Planck's equation

Planck's equation describes the amount of spectral radiance at a certain wavelength radiated by a black body in thermal equilibrium.

Electromagnetics

In terms of wavelength (λ), Planck's [equation] is written [as]

B_{\lambda }(T)={\frac {2hc^{2}}{\lambda ^{5}}}{\frac {1}{e^{\frac {hc}{\lambda k_{\mathrm {B} }T}}-1}}

where B is the spectral radiance, T is the absolute temperature of the black body, k_B is the Boltzmann constant, h is the Planck constant, and c is the speed of light.

This form of the equation contains several constants that are usually not subject to variation with wavelength. These are h, c, and k_B. They may be represented by simple coefficients: c1 = 2h c² and c2 = h c/k_B.

By setting the first partial derivative of Planck's equation in wavelength form equal to zero, iterative calculations may be used to find pairs of (λ,T) that to some significant digits represent the peak wavelength for a given temperature and vice versa.

{\frac {\partial B}{\partial \lambda }}={\frac {c1}{\lambda ^{6}}}{\frac {1}{e^{\frac {c2}{\lambda T}}-1}}[{\frac {c2}{\lambda T}}{\frac {1}{e^{\frac {c2}{\lambda T}}-1}}e^{\frac {c2}{\lambda T}}-5]=0.

Or,

{\frac {c2}{\lambda T}}{\frac {1}{e^{\frac {c2}{\lambda T}}-1}}e^{\frac {c2}{\lambda T}}-5=0.

{\frac {c2}{\lambda T}}{\frac {1}{e^{\frac {c2}{\lambda T}}-1}}e^{\frac {c2}{\lambda T}}=5.

Use c2 = 1.438833 cm K.

Problem 1

For the wavelength, temperature pair (570 nm, 5260 K), and for a simple calculator,

y=(1.48833/(0.0000570*5260))*exp(1.48833/(0.0000570*5260))/(exp(1.48833/(0.0000570*5260))-1)-5, followed by print y, yields a value close to zero (-1.006663E-03). The closer to zero the more accurate the estimate.

sRGB rendering of the spectrum of visible light
Color	Frequency	Wavelength
violet	668–789 THz	380–450 nm
blue	631–668 THz	450–475 nm
cyan	606–630 THz	476–495 nm
green	526–606 THz	495–570 nm
yellow	508–526 THz	570–590 nm
orange	484–508 THz	590–620 nm
red	400–484 THz	620–750 nm

For each color band, pick a wavelength, and calculate the corresponding temperature to complete the pair.

Problem 2

The table below gives approximate ultraviolet wavelength bands. Pick a wavelength from each and calculate the corresponding temperature to complete the pair.

Name	Abbreviation	Wavelength range in nanometers	Energy per photon
Before UV spectrum	Visible light	above 400 nm	below 3.10 eV
Ultraviolet A, long wave, or black light	UVA	400 nm–315 nm	3.10–3.94 eV
Near	NUV	400 nm–300 nm	3.10–4.13 eV
Ultraviolet B or medium wave	UVB	315 nm–280 nm	3.94–4.43 eV
Middle	MUV	300 nm–200 nm	4.13–6.20 eV
Ultraviolet C, short wave, or germicidal	UVC	280 nm–100 nm	4.43–12.4 eV
Far	FUV	200 nm–122 nm	6.20–10.2 eV
Vacuum	VUV	200 nm–100 nm	6.20–12.4 eV
Low	LUV	100 nm–88 nm	12.4–14.1 eV
Super	SUV	150 nm–10 nm	8.28–124 eV
Extreme	EUV	121 nm–10 nm	10.2–124 eV
Beyond UV range	X-rays	below 10 nm	above 124 eV

Problem 3

X-rays span 3 decades in wavelength, frequency and energy. From 10 to 0.1 nanometers (nm) (about 0.12 to 12 keV) they are classified as soft X-rays, and from 0.1 nm to 0.01 nm (about 12 to 120 keV) as hard X-rays.

Super soft X-rays have energies in the 0.09 to 2.5 keV.

For each of the types of X-rays, pick a representative wavelength and calculate the temperature to complete the pair.

Gamma rays have wavelengths less than 10 picometers (less than the diameter of an atom).

10¹⁵ Hz = PHz petahertz

10¹⁸ Hz = EHz exahertz

10²¹ Hz = ZHz zettahertz

10²⁴ Hz = YHz yottahertz

Convert these frequencies to their corresponding wavelengths and calculate the temperature for each to complete the pair.

Problem 4

Astronomers often divide the infrared spectrum as follows:^[1]

Designation	Abbreviation	Wavelength
Near Infrared	NIR	(0.7–1) to 5 µm
Mid Infrared	MIR	5 to (25–40) µm
Far Infrared	FIR	(25–40) to (200–350) µm.

