A heater coil c onnected across a give potential difference has power P . Now , the coil is cut into two equal halves and joined in parallel . Across the same potential difference, this combination has power

Options

A. A. P
B. B. 4P
C. C. P/4
D. D. 2P

Answer

To solve the problem step by step, let's break down the process:

### Step 1: Understand the initial conditions
The heater coil has a power \( P \) when connected across a potential difference \( V \). The power \( P \) can be expressed using the formula:
\[ P = \frac{V^2}{R} \]
where \( R \) is the resistance of the coil.

### Step 2: Determine the resistance of the original coil
From the power formula, we can rearrange it to find the resistance:
\[ R = \frac{V^2}{P} \]

### Step 3: Cut the coil into two equal halves
When the coil is cut into two equal halves, each half will have a resistance of:
\[ R_1 = R_2 = \frac{R}{2} \]

### Step 4: Calculate the equivalent resistance when joined in parallel
When these two halves are connected in parallel, the equivalent resistance \( R' \) can be calculated using the formula for resistors in parallel:
\[ \frac{1}{R'} = \frac{1}{R_1} + \frac{1}{R_2} = \frac{1}{\frac{R}{2}} + \frac{1}{\frac{R}{2}} = \frac{2}{R/2} = \frac{4}{R} \]
Thus, the equivalent resistance \( R' \) is:
\[ R' = \frac{R}{4} \]

### Step 5: Calculate the new power across the same potential difference
Now, we can calculate the new power \( P' \) across the same potential difference \( V \):
\[ P' = \frac{V^2}{R'} \]
Substituting \( R' \):
\[ P' = \frac{V^2}{\frac{R}{4}} = \frac{4V^2}{R} \]

### Step 6: Relate the new power to the original power
Since we know that \( P = \frac{V^2}{R} \), we can express \( P' \) in terms of \( P \):
\[ P' = 4 \cdot \frac{V^2}{R} = 4P \]

### Conclusion
The power of the combination when the coil is cut into two halves and joined in parallel is \( 4P \).