Question

A uniform metallic wire of radius $r$ and length $l$ is heated by passing a current through it. The heat produced can be made 8 times if

• A. $l$ is doubled
• B. both $l$ and $r$ are halved
• C. $l$ is doubled and $r$ is halved
• D. $r$ is doubled
• E. both $l$ and $r$ are doubled

Hint:

Resistance varies directly with length and inversely with square of radius.

Answer 1

The Correct Option is C

Concept:
• Heat produced: \[ H = I^2Rt \]
• Resistance: \[ R = \rho \frac{l}{\pi r^2} \]

Step 1: Relation
\[ H \propto \frac{l}{r^2} \]

Step 2: Apply changes
\[ l \to 2l,\quad r \to \frac{r}{2} \] \[ H \propto \frac{2l}{(r/2)^2} = \frac{2l}{r^2/4} = 8\frac{l}{r^2} \] Final Conclusion:
Option (C)