If `lambda_(m)` denotes the wavelength at which the radiative emission from a black body at a temperature `T K` is maximum, then

Options

A. `lambd_(m) prop T^(4)`
B. `lambda_(m)` is independent of T
C. `lambda_(m) prop T`
D. `lambda_(m) prop T^(-1)`

Answer

To solve the problem regarding the relationship between the wavelength at which the radiative emission from a black body is maximum (denoted as \( \lambda_m \)) and the temperature \( T \), we can use Wien's Displacement Law.

### Step-by-Step Solution:

1. **Understanding Wien's Displacement Law**:
- Wien's Displacement Law states that the wavelength \( \lambda_m \) at which the emission of a black body spectrum is maximum is inversely proportional to the absolute temperature \( T \) of the body. Mathematically, it can be expressed as:
\[ \lambda_m \cdot T = b \]
where \( b \) is a constant known as Wien's constant.

2. **Rearranging the Equation**:
- From the equation \( \lambda_m \cdot T = b \), we can rearrange it to express \( \lambda_m \) in terms of \( T \):
\[ \lambda_m = \frac{b}{T} \]

3. **Identifying the Relationship**:
- The equation \( \lambda_m = \frac{b}{T} \) indicates that \( \lambda_m \) is directly proportional to \( \frac{1}{T} \). This can also be expressed as:
\[ \lambda_m \propto T^{-1} \]

4. **Conclusion**:
- Therefore, the correct option is that \( \lambda_m \) is directly proportional to \( T^{-1} \), which corresponds to option D: \( \lambda_m \propto T^{-1} \).

### Final Answer:
The correct answer is that \( \lambda_m \) is directly proportional to \( T^{-1} \).

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