A wire when connected to `220V` mains sapply power dissipation `P_(1)` Now the wire is cut into two equal pieces which are connected in parallel to the same apply power dissipation in this case is `P_(2)` then `P_(1) : P_(2)` is

Options

A. 1
B. 4
C. 2
D. 3

Answer

To solve the problem, we need to analyze the power dissipation in two scenarios: when the wire is connected to the mains supply and when it is cut into two equal pieces and connected in parallel.

### Step-by-Step Solution:

1. **Understanding Power Dissipation in the Original Wire:**
- The power dissipation \( P_1 \) when the wire is connected to a \( 220V \) mains supply can be expressed using the formula:
\[ P_1 = \frac{V^2}{R} \]
- Here, \( V = 220V \) and \( R \) is the resistance of the wire.

2. **Calculating Power Dissipation \( P_1 \):**
- Substituting the value of \( V \):
\[ P_1 = \frac{(220)^2}{R} \]

3. **Cutting the Wire into Two Equal Pieces:**
- When the wire is cut into two equal pieces, each piece will have half the length of the original wire. Since resistance \( R \) is directly proportional to length, the resistance of each piece becomes:
\[ R' = \frac{R}{2} \]

4. **Connecting the Pieces in Parallel:**
- When two resistances \( R' \) are connected in parallel, the equivalent resistance \( R_{eq} \) can be calculated using the formula for resistors in parallel:
\[ \frac{1}{R_{eq}} = \frac{1}{R'} + \frac{1}{R'} = \frac{2}{R'} \]
- Thus,
\[ R_{eq} = \frac{R'}{2} = \frac{R/2}{2} = \frac{R}{4} \]

5. **Calculating Power Dissipation \( P_2 \):**
- The power dissipation \( P_2 \) when the two pieces are connected in parallel to the same \( 220V \) supply is given by:
\[ P_2 = \frac{V^2}{R_{eq}} \]
- Substituting \( R_{eq} \):
\[ P_2 = \frac{(220)^2}{R/4} = \frac{(220)^2 \cdot 4}{R} \]

6. **Finding the Ratio \( P_1 : P_2 \):**
- Now, we can find the ratio \( \frac{P_1}{P_2} \):
\[ \frac{P_1}{P_2} = \frac{\frac{(220)^2}{R}}{\frac{(220)^2 \cdot 4}{R}} = \frac{1}{4} \]
- Therefore, the ratio \( P_1 : P_2 \) is:
\[ P_1 : P_2 = 1 : 4 \]

### Final Answer:
The ratio \( P_1 : P_2 \) is \( 1 : 4 \).