Some heat is developed in a uniform metallic wire across the ends of…

The correct answer is Option (4) → both the radius and the length of the wire are tripled

At constant voltage the heat developed in time \( t \) is

    \( \displaystyle H = \frac{V^{2}t}{R} \;\Rightarrow\; H \propto \frac{1}{R} \)

so the heat triples when the resistance becomes one-third. The resistance of a wire of length \( L \) and radius \( r \) is

    \( \displaystyle R = \frac{\rho L}{\pi r^{2}} \;\Rightarrow\; R \propto \frac{L}{r^{2}} \)

Option (1) the radius of the wire is tripled — Incorrect. \( r \to 3r \) makes \( R \to R/9 \), so the heat becomes 9 times.

Option (2) the length of the wire is tripled — Incorrect. \( L \to 3L \) makes \( R \to 3R \), so the heat becomes one-third.

Option (3) both the radius and the length of the wire are reduced to one-third — Incorrect. \( R \propto (1/3)/(1/3)^{2} = 3 \), so \( R \to 3R \) and the heat becomes one-third.

Option (4) both the radius and the length of the wire are tripled — Correct. \( R \propto 3/3^{2} = 1/3 \), so \( R \to R/3 \) and the heat is tripled.