Physics-
General
Easy

Question

The figure shows a system of two concentric spheres of radii r subscript 1 end subscript and r subscript 2 end subscript and kept at temperatures T subscript 1 end subscript and T subscript 2 end subscript respectively. The radial rate of flow of heat in a substance between the two concentric spheres, is proportional to

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

The correct answer is: fraction numerator open parentheses r subscript 1 end subscript r subscript 2 end subscript close parentheses over denominator open parentheses r subscript 2 end subscript minus r subscript 1 end subscript close parentheses end fraction


    To measure the radial rate of heat flow, we have to go for integration technique as here the area of the surface through which heat will flow is not constant.

    Let us consider an element (spherical shell) of thickness dx and radius x as shown in figure. Let us first find the equivalent thermal resistance between inner and outer sphere.
    Resistance of shell=d R equals blank fraction numerator d x over denominator K cross times 4 pi x to the power of 2 end exponent end fraction
    open parentheses fraction numerator F r o m blank R equals fraction numerator l over denominator K A end fraction w h e r e comma over denominator K equals t h e r m a l blank c o n d u c t i v i t y end fraction close parentheses
    rightwards double arrow not stretchy integral d R equals R equals not stretchy integral subscript r subscript 1 end subscript end subscript superscript r subscript 2 end subscript end superscript fraction numerator d x over denominator 4 pi K x to the power of 2 blank end exponent end fraction
    = fraction numerator 1 over denominator 4 pi K end fraction open square brackets fraction numerator 1 over denominator r subscript 1 end subscript end fraction minus blank fraction numerator 1 over denominator r subscript 2 end subscript end fraction close square brackets equals blank fraction numerator r subscript 2 end subscript minus r subscript 1 end subscript over denominator 4 pi K left parenthesis r subscript 1 end subscript r subscript 2 end subscript right parenthesis end fraction
    Rate of heat flow = H
    = fraction numerator T subscript 1 end subscript minus T subscript 2 end subscript over denominator R end fraction
    = fraction numerator T subscript 1 end subscript minus T subscript 2 end subscript over denominator r subscript 2 end subscript minus r subscript 1 end subscript end fraction blank cross times 4 pi K open parentheses r subscript 1 end subscript r subscript 2 end subscript close parentheses
    proportional to fraction numerator r subscript 1 end subscript r subscript 2 end subscript over denominator r subscript 2 end subscript minus blank r subscript 1 end subscript end fraction

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