Two heater wires, made of the same material and having the same length and the same radius, are first connected in series and then in parallel to a constant potential difference. If the rate of heat produced in the two cases are Hs and Hp respectively then `(Hs)/(Hp)` will be

Options

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

Explanation

To solve the problem, we need to find the ratio of the heat produced when two heater wires are connected in series (Hs) and when they are connected in parallel (Hp). Let's go through the solution step by step.

### Step 1: Understand the Resistance of Each Wire
Since both heater wires are made of the same material, have the same length (L), and the same radius (r), the resistance (R) of each wire can be expressed using the formula:
\[ R = \rho \frac{L}{A} \]
where \( \rho \) is the resistivity of the material and \( A \) is the cross-sectional area of the wire. The area \( A \) can be calculated as:
\[ A = \pi r^2 \]
Thus, the resistance of each wire is:
\[ R = \rho \frac{L}{\pi r^2} \]

### Step 2: Calculate the Total Resistance in Series
When the two wires are connected in series, the total resistance \( R_s \) is the sum of the individual resistances:
\[ R_s = R + R = 2R \]

### Step 3: Calculate the Heat Produced in Series (Hs)
The power (or rate of heat produced) when connected in series can be calculated using the formula:
\[ H_s = \frac{E^2}{R_s} \]
Substituting the expression for \( R_s \):
\[ H_s = \frac{E^2}{2R} \]

### Step 4: Calculate the Total Resistance in Parallel
When the two wires are connected in parallel, the total resistance \( R_p \) is given by:
\[ \frac{1}{R_p} = \frac{1}{R} + \frac{1}{R} = \frac{2}{R} \]
Thus, the total resistance in parallel is:
\[ R_p = \frac{R}{2} \]

### Step 5: Calculate the Heat Produced in Parallel (Hp)
The power (or rate of heat produced) when connected in parallel can be calculated as:
\[ H_p = \frac{E^2}{R_p} \]
Substituting the expression for \( R_p \):
\[ H_p = \frac{E^2}{\frac{R}{2}} = \frac{2E^2}{R} \]

### Step 6: Calculate the Ratio \( \frac{H_s}{H_p} \)
Now we can find the ratio of the heat produced in series to that produced in parallel:
\[ \frac{H_s}{H_p} = \frac{\frac{E^2}{2R}}{\frac{2E^2}{R}} = \frac{E^2}{2R} \cdot \frac{R}{2E^2} = \frac{1}{4} \]

### Conclusion
Thus, the ratio \( \frac{H_s}{H_p} \) is:
\[ \frac{H_s}{H_p} = \frac{1}{4} \]