Question

One end of a thermally insulated rod is kept at a temperature $T_1$ and the other at $T_2$. The rod is composed of two sections of lengths $\ell_{1}$ and $\ell_{2}$ and thermal conductivities $k_1$ and $k_2$ respectively. The temperature at the interface of the two sections is

• A. $\left(k_{2}\ell_{2}T_{1}+k_{1}\ell_{1}T_{2}\right)/ \left(k_{1}\ell_{1}+k_{2}\ell_{2}\right)$
• B. $\left(k_{2}\ell_{1}T_{1}+k_{1}\ell_{1}T_{2}\right)/ \left(k_{2}\ell_{1}+k_{1}\ell_{2}\right)$
• C. $\left(k_{1}\ell_{2}T_{1}+k_{2}\ell_{1}T_{2}\right)/ \left(k_{1}\ell_{2}+k_{2}\ell_{1}\right)$
• D. $\left(k_{1}\ell_{1}T_{1}+k_{2}\ell_{2}T_{2}\right)/ \left(k_{1}\ell_{1}+k_{2}\ell_{2}\right)$

Answer 1

Answer: (C)

$\left(k_{1}\ell_{2}T_{1}+k_{2}\ell_{1}T_{2}\right)/ \left(k_{1}\ell_{2}+k_{2}\ell_{1}\right)$

Answer 2

$\frac{\left(T_{1}-T\right)k_{1}}{\ell_{1}}=\frac{\left(T-T_{2}\right)k_{2}}{\ell_{2}}$ $T=\frac{T_{1}k_{1}\ell_{2}+T_{2}k_{2}\ell_{1}}{k_{1}\ell_{2}+k_{2}\ell_{1}}$