# Planck's equation

Planck's equation (colored curves) accurately describes black body radiation. Credit: Darth Kule.

"Planck's [equation] describes the amount of [spectral radiance at] a certain wavelength radiated by a black body in thermal equilibrium"[1].

## Evaluation

evaluation activity

## Electromagnetics

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

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

where B is the spectral radiance, T is the absolute temperature of the black body, kB is the Boltzmann constant, h is the Planck constant, and c is the speed of light."[1]

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

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.

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

Or,

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

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.

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:[2]

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.

[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),"[3] "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."[4]

"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.[5] This broad definition includes both UHF and EHF (millimeter waves), and various sources use different boundaries.[6] 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)."[7]

"Radio waves have frequencies from 300 [Gigahertz] GHz to as low as 3 [Kilohertz] kHz, and corresponding wavelengths from 1 millimeter to 100 kilometers."[8]

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

Main source: Hypotheses
1. Planck's equation can be derived without the use of quanta.