`lambda_(m)` is the wavelength of the radiations corresponding to maximum intensity of a very very hot body at temperature T. Which of the following correctly represents the relations between `lambda_(m)` and T:

Options

A. `lambda_(m)` decreases with increase in T
B. `lambda_(m)` increases with increase in T
C. `lambda_(m)` is independent of T
D. None of the above

Explanation

To solve the problem regarding the relationship between the wavelength of radiation corresponding to maximum intensity (\( \lambda_m \)) and temperature (\( T \)), we can use Wien's Displacement Law. Here’s a step-by-step solution:

### Step-by-Step Solution:

1. **Understand Wien's Displacement Law**:
Wien's Displacement Law states that the wavelength (\( \lambda_m \)) at which the intensity of radiation is maximum for a black body is inversely proportional to the absolute temperature (\( T \)). The law can be mathematically expressed as:
\[ \lambda_m \cdot T = b \]
where \( b \) is Wien's constant.

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

3. **Analyzing the Relationship**:
From the equation \( \lambda_m = \frac{b}{T} \), we can see that as the temperature \( T \) increases, the wavelength \( \lambda_m \) decreases. This indicates that \( \lambda_m \) is inversely proportional to \( T \).

4. **Conclusion**:
Therefore, the correct relationship between \( \lambda_m \) and \( T \) is that \( \lambda_m \) decreases as \( T \) increases.

5. **Identify the Correct Option**:
Based on the analysis, the correct option is the one that states that \( \lambda_m \) decreases with increasing temperature \( T \).

### Final Answer:
The correct representation of the relationship between \( \lambda_m \) and \( T \) is that \( \lambda_m \) decreases as \( T \) increases.

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