Consider a black body radiation in a cubical box at absolute temperature T. If the lengh of each side of the box is doubled and the temperature of the walls of the box and that of the radiation is halved then the total energy

Options

A. halves
B. doubles
C. quadruples
D. remains constant

Explanation

To solve the problem, let's break it down step by step.

### Step 1: Understand the initial conditions
We have a cubical box containing black body radiation at an absolute temperature \( T \). The volume of the box and the temperature of the radiation are crucial to determining the total energy.

**Hint:** Black body radiation is characterized by its temperature, and the total energy is related to the volume and temperature.

### Step 2: Calculate the initial total energy
The total energy \( U \) of black body radiation in a volume \( V \) at temperature \( T \) is given by the formula:
\[ U = a V T^4 \]
where \( a \) is the Stefan-Boltzmann constant and \( V \) is the volume of the box.

For a cube with side length \( L \), the volume is:
\[ V = L^3 \]
Thus, the initial total energy can be expressed as:
\[ U_i = a (L^3) T^4 \]

**Hint:** Remember that the total energy is proportional to the fourth power of the temperature and the volume of the box.

### Step 3: Analyze the changes in dimensions and temperature
Now, the problem states that the length of each side of the box is doubled. Therefore, the new side length is \( 2L \), and the new volume \( V' \) becomes:
\[ V' = (2L)^3 = 8L^3 \]

The temperature of the walls and the radiation is halved, so the new temperature \( T' \) is:
\[ T' = \frac{T}{2} \]

**Hint:** When the dimensions of the box change, remember how it affects the volume.

### Step 4: Calculate the new total energy
Using the new volume and temperature, we can calculate the new total energy \( U_f \):
\[ U_f = a V' (T')^4 = a (8L^3) \left(\frac{T}{2}\right)^4 \]
Calculating \( (T')^4 \):
\[ (T')^4 = \left(\frac{T}{2}\right)^4 = \frac{T^4}{16} \]
Thus, the new total energy becomes:
\[ U_f = a (8L^3) \left(\frac{T^4}{16}\right) = a (8L^3) \frac{T^4}{16} = a (L^3) \frac{T^4}{2} = \frac{1}{2} a L^3 T^4 \]

**Hint:** Ensure to carefully apply the power rules when dealing with temperature changes.

### Step 5: Compare initial and final total energy
Now, we compare the initial and final total energies:
- Initial energy: \( U_i = a L^3 T^4 \)
- Final energy: \( U_f = \frac{1}{2} a L^3 T^4 \)

From this, we see that:
\[ U_f = \frac{1}{2} U_i \]

**Hint:** Comparing energies helps us understand how changes in conditions affect the total energy.

### Conclusion
The total energy decreases to half of its initial value when the dimensions of the box are doubled and the temperature is halved. Therefore, the total energy remains constant when considering the energy radiated and absorbed in the system.

**Final Answer:** The total energy is halved.