Earth's surface approximates a blackbody at 288 K. Find the power emitted by Earth's entire sur

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Earth's surface approximates a blackbody at 288 K. Find the...

Earth's surface approximates a blackbody at $288 \mathrm{~K}$. Find the power emitted by Earth's entire surface (a) in the visible wavelength interval from $0.550$ to $0.551 \mu \mathrm{m}$ and (b) in the infrared between $10.000 \mu \mathrm{m}$ and $10.001 \mu \mathrm{m}$. Your answer helps explain why greenhouse gases, which strongly absorb infrared radiation at around $10 \mu \mathrm{m}$, have such a significant effect on Earth's climate. (Note: The wavelength interval here is so small that, for the point this problem is making, you don't need to integrate to apply Equation 34.3.)

Key Concepts

Blackbody Radiation

Blackbody radiation refers to the theoretical spectrum of electromagnetic radiation emitted by an ideal object that absorbs all incident radiation and re?emits energy solely based on its temperature. This concept is fundamental to understanding how objects emit radiation, and it provides the basis for analyzing the emission characteristics of bodies such as Earth, which is often approximated as a blackbody in physics problems.

Planck's Law

Planck's law mathematically describes the spectral radiance of a blackbody as a function of wavelength (or frequency) and temperature. It specifies how the intensity of radiation emitted at different wavelengths deviates with temperature, forming the characteristic curve of blackbody radiation. This law is essential for calculating the emission in specific narrow wavelength intervals, even when full integration is not required.

Wien's Displacement Law

Wien's displacement law states that the wavelength at which the emission of a blackbody peaks is inversely proportional to its temperature. For Earth’s temperature, this peak is located in the infrared region, which explains why the majority of terrestrial radiation is not in the visible range. This concept is key in understanding why even narrow infrared intervals can contain significantly more power compared to similar intervals in the visible spectrum.

Spectral Integration over Narrow Wavelength Intervals

When considering very narrow wavelength intervals, the power emitted can be approximated by multiplying the spectral radiance (given by Planck's law) by the small wavelength interval. This approach simplifies the evaluation of emission in specific bands without needing to perform a full integration, making it a useful approximation in problems dealing with tiny intervals in the spectrum.

Greenhouse Effect

The greenhouse effect involves the absorption and re-emission of infrared radiation by gases in Earth's atmosphere, such as carbon dioxide and water vapor. These gases are particularly effective at absorbing infrared radiation near the wavelengths where Earth’s blackbody emission peaks. This concept explains how small changes or strong absorption in narrow infrared ranges can have a significant impact on Earth's climate by trapping heat.

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