Three bodies `A , B` and `C` have equal surface area and thermal emissivities in the ratio `e_(A) : e_(B) : e_(C) = 1 : (1)/(2): (1)/(4)`. All the three bodies are radiating at same rate. Their wavelengths corresponding to maximum intensity are `alpha_(A), alpha_(B)` and `alpha_(C)` respectively and their temperatures are `T_(A) , T_(B)` and `T_(C)` on kelvin scale , then select the incorrect statement

Options

A. `sqrt(T_(A)T_(C)) = T_(B)`
B. `sqrt(lambda_(A) lambda_(C)) = lambda_(B)`
C. `sqrt(e_(A) T_(A)) sqrt(e_(C)T_(C)) = e_(B)T_(B)`
D. `sqrt(e_(A)lambda_(A)T_(A).e_(B)lambda_(B)T_(B)) = e_(C)lambda_(C)T_(C)`

Explanation

To solve the problem, we need to analyze the relationship between the emissivities, temperatures, and the wavelengths corresponding to maximum intensity for the three bodies A, B, and C.

### Step-by-Step Solution:

1. **Understanding Emissivity and Power Radiated**:
- The emissivities of bodies A, B, and C are given in the ratio:
\[ e_A : e_B : e_C = 1 : \frac{1}{2} : \frac{1}{4} \]
- Let’s denote:
\[ e_A = 1, \quad e_B = \frac{1}{2}, \quad e_C = \frac{1}{4} \]

2. **Using Stefan-Boltzmann Law**:
- The power radiated by each body can be expressed using the Stefan-Boltzmann Law:
\[ P = e \sigma A T^4 \]
- Since all three bodies are radiating at the same rate, we can write:
\[ e_A \sigma T_A^4 = e_B \sigma T_B^4 = e_C \sigma T_C^4 \]

3. **Setting Up the Equations**:
- From the above, we can simplify:
\[ T_A^4 = e_B \cdot T_B^4 / e_A \]
\[ T_B^4 = e_C \cdot T_C^4 / e_B \]
- Substituting the emissivity values:
\[ T_A^4 = \frac{1/2}{1} T_B^4 = \frac{1}{2} T_B^4 \]
\[ T_B^4 = \frac{1/4}{1/2} T_C^4 = \frac{1}{2} T_C^4 \]

4. **Relating Temperatures**:
- From \(T_A^4 = \frac{1}{2} T_B^4\), we get:
\[ T_A = \left(\frac{1}{2}\right)^{1/4} T_B = \frac{T_B}{\sqrt[4]{2}} \]
- From \(T_B^4 = \frac{1}{2} T_C^4\), we get:
\[ T_B = \left(\frac{1}{2}\right)^{1/4} T_C = \frac{T_C}{\sqrt[4]{2}} \]

5. **Finding Relationships**:
- Now substituting back, we can express \(T_A\) in terms of \(T_C\):
\[ T_A = \frac{T_C}{\sqrt[4]{2} \cdot \sqrt[4]{2}} = \frac{T_C}{\sqrt{2}} \]

6. **Wavelengths of Maximum Intensity**:
- The wavelengths corresponding to maximum intensity are given by Wien's displacement law:
\[ \lambda_{max} \propto \frac{1}{T} \]
- Thus, we have:
\[ \alpha_A \propto \frac{1}{T_A}, \quad \alpha_B \propto \frac{1}{T_B}, \quad \alpha_C \propto \frac{1}{T_C} \]
- Substituting the temperature relationships:
\[ \alpha_A \propto \sqrt{2} \cdot \alpha_C \]
\[ \alpha_B \propto \sqrt[4]{2} \cdot \alpha_C \]

7. **Conclusion**:
- The incorrect statement can be identified by checking the relationships derived above. The relationships between the wavelengths and temperatures must hold true based on the derived equations.