When a potential difference "V" is applied across a wire of resistance "R" ,it dissipates energy at a rate "W" .If the wire is cut into two halves and these halves are connected mutually parallel across the same supply,the energy dissipation rate will become:

Options

A. 2W
B. 1/4W
C. 4W
D. 1/2W

Answer

To solve the problem, we need to analyze the situation step by step.

### Step 1: Understand the initial conditions
When a potential difference \( V \) is applied across a wire of resistance \( R \), the power (rate of energy dissipation) \( W \) can be expressed using the formula:
\[ W = \frac{V^2}{R} \]

### Step 2: Cutting the wire
When the wire is cut into two halves, each half will have a resistance of:
\[ R_1 = \frac{R}{2} \]
This is because resistance is directly proportional to the length of the wire, and cutting it in half reduces the length (and hence the resistance) by half.

### Step 3: Connecting the halves in parallel
Now, we connect these two halves in parallel. The equivalent resistance \( R_{eq} \) of two resistors \( R_1 \) and \( R_2 \) in parallel is given by:
\[ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} \]
Substituting \( R_1 = R_2 = \frac{R}{2} \):
\[ \frac{1}{R_{eq}} = \frac{1}{\frac{R}{2}} + \frac{1}{\frac{R}{2}} = \frac{2}{\frac{R}{2}} = \frac{4}{R} \]
Thus, the equivalent resistance \( R_{eq} \) becomes:
\[ R_{eq} = \frac{R}{4} \]

### Step 4: Calculate the new power dissipation
Now, we can find the new power dissipation \( W' \) when the two halves are connected in parallel:
\[ W' = \frac{V^2}{R_{eq}} = \frac{V^2}{\frac{R}{4}} = \frac{4V^2}{R} \]

### Step 5: Relate the new power dissipation to the original
From the original power dissipation \( W = \frac{V^2}{R} \), we can express the new power dissipation as:
\[ W' = 4 \cdot \frac{V^2}{R} = 4W \]

### Conclusion
Therefore, when the wire is cut into two halves and connected in parallel, the energy dissipation rate becomes:
\[ \boxed{4W} \]