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 $T_{2} \, \left(\right.T_{1}>T_{2}\left.\right)$ . Heat current flowing through the rod will be -

• A. $\frac{A \alpha \left(T_{1}^{2} - T_{2}^{2}\right)}{3 l}$
• B. $\frac{A \alpha \left(T_{1}^{2} + T_{2}^{2}\right)}{l}$
• C. $\frac{A \alpha \left(T_{1}^{2} + T_{2}^{2}\right)}{3 l}$
• D. $\frac{A \alpha \left(T_{1}^{2} - T_{2}^{2}\right)}{2 l}$

Answer 1

Answer: (D)

Heat current,
$i=-KA\frac{d T}{d X}$
$i \, dx=-KA \, dT$
$i \, \int \limits_{0}^{l}dX=-A\alpha \int\limits _{T_{1}}^{T_{2}}T \, dT$
$\Rightarrow il=-A\alpha \frac{\left(T_{2}^{2} - T_{1}^{2}\right)}{2}$
$\Rightarrow i=A\alpha \frac{\left(T_{1}^{2} - T_{2}^{2}\right)}{2 l}$