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`

Explanation

To find the ratio \( P_2 : P_1 \), we will follow these steps:

### Step 1: Understand Power Dissipation Formula
The power dissipation \( P \) in an electrical circuit is given by the formula:
\[ P = \frac{V^2}{R} \]
where \( V \) is the voltage and \( R \) is the resistance.

### Step 2: Calculate \( P_1 \)
When the heater wire is connected to a 230 V supply, the power dissipation \( P_1 \) can be expressed as:
\[ P_1 = \frac{V^2}{R} = \frac{230^2}{R} \]

### Step 3: Determine Resistance of Each Piece
When the wire is cut into three equal pieces, the resistance of each piece becomes:
\[ R_{\text{piece}} = \frac{R}{3} \]

### Step 4: Calculate Equivalent Resistance in Parallel
When these three pieces are connected in parallel, the equivalent resistance \( R_{\text{eq}} \) can be calculated using the formula for resistors in parallel:
\[ \frac{1}{R_{\text{eq}}} = \frac{1}{R/3} + \frac{1}{R/3} + \frac{1}{R/3} \]
This simplifies to:
\[ \frac{1}{R_{\text{eq}}} = \frac{3}{R/3} = \frac{9}{R} \]
Thus, the equivalent resistance is:
\[ R_{\text{eq}} = \frac{R}{9} \]

### Step 5: Calculate \( P_2 \)
Now, using the equivalent resistance, we can find the power dissipation \( P_2 \) when the three pieces are connected in parallel:
\[ P_2 = \frac{V^2}{R_{\text{eq}}} = \frac{230^2}{R/9} = \frac{230^2 \cdot 9}{R} \]

### Step 6: Find the Ratio \( P_2 : P_1 \)
Now, we can find the ratio \( P_2 : P_1 \):
\[ \frac{P_2}{P_1} = \frac{\frac{230^2 \cdot 9}{R}}{\frac{230^2}{R}} = \frac{230^2 \cdot 9}{R} \cdot \frac{R}{230^2} = 9 \]
Thus, the ratio \( P_2 : P_1 \) is:
\[ P_2 : P_1 = 9 : 1 \]

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

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