Question

A steady current $I$ flows through a wire of radius $r$, length $L$ and resistivity $\rho$. The current produces heat in the wire. The rate of heat loss in a wire is proportional to its surface area. The steady temperature of the wire is independent of

• A. $L$
• B. $r$
• C. $I$
• D. $\rho$

Answer 1

Answer: (A)

$L$

The resistance of the wire, $R$ can be given as:
$R=\rho \frac{L}{\pi r^2}$
and power, $P$ can be given as

$P=I^2\times R=I^2\times \rho \frac{L}{\pi r^2}$

Now as per the question, the rate of heat loss [Which is power in other words] is proportional to surface area.
let the rate of heat loss be $K\times 2\pi rL\times \delta T$

where $K$ is constant

$P=K\times 2\pi rL\times \delta T=I^2\times \rho \frac{L}{\pi r^2}$

$\Rightarrow K\times 2\pi r\times \delta T=I^2\times \frac{\rho }{\pi r^2}$
$\Rightarrow \delta T=I^2\times \frac{\rho }{K\times 2{\pi }^2{r}^3}$

From the above expression, we can see $\delta T$ is independent of $L$
Hence, the correct option will be $\left(A\right)$