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(A) The rate of heat developed in a wire is given by the power formula: $P = \frac{V^2}{R}$.Initially,the resistance of the wire is $R_1 = \frac{\rho

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Increased $8$ times

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(A) The rate of heat developed in a wire is given by the power formula: $P = \frac{V^2}{R}$.Initially,the resistance of the wire is $R_1 = \frac{\rho L}{A} = \frac{\rho L}{\pi r^2}$.Thus,the initial power is $P_1 = \frac{V^2}{R_1}$.When the length is halved $(L' = L/2)$ and the radius is doubled $(r' = 2r)$,the new resistance $R_2$ becomes:$R_2 = \frac{\rho (L/2)}{\pi (2r)^2} = \frac{\rho L / 2}{\pi (4r^2)} = \frac{\rho L}{8 \pi r^2} = \frac{R_1}{8}$.The new power $P_2$ is given by:$P_2 = \frac{V^2}{R_2} = \frac{V^2}{R_1 / 8} = 8 \left( \frac{V^2}{R_1} \right) = 8 P_1$.Therefore,the rate of heat developed increases $8$ times.

$A$ constant voltage is applied between two ends of a metallic wire. If the length is halved and the radius of the wire is doubled,the rate of heat developed in the wire will be

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