Two bodies A and B have emissivities 0.5 and 0.8 respectively. At some temperature the two bodies have maximum spectral emissive powers at wavelength `8000 Å` and `4000 Å` respectively. The ratio of their emissive powers at these temperatures is :

Options

A. `(5)/(128)`
B. `10`
C. `(5)/(16)`
D. None of these

Explanation

To solve the problem, we need to find the ratio of the emissive powers of two bodies A and B, given their emissivities and the wavelengths at which they have maximum spectral emissive powers.

### Step-by-Step Solution:

1. **Identify Given Data:**
- Emissivity of body A, \( E_A = 0.5 \)
- Emissivity of body B, \( E_B = 0.8 \)
- Wavelength for body A, \( \lambda_A = 8000 \, \text{Å} \)
- Wavelength for body B, \( \lambda_B = 4000 \, \text{Å} \)

2. **Apply Wien’s Displacement Law:**
- According to Wien's displacement law, the product of the wavelength at which the maximum spectral emissive power occurs and the absolute temperature is constant:
\[ \lambda_1 T_1 = \lambda_2 T_2 \]
- Rearranging gives us:
\[ \frac{\lambda_1}{\lambda_2} = \frac{T_2}{T_1} \]

3. **Substituting the Values:**
- Substitute \( \lambda_1 = 8000 \, \text{Å} \) and \( \lambda_2 = 4000 \, \text{Å} \):
\[ \frac{8000}{4000} = \frac{T_2}{T_1} \]
- Simplifying gives:
\[ 2 = \frac{T_2}{T_1} \quad \Rightarrow \quad T_1 = \frac{T_2}{2} \]

4. **Emissive Power Formula:**
- The emissive power \( E \) of a body is given by:
\[ E = \epsilon \sigma T^4 \]
- Where \( \epsilon \) is the emissivity, \( \sigma \) is the Stefan-Boltzmann constant, and \( T \) is the absolute temperature.

5. **Expressing the Ratio of Emissive Powers:**
- The emissive powers for bodies A and B can be expressed as:
\[ E_A = E_A \sigma T_1^4 \quad \text{and} \quad E_B = E_B \sigma T_2^4 \]
- The ratio of emissive powers \( \frac{E_A}{E_B} \) is:
\[ \frac{E_A}{E_B} = \frac{E_A \sigma T_1^4}{E_B \sigma T_2^4} = \frac{E_A}{E_B} \cdot \frac{T_1^4}{T_2^4} \]

6. **Substituting the Values:**
- Substitute \( E_A = 0.5 \) and \( E_B = 0.8 \):
\[ \frac{E_A}{E_B} = \frac{0.5}{0.8} \cdot \left(\frac{T_1}{T_2}\right)^4 \]
- From step 3, we have \( T_1 = \frac{T_2}{2} \):
\[ \frac{T_1}{T_2} = \frac{1}{2} \]
- Therefore:
\[ \left(\frac{T_1}{T_2}\right)^4 = \left(\frac{1}{2}\right)^4 = \frac{1}{16} \]

7. **Final Calculation:**
- Now substituting back:
\[ \frac{E_A}{E_B} = \frac{0.5}{0.8} \cdot \frac{1}{16} = \frac{5}{8} \cdot \frac{1}{16} = \frac{5}{128} \]

### Conclusion:
The ratio of the emissive powers of bodies A and B is:
\[ \frac{E_A}{E_B} = \frac{5}{128} \]