When the temperature of a black body increases, it is observed that the wavelength corresponding to maximum energy changes from `0.26mum` to `0.13mum`. The ratio of the emissive powers of the body at the respective temperatures is

Options

A. `(16)/(1)`
B. `(4)/(1)`
C. `(1)/(4)`
D. `(1)/(16)`

Explanation

To solve the problem, we need to find the ratio of the emissive powers of a black body at two different temperatures, given the wavelengths corresponding to maximum energy. We will use Wien's displacement law and Stefan-Boltzmann law to derive the solution step by step.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Wavelength at temperature T1, \( \lambda_1 = 0.26 \, \mu m \)
- Wavelength at temperature T2, \( \lambda_2 = 0.13 \, \mu m \)

2. **Apply Wien's Displacement Law:**
- According to Wien's displacement law, the product of the wavelength of maximum energy (\( \lambda_m \)) and the temperature (T) is a constant:
\[ \lambda_1 T_1 = \lambda_2 T_2 \]

3. **Rearrange the Equation:**
- From the above equation, we can express the ratio of the temperatures:
\[ \frac{\lambda_1}{\lambda_2} = \frac{T_2}{T_1} \]

4. **Substitute the Values:**
- Substitute the values of \( \lambda_1 \) and \( \lambda_2 \):
\[ \frac{0.26 \, \mu m}{0.13 \, \mu m} = \frac{T_2}{T_1} \]
- This simplifies to:
\[ 2 = \frac{T_2}{T_1} \]
- Thus, we find:
\[ T_2 = 2 T_1 \]

5. **Use Stefan-Boltzmann Law:**
- The emissive power (E) of a black body is proportional to the fourth power of its absolute temperature:
\[ E \propto T^4 \]
- Therefore, the ratio of the emissive powers at temperatures T1 and T2 is:
\[ \frac{E_1}{E_2} = \frac{T_1^4}{T_2^4} \]

6. **Substitute \( T_2 \) in Terms of \( T_1 \):**
- Substitute \( T_2 = 2 T_1 \) into the emissive power ratio:
\[ \frac{E_1}{E_2} = \frac{T_1^4}{(2 T_1)^4} \]

7. **Simplify the Expression:**
- Simplifying the right-hand side:
\[ \frac{E_1}{E_2} = \frac{T_1^4}{16 T_1^4} = \frac{1}{16} \]

8. **Final Result:**
- Therefore, the ratio of the emissive powers of the body at the respective temperatures is:
\[ \frac{E_1}{E_2} = \frac{1}{16} \]