IF the temperature of a black , body is doubled, the wavelength at which the spectral radiancy has its maximum is

Options

A. doubled
B. halved
C. quadrupled
D. unchanged

Explanation

To solve the problem, we will use Wien's Displacement Law, which states that the wavelength at which the spectral radiancy of a black body spectrum is at its maximum (λm) is inversely proportional to the temperature (T) of the body.

### Step-by-Step Solution:

1. **Understand Wien's Displacement Law**:
According to Wien's law, the relationship can be expressed as:
\[ \lambda_m \cdot T = b \]
where \( \lambda_m \) is the wavelength at maximum spectral radiancy, \( T \) is the absolute temperature, and \( b \) is Wien's constant.

2. **Initial Condition**:
Let the initial temperature be \( T \) and the corresponding maximum wavelength be \( \lambda_m \). Therefore, we can write:
\[ \lambda_m \cdot T = b \]

3. **Doubling the Temperature**:
If the temperature is doubled, the new temperature becomes \( 2T \).

4. **Applying Wien's Law Again**:
Using the new temperature in Wien's law:
\[ \lambda_{m_{new}} \cdot (2T) = b \]
where \( \lambda_{m_{new}} \) is the new maximum wavelength.

5. **Relating the Two Conditions**:
Since \( b \) remains constant, we can equate the two expressions:
\[ \lambda_m \cdot T = \lambda_{m_{new}} \cdot (2T) \]

6. **Simplifying the Equation**:
Dividing both sides by \( T \) (assuming \( T \neq 0 \)):
\[ \lambda_m = \lambda_{m_{new}} \cdot 2 \]

7. **Finding the New Wavelength**:
Rearranging gives us:
\[ \lambda_{m_{new}} = \frac{\lambda_m}{2} \]

8. **Conclusion**:
Thus, when the temperature of the black body is doubled, the wavelength at which the spectral radiancy has its maximum becomes half of the original wavelength:
\[ \lambda_{m_{new}} = \frac{\lambda_m}{2} \]

### Final Answer:
The wavelength at which the spectral radiancy has its maximum is halved.