Physics-
General
Easy

Question

The figure shows a system of two concentric spheres of radii r1 and r2 and kept at temperatures T1 and T2, respectively. The radial rate of flow of heat in a substance between the two concentric spheres is proportional to

  1. fraction numerator r subscript 1 end subscript r subscript 2 end subscript over denominator left parenthesis r subscript 1 end subscript minus r subscript 2 end subscript right parenthesis end fraction    
  2. left parenthesis r subscript 2 end subscript minus r subscript 1 end subscript right parenthesis    
  3. left parenthesis r subscript 2 end subscript minus r subscript 1 end subscript right parenthesis left parenthesis r subscript 1 end subscript r subscript 2 end subscript right parenthesis    
  4. In open parentheses fraction numerator r subscript 2 end subscript over denominator r subscript 1 end subscript end fraction close parentheses    

The correct answer is: fraction numerator r subscript 1 end subscript r subscript 2 end subscript over denominator left parenthesis r subscript 1 end subscript minus r subscript 2 end subscript right parenthesis end fraction


    Consider a concentric spherical shell of radius r and thickness dr as shown in fig.

    The radial rate of flow of heat through this shell in steady state will be H equals fraction numerator d Q over denominator d t end fraction equals negative K A fraction numerator d T over denominator d r end fraction equals negative K left parenthesis 4 pi r to the power of 2 end exponent right parenthesis fraction numerator d T over denominator d r end fraction
    Þ not stretchy integral subscript r subscript 1 end subscript end subscript superscript r subscript 2 end subscript end superscript fraction numerator d r over denominator r to the power of 2 end exponent end fraction equals negative fraction numerator 4 pi K over denominator H end fraction not stretchy integral subscript T subscript 1 end subscript end subscript superscript T subscript 1 end subscript end superscript d T
    Which on integration and simplification gives
    H equals fraction numerator d Q over denominator d t end fraction equals fraction numerator 4 pi K r subscript 1 end subscript r subscript 2 end subscript left parenthesis T subscript 1 end subscript minus T subscript 2 end subscript right parenthesis over denominator r subscript 2 end subscript minus r subscript 1 end subscript end fractionÞ fraction numerator d Q over denominator d t end fraction proportional to fraction numerator r subscript 1 end subscript r subscript 2 end subscript over denominator left parenthesis r subscript 2 end subscript minus r subscript 1 end subscript right parenthesis end fraction

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