Question

A constant voltage is applied between the two ends of uniform metallic wire. Some heat is developed in it. If both length and radius of the wire are halved, then the heat developed in the same duration will become

• A. half
• B. twice
• C. one-fourth
• D. same

Answer 1

Answer: (A)

half

Heat developed in metallic wire is
$H=\frac{V^2}{R}t=\frac{V^{2}A}{\rho l}=\frac{V^2\pi r^2}{\rho l}$ .................................................................(1)
when both length and radius of the wire are halved, then the heat developed in the same duration is
$H^{\prime }=\frac{V^2}{R^{\prime }}t=\frac{V^2\pi {\left(\frac{r}{2}\right)}^2}{\rho \frac{l}{2}}$
$H^{\prime }=\frac{2}{4}\frac{V^2\pi r^2}{\rho l}$
$H^{\prime }=\frac{1}{2}H$ ......from (1)