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. $\frac{\alpha }{l}ln\left(\frac{2 T_{1}}{T_{1} + T_{2}}\right)$
• D. $\frac{\alpha }{l}ln\left(\frac{T_{1} + T_{2}}{T_{2}}\right)$

Answer 1

Answer: (B)

$\frac{d Q}{d t}=-K A \frac{(T(x+d x)-T(x))}{d x} \Rightarrow I_{0}=-\frac{\alpha}{T} A \frac{d T}{d x}$
$\Rightarrow \, \int\limits _{0}^{l} \frac{I_{0}}{A \alpha } \, \, dx=-\int\limits _{T_{1}}^{T_{2}} \frac{d T}{T}$
$\therefore \, \left|\frac{I_{0} l}{A \alpha }\right|=ln\frac{T_{1}}{T_{2}}$
$\Rightarrow \, \left|\frac{I_{0}}{A}\right|=\frac{\alpha }{l}ln\frac{T_{1}}{T_{2}}$