Calculating Earth's Maximum Radiation Wavelength

To find the wavelength ($\lambda_{max}$) at which maximum power is radiated by a blackbody, we use Wien's Displacement Law. The law states:

Where:

• $\lambda_{max}$ is the wavelength of maximum radiation.
• $T$ is the absolute temperature in Kelvin (K).
• $b$ is Wien's displacement constant, approximately $2.898 \times 10^{-3} \text{ m}\cdot\text{K}$.

Step 1: Convert Temperature to Kelvin

The given temperature is $15^\circ\text{C}$. We need to convert this to Kelvin:

Step 2: Apply Wien's Displacement Law

Now, substitute the values into Wien's formula to find $\lambda_{max}$:

Step 3: Convert Wavelength to Micrometers

The options are given in micrometers ($\mu\text{m}$). Convert the calculated wavelength:

Since $1 \text{ m} = 10^6 \mu\text{m}$:

Conclusion

The calculated wavelength is approximately $10.1 \mu\text{m}$. This corresponds to the wavelength at which the Earth, considered as a blackbody at $15^\circ\text{C}$, radiates maximum power.