Two heating wires of equal length are first con nected in series and then in parallel to a constant voltage source. The rate of heat produced in the two cases is

Options

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

Answer

To solve the problem regarding the rate of heat produced by two heating wires when connected in series and in parallel, we will follow these steps:

### Step 1: Understand the Setup
We have two heating wires of equal length and resistance \( R \). They are connected first in series and then in parallel to a constant voltage source \( V \).

### Step 2: Calculate the Equivalent Resistance in Series
When the two resistances are connected in series, the equivalent resistance \( R_{eq, series} \) is given by:
\[ R_{eq, series} = R + R = 2R \]

### Step 3: Calculate the Heat Produced in Series
The formula for heat produced \( H \) in a resistor is given by:
\[ H = I^2 R t \]
where \( I \) is the current, \( R \) is the resistance, and \( t \) is the time.

Using Ohm's law, the current \( I \) in the series circuit can be expressed as:
\[ I = \frac{V}{R_{eq, series}} = \frac{V}{2R} \]

Substituting \( I \) into the heat formula:
\[ H_1 = I^2 R_{eq, series} t = \left(\frac{V}{2R}\right)^2 (2R) t \]
\[ H_1 = \frac{V^2}{4R^2} \cdot 2R \cdot t = \frac{V^2 t}{2R} \]

### Step 4: Calculate the Equivalent Resistance in Parallel
When the two resistances are connected in parallel, the equivalent resistance \( R_{eq, parallel} \) is given by:
\[ \frac{1}{R_{eq, parallel}} = \frac{1}{R} + \frac{1}{R} = \frac{2}{R} \implies R_{eq, parallel} = \frac{R}{2} \]

### Step 5: Calculate the Heat Produced in Parallel
Using the same heat formula:
\[ I = \frac{V}{R_{eq, parallel}} = \frac{V}{\frac{R}{2}} = \frac{2V}{R} \]

Substituting \( I \) into the heat formula:
\[ H_2 = I^2 R_{eq, parallel} t = \left(\frac{2V}{R}\right)^2 \left(\frac{R}{2}\right) t \]
\[ H_2 = \frac{4V^2}{R^2} \cdot \frac{R}{2} \cdot t = \frac{2V^2 t}{R} \]

### Step 6: Find the Ratio of Heat Produced
Now, we need to find the ratio of heat produced in series to that in parallel:
\[ \frac{H_1}{H_2} = \frac{\frac{V^2 t}{2R}}{\frac{2V^2 t}{R}} = \frac{1}{4} \]

### Conclusion
The ratio of the heat produced when the wires are connected in series to when they are connected in parallel is:
\[ H_1 : H_2 = 1 : 4 \]