One end of a thermally insulated rod is kept at a temperature T_1...

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 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

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

Options

A $\left(\mathrm{k}_2 \ell_2 \mathrm{T}_1+\mathrm{k}_1 \ell_1 \mathrm{T}_2\right) /\left(\mathrm{k}_1 \ell_1+\mathrm{k}_2 \ell_2\right)$
B $\left(\mathrm{k}_2 \ell_1 \mathrm{T}_1+\mathrm{k}_1 \ell_1 \mathrm{T}_2\right) /\left(\mathrm{k}_2 \ell_1+\mathrm{k}_1 \ell_2\right)$
C $\left(\mathrm{k}_1 \ell_2 \mathrm{T}_1+\mathrm{k}_2 \ell_1 \mathrm{T}_2\right) /\left(\mathrm{k}_1 \ell_2+\mathrm{k}_2 \ell_1\right)$
D $\left(\mathrm{k}_1 \ell_1 \mathrm{T}_1+\mathrm{k}_2 \ell_2 \mathrm{T}_2\right) /\left(\mathrm{k}_1 \ell_1+\mathrm{k}_2 \ell_2\right)$

Correct answer

C. $\left(\mathrm{k}_1 \ell_2 \mathrm{T}_1+\mathrm{k}_2 \ell_1 \mathrm{T}_2\right) /\left(\mathrm{k}_1 \ell_2+\mathrm{k}_2 \ell_1\right)$

Step-by-step 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}

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