Which of the following is v-T graph for spectral distribution of radiance in a perfectly black body? (Where v is the frequency corresponding to maximum wavelength.)

Options

A.
B.
C.
D.

Answer

To solve the question regarding the v-T graph for the spectral distribution of radiance in a perfectly black body, we can follow these steps:

### Step-by-Step Solution:

1. **Understand Wien's Displacement Law**:
Wien's Displacement Law states that the product of the maximum wavelength (λm) of radiation emitted by a black body and its absolute temperature (T) is a constant (b). Mathematically, this is expressed as:
\[ \lambda_m \cdot T = b \]

2. **Relate Wavelength to Frequency**:
The relationship between wavelength (λ) and frequency (ν) is given by the equation:
\[ \lambda = \frac{c}{\nu} \]
where c is the speed of light. Therefore, we can express the maximum wavelength in terms of frequency:
\[ \lambda_m = \frac{c}{\nu_m} \]

3. **Substitute into Wien's Law**:
Substituting the expression for λm into Wien's Law gives:
\[ \frac{c}{\nu_m} \cdot T = b \]
Rearranging this equation to solve for νm yields:
\[ \nu_m = \frac{c}{b} \cdot T \]

4. **Identify the Relationship**:
From the equation \(\nu_m = \frac{c}{b} \cdot T\), we can see that νm is directly proportional to T. This indicates that as the temperature increases, the frequency corresponding to the maximum wavelength also increases linearly.

5. **Graph Interpretation**:
Since νm is proportional to T, the graph of νm versus T will be a straight line with a positive slope. This means that if we were to plot the frequency (ν) on the y-axis and temperature (T) on the x-axis, we would expect a linear graph.

6. **Select the Correct Option**:
Based on the understanding that the graph must be linear with a positive slope, we can identify the correct option from the given choices. The option that represents a linear graph with a positive slope is the correct answer.

### Conclusion:
The correct option for the v-T graph for the spectral distribution of radiance in a perfectly black body is the one that shows a linear relationship with a positive slope.