In the above question 73, when the wires are connected in parallel, the heat produced in thinner wire is `H_(1)` and that in thicker wire is `H_(2).` Which of the following is correct ?

Options

A. `H _(1) gt H _(2)`
B. `H _(1) lt H _(2)`
C. `H _(1) = H _(2)`
D. nothing can be decided

Explanation

To solve the problem regarding the heat produced in two wires connected in parallel, we will follow these steps:

### Step 1: Understand the Setup
We have two wires made of the same material and having the same length but different cross-sectional areas. One wire is thinner (let's call it Wire 1) and the other is thicker (Wire 2).

### Step 2: Determine the Resistance of Each Wire
The resistance \( R \) of a wire is given by the formula:
\[ R = \frac{\rho L}{A} \]
where:
- \( \rho \) is the resistivity of the material (same for both wires),
- \( L \) is the length of the wire (same for both wires),
- \( A \) is the cross-sectional area of the wire.

Since the area of cross-section is different, we can say:
- For the thinner wire (Wire 1), \( R_1 \) is greater because \( A_1 < A_2 \).
- For the thicker wire (Wire 2), \( R_2 \) is less because \( A_2 > A_1 \).

Thus, we conclude that:
\[ R_1 > R_2 \]

### Step 3: Apply the Heat Production Formula
When the wires are connected in parallel, the heat produced in each wire can be calculated using the formula:
\[ H = \frac{V^2}{R} \cdot t \]
where:
- \( H \) is the heat produced,
- \( V \) is the voltage across the wires (same for both in parallel),
- \( R \) is the resistance of the wire,
- \( t \) is the time for which the current flows.

For the thinner wire (Wire 1):
\[ H_1 = \frac{V^2}{R_1} \cdot t \]
For the thicker wire (Wire 2):
\[ H_2 = \frac{V^2}{R_2} \cdot t \]

### Step 4: Find the Ratio of Heat Produced
We can find the ratio of the heat produced in both wires:
\[ \frac{H_1}{H_2} = \frac{\frac{V^2}{R_1} \cdot t}{\frac{V^2}{R_2} \cdot t} = \frac{R_2}{R_1} \]
Since we established that \( R_1 > R_2 \), it follows that:
\[ \frac{R_2}{R_1} < 1 \]
This implies:
\[ H_1 < H_2 \]

### Conclusion
The heat produced in the thinner wire \( H_1 \) is less than the heat produced in the thicker wire \( H_2 \). Therefore, the correct statement is:
\[ H_1 < H_2 \]