1. 10: Electromagnetic Radiation
ENPh257: Thermodynamics
10: Electromagnetic Radiation

2. Thermal radiation
Electric charges radiate electromagnetic radiation when accelerated.
Any object with a non-zero temperatures will radiate (has accelerated charges).
Electromagnetic radiation, quantized as photons, can be in thermal equilibrium with its surroundings in a similar manner to the molecules of an ideal gas.
Photons, unlike gas molecules, can be created and destroyed.
Photons have no mass, their energy is related to their wavelength and frequency, which have a fixed relationship (the speed of light).
Photon energy 𝐸=ħ𝜔, momentum 𝑝 = ħ𝑘, 𝑐 = 𝜔/𝑘 (Planck, 1900). NB: ħ=ℎ/2𝜋
The radiation is isotropic and it has a wavelength (or frequency) distribution, which can be expressed in various ways.
© Chris Waltham, UBC Physics & Astronomy, 2018

3. Planck distribution
Most probable number of particles with energy 𝐸: Maxwell-Boltzmann: 𝑛 𝐸 ~ exp − 𝐸 𝑘 𝐵 𝑇 Bose-Einstein (e.g. photons): 𝑛 𝐸 ~ 1 exp 𝐸 𝑘 𝐵 𝑇 −1 The difference arises from counting statistics of identical spin-one particles (see PHYS403). “Phase-space” factor: Maxwell-Boltzmann: 4𝜋 𝑣 2 𝑑𝑣 Bose-Einstein (e.g. photons): 4𝜋 𝑘 2 𝑑𝑘, 𝐸=ħ𝑐𝑘 For a given energy range, there are more possibilities per unit range as the energy rises.
© Chris Waltham, UBC Physics & Astronomy, 2018

4. Planck distribution
Putting the two factors together: Thermal radiation spectrum will look something like (NB: I haven’t said exactly what this is): 𝑘 3 𝑑𝑘 exp ħ𝑐𝑘 𝑘 𝐵 𝑇 −1 The extra factor of k comes from the energy of each photon. Expression can also be recast in terms of 𝐸, 𝜔, 𝜆, or 𝑓 instead of 𝑘. See
© Chris Waltham, UBC Physics & Astronomy, 2018

5. Solid angle Solid angle (a necessary concept):
Three-dimensional measure of angular size.
Viewed from the centre of a sphere radius r, an area A of the sphere’s surface subtends a solid angle 𝛺:
𝛺= 𝛢 𝑟 2 steradians
Its maximum value is 4𝜋. At large distances (𝑟≫√𝐴), the area can be taken to be that of a flat disk perpendicular to the line of sight.
© Chris Waltham, UBC Physics & Astronomy, 2018

6. Spectral radiance
The spectral radiance of a blackbody is given by Planck’s Law:
𝐵 𝜆 = 2ℎ 𝑐 2 𝜆 exp ℎ𝑐 𝜆 𝑘 𝐵 𝑇 −1
The units are naturally W/(sr.m3), but more commonly expressed as W/(sr.m2.µm) or W/(sr.m2.nm), i.e. power per unit solid angle per unit area (of the emitting body) per unit wavelength range.
ℎ is Planck’s constant, and 𝑐 is the speed of light.
You can find the equivalent formula in terms of frequency etc. on
Different forms have different units, e.g. if given in terms of frequency: W/(sr.m2.Hz),
© Chris Waltham, UBC Physics & Astronomy, 2018

7. Changing units
One can always express the spectral distribution in terms of frequency (𝑓=𝑐/𝜆) rather than wavelength. But bear in mind that:
𝑑𝑓 𝑑𝜆 =− 𝑐 𝜆 2
This means that the peak power frequency 𝑓 𝑚𝑎𝑥 distribution will not correspond to a peak power wavelength 𝜆 𝑚𝑎𝑥 , i.e. 𝑓 𝑚𝑎𝑥 ≠𝑐/ 𝜆 𝑚𝑎𝑥 .
This becomes very important when dealing with photovoltaics, which operate in photon energy space (𝐸=ℎ𝑓) rather than wavelength space.
© Chris Waltham, UBC Physics & Astronomy, 2018

