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

Explanation

To solve the problem, we need to establish the relationship between the temperature \( T \) and the radius \( R \) of the spherical shell during an adiabatic expansion of the black body radiation inside it.

### Step-by-Step Solution:

1. **Internal Energy per Unit Volume**:
The internal energy per unit volume \( u \) is given by:
\[ u = \frac{U}{V} \propto T^4 \]
We can express this as:
\[ u = k T^4 \]
where \( k \) is a constant.

2. **Pressure Relation**:
The pressure \( P \) is related to the internal energy per unit volume as:
\[ P = \frac{1}{3} \left( \frac{U}{V} \right) = \frac{1}{3} u \]
Substituting for \( u \):
\[ P = \frac{1}{3} k T^4 \]

3. **Ideal Gas Law**:
From the ideal gas law, we have:
\[ PV = nRT \]
Rearranging gives:
\[ P = \frac{nRT}{V} \]

4. **Equating Pressures**:
Now, we can equate the two expressions for pressure:
\[ \frac{nRT}{V} = \frac{1}{3} k T^4 \]
Simplifying this, we can cancel one \( T \) from both sides:
\[ \frac{nR}{V} = \frac{1}{3} k T^3 \]

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

6. **Substituting Volume**:
Substituting \( V \) into the equation:
\[ \frac{nR}{\frac{4}{3} \pi R^3} = \frac{1}{3} k T^3 \]
This simplifies to:
\[ \frac{3nR}{4\pi R^3} = \frac{1}{3} k T^3 \]

7. **Rearranging the Equation**:
Rearranging gives:
\[ T^3 = \frac{9nR}{4\pi k} \cdot \frac{1}{R^3} \]
This shows that:
\[ T^3 \propto \frac{1}{R^3} \]

8. **Final Relationship**:
Taking the cube root of both sides, we find:
\[ T \propto \frac{1}{R} \]

### Conclusion:
Thus, the relationship between the temperature \( T \) and the radius \( R \) of the spherical shell during adiabatic expansion is:
\[ T \propto \frac{1}{R} \]