A black body emits radiation at the rate P when its temperature is T. At this temperature the wavelength at which the radiation has maximum intensity is `lamda_0`, If at another temperature `T'` the power radiated is `P'` and wavelength at maximum intensity is `(lamda_0)/(2)` then

Options

A. P'T'=32PT
B. P'T'=16PT
C. P'T'=8PT
D. P'T'=4PT

Answer

To solve the problem, we will use two important laws in thermodynamics related to black body radiation: Stefan-Boltzmann Law and Wien's Displacement Law.

### Step 1: Understand the relationship between power and temperature using Stefan-Boltzmann Law
According to Stefan-Boltzmann Law, the power radiated by a black body is directly proportional to the fourth power of its absolute temperature (T). This can be expressed as:
\[ P \propto T^4 \]
So, we can write:
\[ P = k T^4 \]
where \( k \) is a constant.

### Step 2: Write the power for the second temperature
For the second temperature \( T' \) and the power \( P' \), we can express it similarly:
\[ P' = k (T')^4 \]

### Step 3: Relate the two powers using their temperatures
From the above equations, we can relate the two powers:
\[ \frac{P'}{P} = \frac{(T')^4}{T^4} \]
This implies:
\[ P' = P \left( \frac{T'}{T} \right)^4 \]

### Step 4: Use Wien's Displacement Law to find the relationship between temperatures
Wien's Displacement Law states that the wavelength at which the radiation has maximum intensity is inversely proportional to the temperature. This can be expressed as:
\[ \lambda_0 T = b \]
where \( b \) is a constant.

For the first case:
\[ \lambda_0 T = b \]

For the second case, where the wavelength is \( \frac{\lambda_0}{2} \):
\[ \frac{\lambda_0}{2} T' = b \]

### Step 5: Relate the temperatures using the wavelengths
From the two equations:
1. \( \lambda_0 T = b \)
2. \( \frac{\lambda_0}{2} T' = b \)

We can equate them:
\[ \lambda_0 T = \frac{\lambda_0}{2} T' \]

Dividing both sides by \( \lambda_0 \) (assuming \( \lambda_0 \neq 0 \)):
\[ T = \frac{1}{2} T' \]
Thus:
\[ T' = 2T \]

### Step 6: Substitute \( T' \) back into the power equation
Now substituting \( T' = 2T \) into the power equation:
\[ P' = P \left( \frac{2T}{T} \right)^4 = P \cdot 2^4 = 16P \]

### Conclusion
Thus, we find that:
\[ P' = 16P \]

### Final Answer
The relationship between the powers is \( P' = 16P \).