The ratio of cross-sectional areas of two conducting wires made up of same material and having same length is 1: 2. What will be the ratio of heat produced per second in the wires, when same current is flowing in them?

Options

A. `1:4`
B. `2:1`
C. `1:sqrt2`
D. `1:1`

Answer

To solve the problem, we need to find the ratio of heat produced per second in two conducting wires made of the same material, having the same length, and carrying the same current, given that the ratio of their cross-sectional areas is 1:2.

### Step-by-Step Solution:

1. **Identify Given Information**:
- Let the cross-sectional areas of the two wires be \( A_1 \) and \( A_2 \).
- Given that \( \frac{A_1}{A_2} = \frac{1}{2} \), we can express this as \( A_1 = x \) and \( A_2 = 2x \) for some value \( x \).
- The length of both wires is the same: \( L_1 = L_2 = L \).
- The resistivity of the material is the same for both wires: \( \rho_1 = \rho_2 = \rho \).
- The current flowing through both wires is the same: \( I_1 = I_2 = I \).

2. **Formula for Heat Produced**:
- The heat produced per second (power) in a resistor is given by the formula:
\[ H = I^2 R \]
- The resistance \( R \) of a wire can be calculated using the formula:
\[ R = \frac{\rho L}{A} \]

3. **Calculate Resistance for Both Wires**:
- For wire 1:
\[ R_1 = \frac{\rho L}{A_1} = \frac{\rho L}{x} \]
- For wire 2:
\[ R_2 = \frac{\rho L}{A_2} = \frac{\rho L}{2x} \]

4. **Calculate Heat Produced for Both Wires**:
- For wire 1:
\[ H_1 = I^2 R_1 = I^2 \left(\frac{\rho L}{x}\right) \]
- For wire 2:
\[ H_2 = I^2 R_2 = I^2 \left(\frac{\rho L}{2x}\right) \]

5. **Find the Ratio of Heat Produced**:
- Now, we can find the ratio \( \frac{H_1}{H_2} \):
\[ \frac{H_1}{H_2} = \frac{I^2 \left(\frac{\rho L}{x}\right)}{I^2 \left(\frac{\rho L}{2x}\right)} \]
- Simplifying this gives:
\[ \frac{H_1}{H_2} = \frac{\frac{\rho L}{x}}{\frac{\rho L}{2x}} = \frac{2}{1} \]

6. **Conclusion**:
- Therefore, the ratio of heat produced per second in the two wires is:
\[ \frac{H_1}{H_2} = 2:1 \]