One end of a thermally insulated rod is kept at a temperature $T_1$ and the other at $T_2$. The rod is…

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
  1. $\left(\mathrm{k}_2 \ell_2 \mathrm{T}_1+\mathrm{k}_1 \ell_1 \mathrm{T}_2\right) /\left(\mathrm{k}_1 \ell_1+\mathrm{k}_2 \ell_2\right)$
  2. $\left(\mathrm{k}_2 \ell_1 \mathrm{T}_1+\mathrm{k}_1 \ell_1 \mathrm{T}_2\right) /\left(\mathrm{k}_2 \ell_1+\mathrm{k}_1 \ell_2\right)$
  3. $\left(\mathrm{k}_1 \ell_2 \mathrm{T}_1+\mathrm{k}_2 \ell_1 \mathrm{T}_2\right) /\left(\mathrm{k}_1 \ell_2+\mathrm{k}_2 \ell_1\right)$
  4. $\left(\mathrm{k}_1 \ell_1 \mathrm{T}_1+\mathrm{k}_2 \ell_2 \mathrm{T}_2\right) /\left(\mathrm{k}_1 \ell_1+\mathrm{k}_2 \ell_2\right)$

Solution

$\frac{\left(\mathrm{T}_1-\mathrm{T}\right) \mathrm{k}_1}{\ell_1}=\frac{\left(\mathrm{T}-\mathrm{T}_2\right) \mathrm{k}_2}{\ell_2}$ $\mathrm{~T}=\frac{\mathrm{T}_1 \mathrm{k}_1 \ell_2+\mathrm{T}_2 \mathrm{k}_2 \ell_1}{\mathrm{k}_1 \ell_2+\mathrm{k}_2 \ell_1}$

Asked in: JEE Main 2007

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