Radiation

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© 1998-2008 by Glenn Elert -- A Work in Progress
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Discussion

Heat radiation (as opposed to particle radiation) is the transfer of internal energy in the form of electromagnetic waves. For most bodies on the earth, this radiation lies in the infrared region of the electromagnetic spectrum.

stefan-boltzmann law

One of the first to recognize that heat radiation is related to light was the English astronomer William Herschel, who noticed in 1800 that if a thermometer was moved from one end of a prism produced spectrum to the other, the highest temperatures would register below the red band, where no light was visible. Because of this position, this form of radiation is called infrared (infra being the Latin word for below or within). Sometimes this kind of radiation is called "heat waves" but this is a misnomer. Recall that heat is the transfer of internal energy from one region to another. As all forms of electromagnetic radiation transfer internal energy, they could be called heat.

Stefan-Boltzmann Law

P = εσA(T4 − T04)

where σ [sigma] is called Stefan's constant, which was discovered experimentally by Josef Stefan (1835-1893) Austria in 1879 and ε [epsilon] is the emissivity.

σ =  2π5k4  =  π2k4  = 5.67 × 10−8 W/m2K4
15h3c2 60ℏ3c2

Disconnected thoughts that aren't quotes.


Dark colors absorb more radiant energy than do light colors. The burns on this woman's skin mimic the pattern on her blouse. She was exposed to a monstrous dose of electromagnetic radiation from a nuclear blast. (Source: Unknown)

wien's displacement law

λmax =  hc  =  2.898 × 10−3 Km
(4.96511…)kT T

text

Temperature (or Effective Temperature) of Selected Radiant Sources
kelvin
temperature		 radiant energy source
3 cosmic background radiation
306 human skin
500 household oven at its hottest
660 minimum temperature for incandescence
770 dull red heat
1400 glowing coals, electric stove, electric toaster
1900 candle flame
2000 kerosene lamp
2800 incandescent light bulb, 75 W
2900 incandescent light bulb, 100 W
3000 incandescent light bulb, 200 W
3100 sunrise or sunset (effective)
3200 professional studio lights
3600 one hour after sunrise or one hour before sunset (effective)
4000 two hours after sunrise or two hours before sunset (effective)
5500 direct midday sunlight
6500 daylight (effective)
7000 overcast sky (effective)
20-30,000 lightning bolt

Transition paragraph

Metal Temperature by Color   Color Scale of Temperature
color approximate temperature   color Temperature
°F °C K   °C K
faint red 930 500 770   incipient red heat 500 - 550 770 - 820
blood red 1075 580 855   dark red heat 650 - 750 920 - 1020
dark cherry 1175 635 910   bright red heat 850 - 950 1120 - 1220
medium cherry 1275 0690 0965   yellowish red heat 1050 - 1150 1320 - 1420
cherry 1375 0745 1020   incipient white heat 1250 - 1350 1520 - 1620
bright cherry 1450 0790 1060   white heat 1450 - 1550 1720 - 1820
salmon 1550 0845 1115   "This table is the result of an effort to interpret in terms of thermometric readings, the common expressions used in describing temperatures. It is obvious that these values are only approximations."
dark orange 1630 0890 1160
orange 1725 0940 1215
lemon 1830 1000 1270
light yellow 1975 1080 1355
white 2200 1205 1480
Sources: Process Associates of America & Handbook of Chemistry & Physics, 1924

Transition paragraph

Spectral Classification of Stars
color T (K) class discrete spectra examples
very blue 30,000 O He+, N++, Si+++,
other highly ionized atoms		 naos, mintaka
blue-white 20,000 B weak H;
He, Si+, Si ++, O+, Mg+		 spica, rigel
white 10,000 A strong H;
Mg+, Si+, Fe+, Ti+, Ca+		 sirius, vega
yellow-white 8000 F weak H; Ca+, Fe+, Cr+; Fe, Cr,
and other neutral metals		 canopus, procyon
yellow 6000 G strong Ca+; many neutral
and ionized metals; CH bands		 sun, alpha centauri
orange 4000 K CH bands;
neutral metals		 arcturus, aldebaran
red 3000 M molecular TiO bands;
neutral metals		 antares, betelgeuse

solar energy

greenhouse effect

The basic effect…


[magnify]

Global temperature and atmospheric carbon dioxide trends match. The very long graph made popular by Al Gore in An Inconvenient Truth.


[magnify]

Plot one against the other. The relation is approximately linear. Al Gore never did this one.


[magnify]

Naturally occurring greenhouse gases whose concentrations are increasing due to human activities

Other naturally occurring greenhouse gases of lesser concern.

