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]


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

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.

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.


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

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.

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.

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:

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

These are the main infrared atmospheric windows:

Commonly used sub-divisions are

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

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

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