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(C) According to Fourier's law of heat conduction,the rate of heat flow $H$ through a spherical shell of radius $r$ and thickness $dr$ is given by $H

Vedclass · Accepted Answer

$\frac{(r_1 r_2)}{(r_2 - r_1)}$

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(C) According to Fourier's law of heat conduction,the rate of heat flow $H$ through a spherical shell of radius $r$ and thickness $dr$ is given by $H = -k A \frac{dT}{dr}$,where $A = 4\pi r^2$ is the surface area.Thus,$H = -k (4\pi r^2) \frac{dT}{dr}$.Rearranging the terms,we get $\frac{H}{4\pi k} \frac{dr}{r^2} = -dT$.Integrating both sides from $r_1$ to $r_2$ and $T_1$ to $T_2$:$\frac{H}{4\pi k} \int_{r_1}^{r_2} \frac{dr}{r^2} = - \int_{T_1}^{T_2} dT$.$\frac{H}{4\pi k} \left[ -\frac{1}{r} \right]_{r_1}^{r_2} = -(T_2 - T_1) = T_1 - T_2$.$\frac{H}{4\pi k} \left( \frac{1}{r_1} - \frac{1}{r_2} \right) = T_1 - T_2$.$\frac{H}{4\pi k} \left( \frac{r_2 - r_1}{r_1 r_2} \right) = T_1 - T_2$.Therefore,$H = \frac{4\pi k (T_1 - T_2) r_1 r_2}{r_2 - r_1}$.Since $H$ is proportional to the term $\frac{r_1 r_2}{r_2 - r_1}$,the correct option is $C$.

The figure shows a system of two concentric spheres of radii $r_1$ and $r_2$ kept at temperatures $T_1$ and $T_2$,respectively. The radial rate of flow of heat in a substance between the two concentric spheres is proportional to

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