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`

Explanation

To solve the problem, we will use the concepts of Stefan-Boltzmann law and Wien's displacement law. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the given information
We have a black body that emits radiation at a power \( P \) when its temperature is \( T \). The wavelength at which the radiation has maximum intensity is \( \lambda_0 \). At a new temperature \( T' \), the power radiated is \( P' \) and the wavelength at maximum intensity is \( \frac{\lambda_0}{2} \).

### Step 2: Apply Stefan-Boltzmann Law
According to the Stefan-Boltzmann law, the power radiated by a black body is given by:
\[ P = e \sigma A T^4 \]
For a black body, emissivity \( e = 1 \), so:
\[ P = \sigma A T^4 \]
For the new temperature \( T' \):
\[ P' = \sigma A (T')^4 \]

### Step 3: Apply Wien's Displacement Law
Wien's displacement law states that the wavelength of maximum intensity (\( \lambda \)) is inversely proportional to the absolute temperature (\( T \)):
\[ \lambda = \frac{B}{T} \]
where \( B \) is Wien's constant. For the original temperature \( T \):
\[ \lambda_0 = \frac{B}{T} \]
For the new temperature \( T' \) where the maximum wavelength is \( \frac{\lambda_0}{2} \):
\[ \frac{\lambda_0}{2} = \frac{B}{T'} \]

### Step 4: Relate the temperatures
From the equation above, we can express \( T' \):
\[ T' = \frac{B}{\frac{\lambda_0}{2}} = \frac{2B}{\lambda_0} \]
Now, substituting \( \lambda_0 = \frac{B}{T} \) into this equation gives:
\[ T' = \frac{2B}{\frac{B}{T}} = 2T \]

### Step 5: Substitute \( T' \) into the power equation
Now we substitute \( T' = 2T \) into the power equation:
\[ P' = \sigma A (T')^4 = \sigma A (2T)^4 = \sigma A \cdot 16T^4 = 16\sigma A T^4 \]
Thus, we can relate \( P' \) to \( P \):
\[ P' = 16P \]

### Step 6: Relate \( P' \) and \( T' \)
Now, using the relationship we obtained:
\[ P' T' = 16P \cdot 2T = 32PT \]

### Conclusion
Thus, the relationship we derived is:
\[ P' T' = 32PT \]
This means that the correct answer is option 1.

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