Question

A rod of length $l$ and cross-section area $A$ has a variable thermal conductivity given by $K=\alpha T$ , where $\alpha$ is a positive constant and $T$ is the temperature in Kelvin. Two ends of the rod are maintained at temperatures $T_1$ and $T_2(T_1>T_2)$ . Heat current flowing through the rod will be -

• A. $\frac{A\alpha \left(T_1^2-T_2^2\right)}{3l}$
• B. $\frac{A\alpha \left(T_1^2+T_2^2\right)}{l}$
• C. $\frac{A\alpha \left(T_1^2+T_2^2\right)}{3l}$
• D. $\frac{A\alpha \left(T_1^2-T_2^2\right)}{2l}$

Answer 1

Answer: (D)

Heat current,
$i=-KA\frac{dT}{dX}$
$idx=-KAdT$
$i\underset{0}{\overset{l}{\int }}dX=-A\alpha \underset{T_1}{\overset{T_2}{\int }}TdT$
$\Rightarrow il=-A\alpha \frac{\left(T_2^2-T_1^2\right)}{2}$
$\Rightarrow i=A\alpha \frac{\left(T_1^2-T_2^2\right)}{2l}$