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Physics-

General

Easy

Question

One end of a thermally insulated rod is kept at a temperature $T subscript 1 end subscript$ and other at $T subscript 2 end subscript$ . The rod is composed of two sections of lengths $l subscript 1 end subscript$ and $l subscript 2 end subscript$ and thermal conductivities $K subscript 1 end subscript$ and $K subscript 2 end subscript$ respectively. The temperature at the interface of the two sections is

${(K}_{2} l_{2} T_{1} + K_{1} l_{1} T_{2}) ∕ (K_{1} l_{1} + K_{2} l_{2})$
${(K}_{2} l_{1} T_{1} + K_{1} l_{2} T_{2}) ∕ (K_{2} l_{1} + K_{1} l_{2})$
${(K}_{1} l_{2} T_{1} + K_{2} l_{1} T_{2}) ∕ (K_{1} l_{2} + K_{2} l_{1})$
${(K}_{1} l_{1} T_{1} + K_{2} l_{2} T_{2}) ∕ (K_{1} l_{1} + K_{2} l_{2})$

The correct answer is: $left parenthesis K subscript 1 end subscript l subscript 2 end subscript T subscript 1 end subscript plus K subscript 2 end subscript l subscript 1 end subscript T subscript 2 end subscript right parenthesis ∕ left parenthesis K subscript 1 end subscript l subscript 2 end subscript plus K subscript 2 end subscript l subscript 1 end subscript right parenthesis$

Let temperature at the interface is T.
For part AB,

$fraction numerator Q subscript 1 end subscript over denominator t end fraction proportional to fraction numerator open parentheses T subscript 1 end subscript minus T close parentheses K subscript 1 end subscript over denominator l subscript 1 end subscript end fraction$
For part $B C comma$
$fraction numerator Q subscript 2 end subscript over denominator t end fraction proportional to fraction numerator open parentheses T minus T subscript 2 end subscript close parentheses K subscript 2 end subscript over denominator l subscript 2 end subscript end fraction$
At equilibrium, $fraction numerator Q subscript 1 end subscript over denominator t end fraction equals fraction numerator Q subscript 2 end subscript over denominator t end fraction$
$therefore fraction numerator open parentheses T subscript 1 end subscript minus T close parentheses K subscript 1 end subscript over denominator l subscript 1 end subscript end fraction equals fraction numerator open parentheses T minus T subscript 2 end subscript close parentheses K subscript 2 end subscript over denominator l subscript 2 end subscript end fraction$
$rightwards double arrow T equals fraction numerator T subscript 1 end subscript K subscript 1 end subscript l subscript 2 end subscript plus T subscript 2 end subscript K subscript 2 end subscript l subscript 1 end subscript over denominator K subscript 1 end subscript l subscript 2 end subscript plus K subscript 2 end subscript l subscript 1 end subscript end fraction$

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