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Chapter 34: Problem 17

Find \(\lambda_{\text {peak }}\) and \(\lambda_{\text {median }}\) for Earth, considered a \(288-\mathrm{K}\) blackbody.

Short Answer

Expert verified
Based on the Wien's Displacement Law, the peak wavelength \(\lambda_{\text {peak }}\) of radiation emitted by the Earth is approximately \(10.1\) µm. The median wavelength \(\lambda_{\text {median }}\) can be approximated as around \(5.4\) µm.

Step by step solution

01

Calculate Peak Wavelength

Use Wien's Displacement Law to calculate the peak wavelength \(\lambda_{\text {peak }}\). Wien's Displacement Law \( \lambda_{\text{peak}} = \frac{b}{T} \), where b is Wien's displacement constant \( b \approx 2.9 \times 10^{-3} \, m \cdot K \), and T is the temperature in Kelvin. For Earth, this means that \( \lambda_{\text {peak }} = \frac{2.9 \times 10^{-3}}{288} \) m.
02

Express Result in Micrometers

Next, convert the peak wavelength to micrometers for easier interpretation. Since 1 meter equals \(1 \times 10^{6}\) micrometers, \(\lambda_{\text {peak }} =\frac{2.9 \times 10^{-3}}{288} \times 1 \times 10^{6}\) micrometers.
03

Calculate Median Wavelength

The exact value of \(\lambda_{\text {median }}\) is less straightforward to calculate, as it involves integration over all possible wavelengths to find the point that divides the total power into two equal halves. Assuming that the distribution of wavelengths resembles a normal distribution, a good approximation is that the median falls about halfway between the peak and the edge of the 'visible spectrum' (approximately 700 nm, or \(0.7\) µm). Thus, \(\lambda_{\text {median }}\) can be approximated as \((\lambda_{\text {peak}} + 0.7) / 2\).

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Most popular questions from this chapter

Spacecraft instruments measure the radiation from an asteroid, and the data show that the power per unit wavelength peaks at \(40 \mu \mathrm{m} .\) Assuming the asteroid is a blackbody, find its surface temperature.

Find the rate of photon production by (a) a radio antenna broadcasting \(1.0 \mathrm{kW}\) at \(89.5 \mathrm{MHz},\) (b) a laser producing \(1.0 \mathrm{mW}\) of 633-nm light, and (c) an X-ray machine producing 0.10-nm X rays with total power \(2.5 \mathrm{kW}\)

The radiance of a blackbody peaks at \(660 \mathrm{nm}\). (a) What's its temperature? (b) How does its radiance at 400 nm compare with that at \(700 \mathrm{nm} ?\)

A photon's wavelength is equal to the Compton wavelength of a particle with mass \(m .\) Show that the photon's energy is equal to the particle's rest energy.

For a \(2.0-\mathrm{kK}\) blackbody, by what percentage is the Rayleigh-Jeans law in error at wavelengths of (a) \(1.0 \mathrm{mm},\) (b) \(10 \mu \mathrm{m},\) and (c) \(1.0 \mu \mathrm{m} ?\)
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