The earth is much cooler than the Sun, but it also emits radiation. The earth’s peak wavelength of emission is _______ compared to the Sun’s peak wavelength.

Question

The earth is much cooler than the Sun, but it also emits radiation.  The earth’s peak wavelength of emission is _______ compared to the Sun’s peak wavelength.

BlackTom AI · Accepted Answer

Wien’s displacement law relates the temperature of a blackbody to its peak emission wavelength: λ_max = b / T, where b ≈ 2.89777 × 10^−3 m·K.

1) Relevant knowledge points and concepts
- Wien’s displacement law (λ_max inversely proportional to temperature).
- The Sun’s effective temperature is ≈ 5800 K; the Earth’s effective radiating temperature is ≈ 288 K (≈15 °C is surface, but radiating temperature ~255 K; using 288 K or 255 K both give the same qualitative result: Earth peaks at much longer wavelengths).
- Shorter wavelength corresponds to higher temperature; longer wavelength corresponds to lower temperature.

2) Step-by-step reasoning and calculations
- For the Sun: λ_max,Sun = b / T_Sun = 2.89777×10^−3 m·K / 5800 K.
  Intermediate step: 2.89777×10^−3 / 5800 = 4.9965×10^−7 m ≈ 5.0×10^−7 m = 0.50 μm (visible light, green).
- For the Earth (use 288 K for demonstration): λ_max,Earth = b / T_Earth = 2.89777×10^−3 / 288 ≈ 1.005×10^−5 m = 10.05 μm (thermal infrared).
- Ratio: λ_max,Earth / λ_max,Sun ≈ (1.005×10^−5) / (4.9965×10^−7) ≈ 20.1. So Earth’s peak wavelength is roughly twenty times longer than the Sun’s.

3) Why the correct answer is correct
Because the Earth is much cooler than the Sun, Wien’s law predicts its peak emission occurs at significantly longer wavelengths (infrared) compared with the Sun’s peak in the visible. The numerical calculation demonstrates this (≈10 μm vs ≈0.5 μm).

4) Why the other options are incorrect
- "The same as": Incorrect because identical peak wavelengths would require identical temperatures; the Earth and Sun have very different temperatures.
- "Shorter": Incorrect because a shorter peak wavelength would imply the Earth is hotter than the Sun, which is not true.

Conclusion: The Earth’s peak wavelength of emission is Longer, which matches the provided answer.

The earth is much cooler than the Sun, but it also emits radiation. The earth’s peak wavelength of emission is _______ compared to the Sun’s peak wavelength.单项选择题

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