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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

  1. left parenthesis K subscript 2 end subscript l subscript 2 end subscript T subscript 1 end subscript plus K subscript 1 end subscript l subscript 1 end subscript T subscript 2 end subscript right parenthesis ∕ left parenthesis K subscript 1 end subscript l subscript 1 end subscript plus K subscript 2 end subscript l subscript 2 end subscript right parenthesis    
  2. left parenthesis K subscript 2 end subscript l subscript 1 end subscript T subscript 1 end subscript plus K subscript 1 end subscript l subscript 2 end subscript T subscript 2 end subscript right parenthesis ∕ left parenthesis K subscript 2 end subscript l subscript 1 end subscript plus K subscript 1 end subscript l subscript 2 end subscript right parenthesis    
  3. 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    
  4. left parenthesis K subscript 1 end subscript l subscript 1 end subscript T subscript 1 end subscript plus K subscript 2 end subscript l subscript 2 end subscript T subscript 2 end subscript right parenthesis ∕ left parenthesis K subscript 1 end subscript l subscript 1 end subscript plus K subscript 2 end subscript l subscript 2 end subscript right parenthesis    

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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