Q

# Confused! kindly explain, - Properties of Solids and Liquids - JEE Main

The figure shows a system of two concentric spheres of radii $r_{1}\: and\: r_{2}$ and 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

• Option 1)

$\frac{r_{1}r_{2}}{\left ( r_{2}-r_{1} \right )}$

• Option 2)

$\left ( r_{2}-r_{1} \right )$

• Option 3)

$\frac{\left ( r_{2}-r_{1} \right )}{r_{1}r_{2}}$

• Option 4)

$\ln \left ( \frac{r_{2}}{r_{1}} \right )$

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As we have learned

Thermal Conductivity -

$Q=\frac{KA(\theta_{1}-\theta_{2})t}{l}$

K = thermal conductivity

- wherein

Rate of flow of heat $\frac{d\theta }{dt} = \frac{KA\Delta T}{L}$

We have a spherical shell of radius r and thickness dx

$A = 4 \pi r^2 , L =dr \\ \frac{d\theta }{dt}= \frac{K4 \pi r^2 dt }{dr }$

$4 \pi K \int_{T_1}^{T_2}dT= \frac{d\theta }{dt} \int_{r_1}^{r_2}\frac{dr}{r^2} = \frac{d\theta }{dt} \left ( \frac{1}{r_1}-\frac{1}{r_2} \right )$

$4 \pi K(T_2-T_1)= \frac{d\theta }{dt} \left ( \frac{r_2r_1}{r_2-r_1} \right )$

$\therefore \frac{d\theta }{dt} \propto \frac{r_2r_1}{r_2-r_1}$

Option 1)

$\frac{r_{1}r_{2}}{\left ( r_{2}-r_{1} \right )}$

Option 2)

$\left ( r_{2}-r_{1} \right )$

Option 3)

$\frac{\left ( r_{2}-r_{1} \right )}{r_{1}r_{2}}$

Option 4)

$\ln \left ( \frac{r_{2}}{r_{1}} \right )$

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