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

Answer

To solve the problem step by step, we can follow these steps:

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

### Step 2: Determine the resistance of the cut coil
When the coil is cut into two equal halves, the resistance of each half will be:
\[ R' = \frac{R}{2} \]
This is because resistance is directly proportional to the length of the conductor, and cutting the coil in half reduces its length by half.

### Step 3: Calculate the equivalent resistance when joined in parallel
When the two halves are connected in parallel, the equivalent resistance \( R_{eq} \) can be calculated using the formula for resistances in parallel:
\[ \frac{1}{R_{eq}} = \frac{1}{R'} + \frac{1}{R'} \]
Substituting \( R' = \frac{R}{2} \):
\[ \frac{1}{R_{eq}} = \frac{1}{\frac{R}{2}} + \frac{1}{\frac{R}{2}} = \frac{2}{R/2} = \frac{4}{R} \]
Thus, the equivalent resistance is:
\[ R_{eq} = \frac{R}{4} \]

### Step 4: Calculate the new power across the same potential difference
Now, we can find the new power \( P' \) across the same potential difference \( V \) using the equivalent resistance:
\[ P' = \frac{V^2}{R_{eq}} = \frac{V^2}{\frac{R}{4}} = \frac{4V^2}{R} \]

### Step 5: Relate the new power to the initial power
Since we know that \( P = \frac{V^2}{R} \), we can express \( P' \) in terms of \( P \):
\[ P' = 4 \cdot \frac{V^2}{R} = 4P \]

### Final Answer
Thus, the power of the combination when the coil is cut into two halves and connected in parallel is:
\[ P' = 4P \]

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