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Step-by-Step Solution
Step 1: Understand the Problem
We have two concentric spherical shells with radii $r_1$ and $r_2$ ($r_1 < r_2$), filled with a material of thermal conductivity $K$. The inner shell is at temperature $\theta_1$ and the outer shell is at temperature $\theta_2$, and we want to find the rate of heat flow from the region at higher temperature to the region at lower temperature.
Step 2: Concept of Thermal Resistance in a Spherical Shell
For radial heat conduction through a thin spherical shell of thickness $dr$ at radius $r$, the thermal resistance $dR$ is given by:
$dR = \frac{dr}{K \cdot 4\pi r^2}.$
This formula arises from the general law of heat conduction, considering the cross-sectional area for heat flow at radius $r$ is $4\pi r^2$.
Step 3: Integrate the Resistance from $r_1$ to $r_2$
To find the total thermal resistance $R$ between the two shells, we integrate $dR$ from $r_1$ to $r_2$:
$R = \int_{r_1}^{r_2} \frac{dr}{K \, 4\pi r^2}.$
Carrying out this integral:
$R = \frac{1}{4\pi K} \int_{r_1}^{r_2} \frac{dr}{r^2}
= \frac{1}{4\pi K} \Bigl[ -\frac{1}{r} \Bigr]_{r_1}^{r_2}
= \frac{1}{4\pi K} \left( \frac{1}{r_1} - \frac{1}{r_2} \right).$
This can be rewritten as:
$R = \frac{1}{4\pi K} \left(\frac{r_2 - r_1}{r_1\, r_2}\right).$
Step 4: Calculate the Rate of Heat Flow
Heat flow (often referred to as the “thermal current” $i$) is given by the temperature difference divided by the total thermal resistance:
$i = \frac{\theta_2 - \theta_1}{R}.$
Substitute the expression for $R$:
$i = \left(\theta_2 - \theta_1\right) \Bigl/ \left( \frac{1}{4\pi K} \frac{r_2 - r_1}{r_1\, r_2} \right).$
Simplifying this:
$i = \frac{4\pi K\, r_1\, r_2}{r_2 - r_1}\, \bigl(\theta_2 - \theta_1\bigr).$
Step 5: Final Answer
Therefore, the rate at which heat flows radially through the material is:
$\displaystyle i = \frac{4\pi K\, r_1\, r_2 \,\bigl(\theta_2 - \theta_1\bigr)}{r_2 - r_1}.$
This matches the given correct option:
$\displaystyle \frac{4\pi K\, r_1\, r_2\, (\theta_2 - \theta_1)}{r_2 - r_1}.$
