A point source of heat of power P is placed at the centre of a spherical shell of mean radius R. The material of the shell has thermal conductivity K. If the temperature difference between the outer and inner surface of the shell in not to exceed T, the thickness of the shell should not be less than .......

Explanation

To solve the problem, we need to determine the minimum thickness of the spherical shell that ensures the temperature difference between the inner and outer surfaces does not exceed T.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a point source of heat with power \( P \) located at the center of a spherical shell with mean radius \( R \).
- The shell has a thermal conductivity \( K \).
- We need to find the minimum thickness \( d \) of the shell such that the temperature difference between the inner and outer surfaces does not exceed \( T \).

2. **Heat Transfer through the Shell**:
- The heat transfer through the shell can be modeled using Fourier's law of heat conduction. The rate of heat transfer \( Q \) through a spherical shell is given by:
\[ Q = \frac{K \cdot A \cdot \Delta T}{d} \]
- Here, \( A \) is the surface area of the sphere, \( \Delta T \) is the temperature difference, and \( d \) is the thickness of the shell.

3. **Surface Area of the Shell**:
- The surface area \( A \) of a sphere with radius \( R \) is:
\[ A = 4\pi R^2 \]

4. **Relating Heat Transfer to Power**:
- Since the point source of heat is supplying power \( P \), we can set \( Q = P \):
\[ P = \frac{K \cdot (4\pi R^2) \cdot T}{d} \]

5. **Rearranging for Thickness \( d \)**:
- Rearranging the equation to solve for \( d \):
\[ d = \frac{K \cdot (4\pi R^2) \cdot T}{P} \]

6. **Final Expression**:
- Therefore, the minimum thickness \( d \) of the shell should not be less than:
\[ d \geq \frac{4\pi K R^2 T}{P} \]

### Conclusion:
The minimum thickness of the shell required to ensure that the temperature difference does not exceed \( T \) is given by:
\[ d \geq \frac{4\pi K R^2 T}{P} \]