The power radiated by a black body is P, and it radiates maximum energy around the wavelength `lambda_(0)`. If the temperature of the black body is now changed so that it radiates maximum energy around a wavelength `3lambda_(0)//4`, the power radiated by it will increase by a factor of

Options

A. `256/81`
B. `4/3`
C. `3/4`
D. `84/256

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand Wien's Displacement Law
Wien's Displacement Law states that the wavelength of maximum radiation (\( \lambda_{max} \)) is inversely proportional to the temperature (\( T \)) of the black body. Mathematically, it can be expressed as:
\[ \lambda_{max} \cdot T = b \]
where \( b \) is a constant.

### Step 2: Set Up Initial Conditions
Let the initial wavelength of maximum radiation be \( \lambda_0 \) and the initial temperature be \( T_0 \). According to Wien's law:
\[ \lambda_0 \cdot T_0 = b \]

### Step 3: Set Up Final Conditions
Now, the black body radiates maximum energy around a new wavelength \( \frac{3\lambda_0}{4} \) at a new temperature \( T' \). According to Wien's law for the final conditions:
\[ \frac{3\lambda_0}{4} \cdot T' = b \]

### Step 4: Relate the Two Temperatures
From the two equations derived from Wien's law, we can equate them:
\[ \lambda_0 \cdot T_0 = \frac{3\lambda_0}{4} \cdot T' \]
We can simplify this by canceling \( \lambda_0 \) (assuming \( \lambda_0 \neq 0 \)):
\[ T_0 = \frac{3}{4} T' \]
Rearranging gives:
\[ T' = \frac{4}{3} T_0 \]

### Step 5: Use Stefan-Boltzmann Law for Power Calculation
The power radiated by a black body is given by the Stefan-Boltzmann law:
\[ P = \sigma A T^4 \]
where \( \sigma \) is the Stefan-Boltzmann constant and \( A \) is the surface area.

### Step 6: Calculate Initial Power
The initial power \( P_0 \) when the temperature is \( T_0 \) is:
\[ P_0 = \sigma A T_0^4 \]

### Step 7: Calculate Final Power
The final power \( P' \) when the temperature is \( T' = \frac{4}{3} T_0 \) is:
\[ P' = \sigma A \left(\frac{4}{3} T_0\right)^4 \]
Calculating this gives:
\[ P' = \sigma A \cdot \frac{256}{81} T_0^4 \]

### Step 8: Relate Final Power to Initial Power
Now, we can express the final power in terms of the initial power:
\[ P' = \frac{256}{81} P_0 \]
Since \( P_0 = P \), we have:
\[ P' = \frac{256}{81} P \]

### Step 9: Conclusion
Thus, the power radiated by the black body increases by a factor of:
\[ \frac{256}{81} \]