Consider a spherical shell of radius R at temperature T. The black body radiation inside it can be considered as an ideal gas of photons with internal energy per unit volume `u=U/V propT^4` and pressure `P=1/3(U/V)`. If the shell now undergoes an adiabatic expansion the relation between T and R is :

Options

A. `T prop e^(-3R)`
B. `T prop 1/R`
C. `T prop 1/R^(3)`
D. `T prop e^(-R)`

Answer

To solve the problem, we need to find the relationship between the temperature \( T \) and the radius \( R \) of a spherical shell undergoing adiabatic expansion, given the properties of black body radiation inside it.

### Step-by-step Solution:

1. **Understanding the Internal Energy and Pressure**:
- The internal energy per unit volume \( u \) is given as \( u = k T^4 \), where \( k \) is a proportionality constant.
- The pressure \( P \) is given by \( P = \frac{1}{3} \frac{U}{V} = \frac{1}{3} u \).

2. **Expressing Pressure in Terms of Temperature**:
- From the internal energy expression, we can write:
\[ P = \frac{1}{3} k T^4 \]
- This is our equation (3).

3. **Using the Ideal Gas Law**:
- For an ideal gas, the relationship between pressure, volume, and temperature is given by:
\[ PV = nRT \]
- Assuming we have 1 mole of gas (n = 1), we can rewrite this as:
\[ P = \frac{RT}{V} \]
- This is our equation (4).

4. **Equating the Two Expressions for Pressure**:
- From equations (3) and (4), we can set them equal to each other:
\[ \frac{RT}{V} = \frac{1}{3} k T^4 \]

5. **Rearranging the Equation**:
- Rearranging gives:
\[ 3RT = k T^4 \cdot V \]
- Thus, we can express \( V \) in terms of \( T \):
\[ V = \frac{3R}{k} \cdot \frac{T}{T^4} = \frac{3R}{k} \cdot \frac{1}{T^3} \]

6. **Volume of the Spherical Shell**:
- The volume \( V \) of a spherical shell of radius \( R \) is given by:
\[ V = \frac{4}{3} \pi R^3 \]

7. **Setting the Two Volume Expressions Equal**:
- Now we can set the two expressions for volume equal to each other:
\[ \frac{4}{3} \pi R^3 = \frac{3R}{k} \cdot \frac{1}{T^3} \]

8. **Simplifying the Equation**:
- Rearranging gives:
\[ T^3 = \frac{3R}{k} \cdot \frac{4}{3} \pi R^3 \]
- Thus, we can express \( T^3 \) in terms of \( R^3 \):
\[ T^3 \propto R^3 \]

9. **Finding the Relationship Between T and R**:
- From the above, we can conclude that:
\[ T \propto \frac{1}{R} \]
- Therefore, the relationship between temperature \( T \) and radius \( R \) during adiabatic expansion is:
\[ T \propto \frac{1}{R} \]

### Final Result:
The relation between temperature \( T \) and radius \( R \) is:
\[ T \propto \frac{1}{R} \]