_1 \) can be calculated as:
\[ R_1 = \frac{R}{3} \]

3. **Calculate the Equivalent Resistance When Connected in Parallel**:
- When three resistances \( R_1 \) are connected in parallel, the equivalent resistance \( R_{eq} \) is given by:
\[ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_1} + \frac{1}{R_1} = \frac{3}{R_1} \]
- Substituting \( R_1 = \frac{R}{3} \):
\[ \frac{1}{R_{eq}} = \frac{3}{\frac{R}{3}} = \frac{9}{R} \implies R_{eq} = \frac{R}{9} \]

4. **Identify the Power Dissipation in the Second Case (P2)**:
- The power dissipation \( P_2 \) when the three pieces are connected in parallel can be calculated using:
\[ P_2 = \frac{V^2}{R_{eq}} = \frac{V^2}{\frac{R}{9}} = \frac{9V^2}{R} \]

5. **Calculate the Ratio of Power Dissipation (P2/P1)**:
- Now, we can find the ratio of \( P_2 \) to \( P_1 \):
\[ \frac{P_2}{P_1} = \frac{\frac{9V^2}{R}}{\frac{V^2}{R}} = \frac{9V^2}{R} \cdot \frac{R}{V^2} = 9 \]

6. **Final Result**:
- Therefore, the ratio \( \frac{P_2}{P_1} \) is:
\[ \frac{P_2}{P_1} = 9 \]

### Conclusion:
The ratio \( P_2 : P_1 \) is \( 9 : 1 \).