3.6 Thermal Radiation

Pre-Lecture Reading 3.6

Video Lecture

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Thermal Radiation (27:10)

Supplementary Notes

Thermal Radiation

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See Thermal Radiation.
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Temperature measures the average motion of a collection of particles.
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Absolute zero corresponds to no motion.
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Moving particles bounce off of one another.
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These bounces create ripples in the electromagnetic fields emanating from these particles.
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These ripples are light (see Light and the Electromagnetic Field).
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Consequently, a distribution of particle speeds results in a distribution of bounce speeds, and consequently in a distribution of ripple frequencies, and consequently in a distribution of light frequencies.
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This distribution of light frequencies is called a Planck, or blackbody, or thermal distribution.

Figure 1

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Higher temperatures correspond to faster particle speeds, and consequently faster bounce speeds, and consequently higher-frequency ripples, and consequently higher-frequency light: The distribution of light shifts to higher frequencies. This is called Wien's Law.
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Higher temperatures correspond to faster particle motions, and consequently more particle bounces, and consequently more ripples in the electromagnetic field, and consequently more light: The distribution of light becomes more intense at all frequencies. This is called Stefan's Law.

Wien's Law

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See Wien's Law.

( 10 )

λ_peak =

2.9 mm

(T/1 K)

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λ_peak = wavelength
at which a blackbody emits the most light
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T = temperature of the blackbody
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For stars, λ_peak is usually measured in nm and T in 1,000s of K. Hence, you might find this equivalent form of Wien's law more convenient to use:

( 11 )

λ_peak =

2,900 nm

(T/1,000 K)

Example:
The sun's surface temperature is 5,800 K. Hence, it emits most of its light at

λ_peak =

2,900 nm

(5,800 K/1,000 K)

= 500 nm,

which is in the visible part of the electromagnetic spectrum.

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Reminder: The visible part of the electromagnetic spectrum spans wavelengths of ≈700 nm (red) to ≈400 nm (violet).

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If λ_peak is in the ultraviolet, X-ray, or gamma-ray parts of the electromagnetic spectrum, its thermal distribution will still cross the visible part of the spectrum, and will be blue in color.
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If λ_peak is in the near-infrared (infrared, but almost visible) part of the electromagnetic spectrum, its thermal distribution will still cross the red side of the visible part of the spectrum, and will consequently be red in color (e.g., the
T = 1,000 distribution in Figure 1).
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If λ_peak is in the far-infrared or radio parts of the electromagnetic spectrum, its distribution will not cross the visible part of the spectrum, and will consequently be invisible (although detectable in the far-infrared and radio; e.g., the T = 300 distribution in Figure 1).

Stefan's Law

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See Stefan's Law.

( 12 )

F = σT⁴

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F = energy flux (energy emitted per unit area and per unit time) of a blackbody
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σ (Greek letter "sigma") = Stefan–Boltzmann constant
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T = temperature of the blackbody

In this course, you will never need to use the Stefan–Boltzmann constant to solve a problem.

Example:
Person A has a fever and is 1.01 times hotter than Person B. The energy flux coming off of Person A is how many times greater than the energy flux coming off of Person B? Solution: Let T_A and F_A be the temperature and energy flux of Person A. Let T_B and F_B be the temperature and energy flux of Person B. Then, F_A = σT_A⁴ and F_B = σT_B⁴. Dividing the latter equation into the former equation yields:

F_A

F_B

σT_A⁴

σT_B⁴

T_A

T_B

	4

1.01⁴ ≈ 1.04.

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Notice that we did not need to know the constant of proportionality, in this case σ, to solve this problem. This is what is called a ratio problem.

Most of the math problems in this course are ratio problems.

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If the constant of proportionality is not needed, we can more simply write:

F ∝ T⁴

Example:
Person A has a fever and is 1.01 times hotter than Person B. The energy flux coming off of Person A is 1.01⁴ = 1.04 times greater than the energy flux coming off of Person B.

Assignment 3

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Do Questions 4 and 5.