Question

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

• A. four times
• B. one forth
• C. twice
• D. half

Answer 1

The correct option is D half

Heat developed in
$H=\frac{V^2}{R}$
But, $R=\frac{\rho L}{\pi r^2}$
where variables have their usual meanings.
If both the length and the radius of the wire halved,
$R^{\prime }=\frac{\rho \frac{L}{2}}{\pi (\frac{r}{2}{)}^2}$
Hence, the heat developed is
$H^{\prime }=\frac{\left(\frac{\pi r^2}{4}\right)V^2}{\frac{L}{2}}$
$H^{\prime }=\frac{1}{2}H$