A long resistance wire is divided into 2n parts. Then n parts are connected in series and the other n parts in parallel separately. Both combinations are connected to identical supplies. Then the ratio of heat produced in series to parallel combinations will be -

Options

A. `1:1`
B. `1:n^2`
C. `1:n^4`
D. `n^2:1`

Answer

To solve the problem, we need to determine the ratio of heat produced in the series combination to that produced in the parallel combination of the resistance wire divided into 2n parts. Here’s a step-by-step solution:

### Step 1: Define the Resistance of Each Part
Let the resistance of each part of the wire be \( R \). Since the wire is divided into \( 2n \) parts, each part has a resistance of \( R \).

### Step 2: Calculate the Resistance in Series
When \( n \) parts are connected in series, the total resistance \( R_s \) is given by:
\[ R_s = R + R + R + \ldots + R \quad (n \text{ times}) = nR \]

### Step 3: Calculate the Resistance in Parallel
When \( n \) parts are connected in parallel, the total resistance \( R_p \) is given by:
\[ \frac{1}{R_p} = \frac{1}{R} + \frac{1}{R} + \ldots + \frac{1}{R} \quad (n \text{ times}) = \frac{n}{R} \]
Thus, the equivalent resistance in parallel is:
\[ R_p = \frac{R}{n} \]

### Step 4: Determine Heat Produced in Each Configuration
The heat produced \( H \) in a resistor is given by the formula:
\[ H = I^2 R \quad \text{or} \quad H = \frac{V^2}{R} \cdot t \]
where \( V \) is the voltage across the resistor, \( I \) is the current, and \( t \) is the time.

#### Heat in Series Combination
Using \( H_s \) for heat in the series combination:
\[ H_s = \frac{V^2}{R_s} \cdot t = \frac{V^2}{nR} \cdot t \]

#### Heat in Parallel Combination
Using \( H_p \) for heat in the parallel combination:
\[ H_p = \frac{V^2}{R_p} \cdot t = \frac{V^2}{\frac{R}{n}} \cdot t = \frac{nV^2}{R} \cdot t \]

### Step 5: Calculate the Ratio of Heat Produced
Now, we find the ratio of heat produced in series to that in parallel:
\[ \frac{H_s}{H_p} = \frac{\frac{V^2}{nR} \cdot t}{\frac{nV^2}{R} \cdot t} \]
The \( V^2 \) and \( t \) terms cancel out:
\[ \frac{H_s}{H_p} = \frac{1/nR}{n/R} = \frac{1}{n^2} \]

### Conclusion
Thus, the ratio of heat produced in the series combination to that in the parallel combination is:
\[ \frac{H_s}{H_p} = \frac{1}{n^2} \]

### Final Answer
The answer is \( \frac{1}{n^2} \).