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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

$\frac{r_{1} r_{2}}{(r_{1} - r_{2})}$
$(r_{2} - r_{1})$
$(r_{2} - r_{1}) (r_{1} r_{2})$
In $(\frac{r_{2}}{r_{1}})$

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
$H equals fraction numerator d Q over denominator d t end fraction equals fraction numerator 4 pi K r subscript 1 end subscript r subscript 2 end subscript left parenthesis T subscript 1 end subscript minus T subscript 2 end subscript right parenthesis over denominator r subscript 2 end subscript minus r subscript 1 end subscript end fraction$ Þ $fraction numerator d Q over denominator d t end fraction proportional to fraction numerator r subscript 1 end subscript r subscript 2 end subscript over denominator left parenthesis r subscript 2 end subscript minus r subscript 1 end subscript right parenthesis end fraction$

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