Question

A current $i$ passes through a wire of length $L$, radius $R$ and resistively $\rho$. The rate of heat generated is

• A. $\frac{i\rho L}{\pi r^2}$
• B. $i^2\left(\frac{L\rho }{\pi r^2}\right)$
• C. $\left(\frac{i^2\rho L}{r}\right)$
• D. $noneofthese$

Answer 1

The correct option is B $i^2\left(\frac{L\rho }{\pi r^2}\right)$

A current $i$ passes through a wire of length $L$, radius $R$ and resistivity . The rate of heat generated is
$H=i^{2}R$ ..................................................(1)
The resistance of a wire is proportional to the length directly and to the area of cross section inversely i.e.
$R\propto \frac{L}{A}$
$R=\rho \frac{L}{A}$
Also, if $r$ is the radius of wire then $A=\pi r^2$
Hence, $R=\frac{L\rho }{\pi r^2}$
$H=i^2\left(\frac{L\rho }{\pi r^2}\right)$