Greenhouse gases that do not occur naturally.

key infrared absorption bands in the atmosphere correspond to H2O, CO2, O3

Global Warming Potential of Selected Greeenhouse Gases
molecule potential lifetime (years)
CO2 carbon dioxide 1 120
CH4 methane 23 12
N2O nitrous oxide 296 114
CO carbon monoxide   0.25
CCl3F CFC11 3,800 50
CF2Cl2 CFC12 8,100 102
CCl2FCClF2 CFC113 4,800 85
CCl4 carbon tetrachloride 1,400 42
Source: unknown

Summary

Problems

practice

  1. Dyson Sphere
    1. Given a Dyson sphere as big as the earth's orbit surrounding the sun, determine …
      1. its surface temperature and
      2. the peak wavelength of the radiation emitted.
    2. Given a Dyson sphere surrounding the sun with an interior temperature suitable for human habitation, determine …
      1. its radius and
      2. the peak wavelength of the radiation emitted.
    Solutions …
    1. A Dyson Sphere as Big as the Earth
      1. Once equilibrium has been reached, a Dyson sphere will radiate energy at the same rate as the star it surrounds. Assume that both the sphere and the star are blackbody radiators. Using Stefan's law it can be shown that the temperature of a Dyson sphere is inversely proportional to the square root of its radius. For a given star, larger spheres have more surface area, do not need to radiate energy away so quickly, and will thus be cooler.
             
        P =  Q  = σεDADTD4 = σε☉A☉T☉4 = constant
        t
             
        ⎛
        ⎜
        ⎝        		 TD ⎞4
        ⎟
        ⎠        		  =  σε☉A☉  =  4πr☉2  =  ⎛
        ⎜
        ⎝        		 r☉ ⎞2
        ⎟
        ⎠
        T☉ σεDAD 4πrD2 rD
        TD  =  ⎛
        ⎜
        ⎝        		 r☉ ⎞
        ⎟
        ⎠        		 ½
        T☉ rD  
                   
        Using the physical data for the sun and earth we can now determine the sphere's surface temperature
                         
        TD = T☉  ⎛
        ⎜
        ⎝        		 r☉ ⎞½
        ⎟
        ⎠        		  = 5500 K  ⎛
        ⎜
        ⎝        		 6.96 × 108 m ⎞½
        ⎟
        ⎠        		  = 375  K = 102 °C
        rD 1.496 × 1011 m
                         
      2. The peak wavelength of the thermal radiation emitted comes from Wien's displacement law.
                 
        λmax =  2.898 × 10−3 mK  =  2.898 × 10−3 mK  = 7700 nm
        T 375 K
                 
        No surprise here, the peak would lie in the infrared region of the electromagnetic spectrum. Had an alien civilization built a Dyson sphere with these dimensions, it would emit characteristic blackbody radiation with a peak wavelength of 7700 nm.
    2. A More Comfortable Dyson Sphere
      1. Unfortunately for us humans, however, a Dyson sphere of this type would be uninhabitable. The interior would be 2 °C hotter than the normal boiling point of water. A more comfortable temperature would be about 300 K (27 °C). Were an advanced human civilization to build such a structure, it would have to be larger than the orbit of the earth.
                         
        rD = r☉  ⎛
        ⎜
        ⎝        		 T☉ ⎞2
        ⎟
        ⎠        		  = 6.96 × 108 m  ⎛
        ⎜
        ⎝        		 550 K ⎞2
        ⎟
        ⎠        		  = 2.34 × 1011 m = 1.56 AU
        TD 300 K
                         
        This is just a bit bigger than the orbit of Mars (which is 1.52 astronomical units). Temperature would be fine inside this sphere, but the increased distance to the sun would make it seem a bit dim. From the outside, however, such a sphere would be invisible (invisible to creatures with eyes like our, that is).
      2. Using Wien's law, we again find a peak wavelength in the infrared.
                 
        λmax =  2.898 × 10−3 mK  =  2.898 × 10−3 mK  = 9700 nm
        T 300 K
                 
  2. Write something else.
    • Answer it
  3. Write something different.
    • Answer it
  4. Write something completely different.
    • Answer it.

investigative

  1. Compute the rate at which your body radiates heat to a 20 °C room when unclothed. Assume a skin temperature of 34 °C. Use one of the following empirical formulas to estimate your body surface area. (Be careful to use the correct units for the equation you choose. A = surface area, h = height, m = mass.)

    Empirical Formulas for Estimating Body Surface Area
    source formula
    Boyd (1935) A[m2] = 0.0003207 · h[cm]0.3 · m[g](0.7285 - 0.0188 · log(m[g]))
    DuBois & DuBois (1916) A[m2] = 0.20247 · h[m]0.725 · m[kg]0.425
    Gehan & George (1970) A[m2] = 0.0235 · h[cm]0.42246 · m[kg]0.51456
    Haycock, Schwartz, Wisotsky (1978) A[m2] = 0.024265 · h[cm]0.3964 · m[kg]0.5378
    Mosteller (1987) A[m2] = √(h[cm] · m[kg] / 3600)
    Current (1998)* A[m2] = 0.1321 + 0.03433 · m[kg] ≈ (m[kg] + 4) / 30
    * Use this formula for children 3 - 30 kg.


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