Power radiated by a black body is `P_0` and the wavelength corresponding to maximum energy is around `lamda_0`, On changing the temperature of the black body, it was observed that the power radiated becames `(256)/(81)P_0`. The shift in wavelength corresponding to the maximum energy will be

Options

A. `+(lamda_0)/(4)`
B. `+(lamda_0)/(2)`
C. `-(lamda_0)/(4)`
D. `-(lamda_0)/(2)`

Explanation

To solve the problem, we will use two important laws: Wien's Law and Stefan-Boltzmann Law. Let's break down the solution step by step.

### Step 1: Understand the given information
We know:
- Initial power radiated by the black body: \( P_0 \)
- Wavelength corresponding to maximum energy: \( \lambda_0 \)
- New power radiated after changing temperature: \( P = \frac{256}{81} P_0 \)

### Step 2: Apply Stefan-Boltzmann Law
According to the Stefan-Boltzmann Law:
\[ \frac{P_0}{P} = \left(\frac{T_0}{T}\right)^4 \]
Substituting the new power \( P \):
\[ \frac{P_0}{\frac{256}{81} P_0} = \left(\frac{T_0}{T}\right)^4 \]
This simplifies to:
\[ \frac{81}{256} = \left(\frac{T_0}{T}\right)^4 \]

### Step 3: Solve for the temperature ratio
Taking the fourth root of both sides:
\[ \frac{T_0}{T} = \left(\frac{81}{256}\right)^{\frac{1}{4}} = \frac{3}{4} \]
This means:
\[ T = \frac{4}{3} T_0 \]

### Step 4: Apply Wien's Law
Wien's Law states:
\[ \lambda_0 T_0 = \lambda T \]
Rearranging gives:
\[ \frac{\lambda}{\lambda_0} = \frac{T_0}{T} \]
Substituting \( T \) from the previous step:
\[ \frac{\lambda}{\lambda_0} = \frac{T_0}{\frac{4}{3} T_0} = \frac{3}{4} \]
Thus, we have:
\[ \lambda = \frac{3}{4} \lambda_0 \]

### Step 5: Calculate the shift in wavelength
The shift in wavelength \( \Delta \lambda \) is given by:
\[ \Delta \lambda = \lambda - \lambda_0 \]
Substituting for \( \lambda \):
\[ \Delta \lambda = \frac{3}{4} \lambda_0 - \lambda_0 = -\frac{1}{4} \lambda_0 \]

### Final Answer
The shift in wavelength corresponding to the maximum energy is:
\[ \Delta \lambda = -\frac{1}{4} \lambda_0 \]