Heat flows radially outwards through a spherical shell of outside radius `R_2` and inner radius `R_1`. The temperature of inner surface of shell is `theta_1` and that of outer is `theta_2`. At what radial distance from centre of shell the temperature is just half way between `theta_1 and theta_2`?

Explanation

To find the radial distance from the center of a spherical shell where the temperature is halfway between the temperatures at the inner and outer surfaces, we can follow these steps:

### Step 1: Understand the Problem
We have a spherical shell with:
- Inner radius \( R_1 \)
- Outer radius \( R_2 \)
- Temperature at the inner surface \( \theta_1 \)
- Temperature at the outer surface \( \theta_2 \)

We need to find the radial distance \( R \) from the center of the shell where the temperature \( T \) is equal to the average of \( \theta_1 \) and \( \theta_2 \):
\[ T = \frac{\theta_1 + \theta_2}{2} \]

### Step 2: Set Up the Heat Transfer Equation
The heat transfer through the spherical shell can be described using the concept of thermal resistance. The heat flow \( H \) can be expressed as:
\[ H = \frac{\Delta T}{R} \]
where \( \Delta T \) is the temperature difference and \( R \) is the thermal resistance.

### Step 3: Calculate the Thermal Resistance
For a spherical shell, the thermal resistance \( R \) can be calculated using:
\[ R = \frac{1}{4\pi k} \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
where \( k \) is the thermal conductivity of the material.

### Step 4: Write the Heat Flow Equations
For the inner and outer surfaces, we can write the heat flow equations:
1. For the inner surface:
\[ H_1 = \frac{T - \theta_1}{\frac{1}{4\pi k} \left( \frac{1}{R} - \frac{1}{R_1} \right)} \]
2. For the outer surface:
\[ H_2 = \frac{T - \theta_2}{\frac{1}{4\pi k} \left( \frac{1}{R_2} - \frac{1}{R} \right)} \]

### Step 5: Set the Heat Flows Equal
Since there is no heat generation within the shell, we set \( H_1 + H_2 = 0 \):
\[ \frac{T - \theta_1}{\frac{1}{R} - \frac{1}{R_1}} + \frac{T - \theta_2}{\frac{1}{R_2} - \frac{1}{R}} = 0 \]

### Step 6: Substitute the Average Temperature
Substituting \( T = \frac{\theta_1 + \theta_2}{2} \) into the equation:
\[ \frac{\frac{\theta_1 + \theta_2}{2} - \theta_1}{\frac{1}{R} - \frac{1}{R_1}} + \frac{\frac{\theta_1 + \theta_2}{2} - \theta_2}{\frac{1}{R_2} - \frac{1}{R}} = 0 \]
This simplifies to:
\[ \frac{\frac{\theta_2 - \theta_1}{2}}{\frac{1}{R} - \frac{1}{R_1}} + \frac{\frac{\theta_1 - \theta_2}{2}}{\frac{1}{R_2} - \frac{1}{R}} = 0 \]

### Step 7: Solve for R
After simplifying and rearranging the equation, we can find:
\[ R = \frac{R_1 R_2}{R_1 + R_2} \]

### Final Answer
The radial distance from the center of the shell where the temperature is halfway between \( \theta_1 \) and \( \theta_2 \) is:
\[ R = \frac{R_1 R_2}{R_1 + R_2} \]