Question

A rod of length $l$ with thermally insulated lateral surface consists of material whose heat conductivity coefficient varies with temperature as $K\left(x\right)=\frac{\alpha }{T}$ (here $\alpha$ is a positive constant). The ends of the rod are kept at temperature $T_1$ and $T_2$ ( $T_1>T_2$ ).

Find the heat current per unit cross-sectional area in the rod

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

Answer 1

Answer: (B)

$\frac{dQ}{dt}=-KA\frac{(T(x+dx)-T(x))}{dx}\Rightarrow I_0=-\frac{\alpha }{T}A\frac{dT}{dx}$
$\Rightarrow \underset{0}{\overset{l}{\int }}\frac{I_0}{A\alpha }dx=-\underset{T_1}{\overset{T_2}{\int }}\frac{dT}{T}$
$\therefore \left|\frac{I_{0}l}{A\alpha }\right|=ln\frac{T_1}{T_2}$
$\Rightarrow \left|\frac{I_0}{A}\right|=\frac{\alpha }{l}ln\frac{T_1}{T_2}$