8. Spectral radiance
The wavelength at peak power is given by Wien’s displacement law:
𝜆 𝑚𝑎𝑥 =𝑏/𝑇
𝑏 ≈ m.K
6000 K gives 𝜆 𝑚𝑎𝑥 =500 nm (the Sun)
300 K gives 𝜆 𝑚𝑎𝑥 =10 μm (you).
© Chris Waltham, UBC Physics & Astronomy, 2018

9. Radiance, power
The radiance L of a blackbody is the spectral radiance integrated over all wavelengths, measured in W/(sr.m2).
𝐿= 0 ∞ 𝐵 𝜆 𝑑𝜆
Power P of a blackbody is the spectral radiance integrated over all wavelengths and solid angle, in W:
𝑃=𝐴 0 ∞ 𝐵 𝜆 𝑑𝜆 𝑑𝛺=𝜎𝐴 𝑇 4
Integration over solid angle is equivalent to multiplying by 𝜋 (see Lambert’s Law).
where 𝜎 is the Stefan-Boltzmann constant = 5.67 x 10-8 W/(m2.K4) and can be expressed in fundamental units.
Real surfaces radiate less than this: 𝑃=𝜀𝜎𝐴 𝑇 4 , where 𝜀 is the emissivity, which is a function of wavelength.
e.g. white paint has 𝜀 ≈ 0.1 (i.e. a reflectance of 0.9) for visible light but 𝜀 ≈ 0.9 for its own thermal radiation.
“Blackbody” is defined to have 𝜀 = 1 (I know, circular argument).
© Chris Waltham, UBC Physics & Astronomy, 2018

10. Sun and earth
These two curves have very little overlap: the infrared part of the solar spectrum (“near IR”) has very little in common with the thermal IR radiated by the Earth or us.
We will see in a moment that the yellow and red parts of this plot will have more-or-less the same total power once we have accounted for solid angle and emissivity.
scienceofdoom.com
© Chris Waltham, UBC Physics & Astronomy, 2018

11. Sun-earth system
Not to scale!
We want the solar intensity I, (in W/m2) measured at the Earth’s orbit, so we need to multiply the solar radiance 𝐿 (at the surface of the Sun, in W/m2/sr) by the area of the Sun’s disc and the solid angle 𝛺 of an area 𝛿𝑆 facing the Sun at the Earth’s orbit. 𝛺= δ𝑆 𝑅 𝑜𝑟𝑏𝑖𝑡 2
We can approximate the relevant area of the Sun as its projected area 𝜋 𝑟 𝑠𝑢𝑛 2 , as that is all we can perceive from the Earth, and the radius of the Sun is tiny compared to its distance from us:
𝐼= 𝐿𝜋 𝑟 𝑠𝑢𝑛 2 𝛺 𝛿𝑆 = 𝐿𝜋 𝑟 𝑠𝑢𝑛 2 𝑅 𝑜𝑟𝑏𝑖𝑡 2
The factor 𝜋 𝑟 𝑠𝑢𝑛 2 𝑅 𝑜𝑟𝑏𝑖𝑡 2 is just the solid angle of the Sun’s disc viewed from the Earth (i.e. reversing our vision to look at the Sun from the Earth rather than the reverse), i.e. 6.87×10−5 steradians.
© Chris Waltham, UBC Physics & Astronomy, 2018

12. Solar intensity at earth
At the Earth’s orbital distance from the Sun, for a surface perpendicular to the Sun’s rays: 𝐼≈ 1370 W/m2 At the bottom of the Earth’s atmosphere: 𝐼<~1000 W/m2 Depending on angle (solar elevation and detector/absorber), atmospheric conditions etc.
© Chris Waltham, UBC Physics & Astronomy, 2018