Heating Time for Water with Parallel and Series Wire Configurations

To find the time needed to raise the temperature of the water with the new wire configurations, we first establish the relationship between resistance, power, and the heat required.

1. Initial Configuration

Let the resistivity of the wire material be $\rho$ and the operating voltage of the kettle be $V$.

• The initial wire has length $L$ and diameter $d$. Its cross-sectional area is: $A_1 = \frac{\pi d^2}{4}$

• The resistance of the original heating element is: $R_1 = \rho \frac{L}{A_1} = \frac{4\rho L}{\pi d^2}$

• The heat $Q$ required to raise the temperature of $0.5\text{ kg}$ of water by $40\text{ K}$ in time $t_1 = 4\text{ minutes}$ is generated by the electric power $P_1$: $Q = P_1 \cdot t_1 = \frac{V^2}{R_1} t_1 \quad \implies \quad Q = \frac{V^2}{R_1} \times 4\text{ min}$

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2. Resistance of a Single New Wire

Each new wire has length $L$ and diameter $2d$.

• The cross-sectional area of a new wire is: $A_2 = \frac{\pi (2d)^2}{4} = \pi d^2 = 4 A_1$

• The resistance $R'$ of one single new wire is: $R' = \rho \frac{L}{A_2} = \rho \frac{L}{4 A_1} = \frac{R_1}{4}$

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3. Evaluation of Connection Schemes

Case I: Wires connected in Series

• The equivalent resistance of two such wires connected in series is: $R_{\text{series}} = R' + R' = 2R' = 2 \left(\frac{R_1}{4}\right) = \frac{R_1}{2}$

• Equating the heat required $Q$: $Q = \frac{V^2}{R_{\text{series}}} \cdot t_{\text{series}}$ $\frac{V^2}{R_1} \times 4 = \frac{V^2}{R_1 / 2} \cdot t_{\text{series}}$ $4 = 2 \cdot t_{\text{series}} \implies t_{\text{series}} = 2\text{ minutes}$

Thus, Option (B) is correct.

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Case II: Wires connected in Parallel

• The equivalent resistance of two such wires connected in parallel is: $R_{\text{parallel}} = \frac{R'}{2} = \frac{R_1 / 4}{2} = \frac{R_1}{8}$

• Equating the heat required $Q$: $Q = \frac{V^2}{R_{\text{parallel}}} \cdot t_{\text{parallel}}$ $\frac{V^2}{R_1} \times 4 = \frac{V^2}{R_1 / 8} \cdot t_{\text{parallel}}$ $4 = 8 \cdot t_{\text{parallel}} \implies t_{\text{parallel}} = 0.5\text{ minutes}$

Thus, Option (D) is correct.

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Conclusion

The correct choices are B and D.