Question

A rod of length $l$ with thermally insulated lateral surface consists of material whose heat conductivity coefficient varies with temperature as $k=\alpha /T$, where $\alpha$ is a constant. The ends of the rod are kept at temperature $T_1$ and $T_2$. Find the function $T\left(x\right)$, where $x$ is the distance from the end whose temperature is $T_1$, and the heat flow density.

• A. $T_{\left(x\right)}=2T_1{\left(\frac{T_2}{T_1}\right)}^{xf};H=\frac{\alpha }{l}ln\left(\frac{T_2}{T_1}\right)$.
• B. $T_{\left(x\right)}=T_1{\left(\frac{T_2}{T_1}\right)}^{xf};H=\frac{\alpha }{l}ln\left(\frac{T_2}{T_1}\right)$.
• C. $T_{\left(x\right)}=3T_1{\left(\frac{T_2}{T_1}\right)}^{xf};H=\frac{\alpha }{l}ln\left(\frac{T_2}{T_1}\right)$.
• D. $T_{\left(x\right)}=4T_1{\left(\frac{T_2}{T_1}\right)}^{xf};H=\frac{\alpha }{l}ln\left(\frac{T_2}{T_1}\right)$.

Answer 1

The correct option is B $T_{\left(x\right)}=T_1{\left(\frac{T_2}{T_1}\right)}^{xf};H=\frac{\alpha }{l}ln\left(\frac{T_2}{T_1}\right)$.