Question

Consider the Earth to be a blackbody with an average temperature of 15°C. Find out the wavelength ($\lambda_{\text{max}}$) at which maximum power is radiated:

• A. $\sim$ 10.1 $\mu$m
• B. $\sim$ 1.1 $\mu$m
• C. $\sim$ 20 $\mu$m
• D. $\sim$ 0.11 $\mu$m

Hint:

Remember to always convert temperatures from Celsius to Kelvin when using Wien's Displacement Law, and use the correct value for Wien's displacement constant.

Answer 1

The Correct Option is A

Step 1: Concept
Wien's Displacement Law states that the wavelength ($\lambda_{\text{max}}$) at which a blackbody radiates maximum energy is inversely proportional to its absolute temperature ($T$). Mathematically: \[ \lambda_{\text{max}} = \frac{b}{T} \] where $b$ is Wien's displacement constant ($2.89777 \times 10^{-3}\ \text{m}\cdot\text{K}$).

Step 2: Meaning
Given the average temperature of Earth as 15°C, we convert to Kelvin and apply Wien's Displacement Law.

Step 3: Analysis
Convert the temperature from Celsius to Kelvin: \[ T = 15 + 273.15\ \text{K} = 288.15\ \text{K} \] Using Wien's Displacement Law: \[ \lambda_{\text{max}} = \frac{2.89777 \times 10^{-3}\ \text{m}\cdot\text{K}}{288.15\ \text{K}} \approx 1.006 \times 10^{-5}\ \text{m} = 10.06\ \mu\text{m} \] This value is closest to option (A): $\sim$10.1~$\mu$m.

Step 4: Conclusion
The wavelength at which maximum power is radiated by the Earth as a blackbody at 15°C is approximately 10.1 $\mu$m. Final Answer: (A)