These are the approximate ranges for photon energies of the infrared bands:

Division Name	Wavelength	Photon Energy
Near-infrared	0.75-1.4 µm	0.9-1.7 eV
Short-wavelength infrared	1.4-3 µm	0.4-0.9 eV
Mid-wavelength infrared	3-8 µm	150-400 meV
Long-wavelength infrared	8–15 µm	80-150 meV
Far infrared	15 - 1,000 µm	1.2-80 meV

These are the main infrared atmospheric windows:

Wavelength range (micrometres)	Astronomical bands	Telescopes
0.65 to 1.0	R and I bands	All major optical telescopes
1.1 to 1.4	J band	Most major optical telescopes and most dedicated infrared telescopes
1.5 to 1.8	H band	Most major optical telescopes and most dedicated infrared telescopes
2.0 to 2.4	K band	Most major optical telescopes and most dedicated infrared telescopes
3.0 to 4.0	L band	Most dedicated infrared telescopes and some optical telescopes
4.6 to 5.0	M band	Most dedicated infrared telescopes and some optical telescopes
7.5 to 14.5	N band	Most dedicated infrared telescopes and some optical telescopes
17 to 25	Q band	Some dedicated infrared telescopes and some optical telescopes
28 to 40	Z band	Some dedicated infrared telescopes and some optical telescopes
330 to 370		Some dedicated infrared telescopes and some optical telescopes
450	submillimeter	Submillimeter telescopes

Commonly used sub-divisions are

Division Name	Abbreviation	Wavelength	Characteristics
Near-infrared	NIR, IR-A DIN	0.75-1.4 µm	Defined by the water absorption, and commonly used in fiber optic telecommunication because of low attenuation losses in the SiO₂ glass (silica) medium. Image intensifiers are sensitive to this area of the spectrum. Examples include night vision devices such as night vision goggles.
Short-wavelength infrared	SWIR, IR-B DIN	1.4-3 µm	Water absorption increases significantly at 1,450 nm. The 1,530 to 1,560 nm range is the dominant spectral region for long-distance telecommunications.
Mid-wavelength infrared	MWIR, IR-C DIN. Also called intermediate infrared (IIR)	3-8 µm	In guided missile technology the 3-5 µm portion of this band is the atmospheric window in which the homing heads of passive IR 'heat seeking' missiles are designed to work, homing on to the Infrared signature of the target aircraft, typically the jet engine exhaust plume
Long-wavelength infrared	LWIR, IR-C DIN	8–15 µm	This is the "thermal imaging" region, in which sensors can obtain a completely passive picture of the outside world based on thermal emissions only and requiring no external light or thermal source such as the sun, moon or infrared illuminator. Forward-looking infrared (FLIR) systems use this area of the spectrum. This region is also called the "thermal infrared."
Far infrared	FIR	15 - 1,000 µm	(see also far-infrared laser).

Pick a representative wavelength from each band and calculate its temperature to complete the pair.

Problem 5

Submillimeter waves lie at the far end of the infrared band, just before the start of the microwave band. Credit: Tatoute.{{free media}}

“[T]erahertz radiation refers to electromagnetic waves propagating at frequencies in the terahertz range. It is synonymously termed submillimeter radiation, terahertz waves, terahertz light, T-rays, T-waves, T-light, T-lux, THz. The term typically applies to electromagnetic radiation with frequencies between high-frequency edge of the microwave band, 300 gigahertz (3 x 10¹¹ Hz),"^[2] and the long-wavelength edge of far-infrared light, 3000 GHz (3 x 10¹² Hz or 3 THz). In wavelengths, this range corresponds to 0.1 mm (or 100 µm) infrared to 1.0 mm microwave.

Microwaves, a subset of radio waves, have wavelengths ranging from as long as one meter to as short as one millimeter, or equivalently, with frequencies between 300 MHz (0.3 GHz) and 300 GHz.^[3] This broad definition includes both UHF and EHF (millimeter waves), and various sources use different boundaries.^[4] In all cases, microwave includes the entire SHF band (3 to 30 GHz, or 10 to 1 cm) at minimum, with RF engineering often putting the lower boundary at 1 GHz (30 cm), and the upper around 100 GHz (3 mm).

Radio waves have frequencies from 300 [Gigahertz] GHz to as low as 3 [Kilohertz] kHz, and corresponding wavelengths from 1 millimeter to 100 kilometers.

Pick representative wavelengths from each band or sub-band and calculate a matching temperature to complete the pair.

Problem 6

The first equation at top for the problem set allows the calculation of spectral radiance. Using a simple graphics routine calculate representative spectra for each wavelength, temperature pair.

Try integrating Planck's equation at the top of the resource to give the area under any such curve. Calculate the areas under each curve you've plotted. What is this area under the curve called?

Hypotheses

Planck's equation can be derived without the use of quanta.

References

↑ IPAC Staff. "Near, Mid and Far-Infrared". NASA ipac. Retrieved 2007-04-04.
↑ Hankwang (October 31, 2008). Terahertz radiation. pp. 1. http://en.wikipedia.org/w/index.php?title=Terahertz_radiation&diff=248835142&oldid=248831987. Retrieved 2012-08-05.
↑ Pozar, David M. (1993). Microwave Engineering Addison-Wesley Publishing Company. ISBN 0-201-50418-9.
↑ http://www.google.com/search?hl=en&defl=en&q=define:microwave&ei=e6CMSsWUI5OHmQee2si1DQ&sa=X&oi=glossary_definition&ct=title

External links

[IPAC-1] IPAC Staff. "Near, Mid and Far-Infrared". NASA ipac. Retrieved 2007-04-04.

[Hankwang-2] Hankwang (October 31, 2008). Terahertz radiation. pp. 1. http://en.wikipedia.org/w/index.php?title=Terahertz_radiation&diff=248835142&oldid=248831987. Retrieved 2012-08-05.

[Pozer-3] Pozar, David M. (1993). Microwave Engineering Addison-Wesley Publishing Company. ISBN 0-201-50418-9.

[googlegloss-4] ttp://www.google.com/search?hl=en&defl=en&q=define:microwave&ei=e6CMSsWUI5OHmQee2si1DQ&sa=X&oi=glossary_definition&ct=title

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