The spectral density of blackbody radiation is a universal function of wavelength and temperature. This means that the spectral composition and radiation energy of a completely black body do not depend on the nature of the body.
Formulas (1.1) and (1.2) show that knowing the spectral and integral radiation density of an absolutely black body, they can be calculated for any non-black body if the absorption coefficient of the latter is known, which must be determined experimentally.
Research led to the following laws of black body radiation.
1. Stefan-Boltzmann law: The integral radiation density of an absolutely black body is proportional to the fourth power of its absolute temperature
Magnitude σ called Stefan's constant- Boltzmann:
σ = 5.6687·10 -8 J m - 2 s - 1 K – 4.
Energy emitted over time t absolutely black body with a radiating surface S at constant temperature T,
W=σT 4 St
If the body temperature changes over time, i.e. T = T(t), That
Stefan-Boltzmann's law indicates extremely fast growth radiation power with increasing temperature. For example, when the temperature increases from 800 to 2400 K (i.e. from 527 to 2127 ° C), the radiation of a completely black body increases by 81 times. If a completely black body is surrounded by a medium with a temperature T 0, then the eye will absorb the energy emitted by the environment itself.
In this case, the difference between the power of emitted and absorbed radiation can be approximately expressed by the formula
U=σ(T 4 – T 0 4)
The Stefan-Boltzmann law is not applicable to real bodies, as observations show a more complex relationship R on temperature, as well as on the shape of the body and the condition of its surface.
2. Wien's law of displacement. Wavelength λ 0, which accounts for the maximum spectral density black body radiation is inversely proportional to the absolute temperature of the body:
λ 0 = or λ 0 T = b.
Constant b, called Wien's law constant, equal to b = 0.0028978 m K ( λ expressed in meters).
Thus, with increasing temperature, not only the total radiation increases, but, in addition, the distribution of energy across the spectrum changes. For example, at low body temperatures, mainly infrared rays are studied, and as the temperature increases, the radiation becomes reddish, orange and, finally, white. In Fig. Figure 2.1 shows the empirical distribution curves of the radiation energy of a black body over wavelengths at different temperatures: it is clear from them that the maximum spectral density of radiation shifts towards shorter waves with increasing temperature.
3. Planck's law. The Stefan-Boltzmann law and the Wien displacement law do not solve the main problem of how large the spectral radiation density is at each wavelength in the spectrum of a black body at temperature T. To do this you need to install functional dependence And from λ And T.
Based on the idea of the continuous nature of the emission of electromagnetic waves and on the law of uniform distribution of energy over degrees of freedom (accepted in classical physics), two formulas were obtained for the spectral density and radiation of a black body:
1) Wine formula
Where a And b- constant values;
2) Rayleigh-Jeans formula
u λT = 8πkT λ – 4 ,
Where k- Boltzmann constant. Experimental testing has shown that for a given temperature Wien's formula is correct for short waves (when λT very little and gives sharp convergences of experience in the field long waves. The Rayleigh-Jeans formula turned out to be true for long waves and is completely inapplicable for short ones (Fig. 2.2).
Thus, classical physics was unable to explain the law of energy distribution in the radiation spectrum of an absolutely black body.
To determine the type of function u λТ completely new ideas about the mechanism of light emission were needed. In 1900, M. Planck hypothesized that absorption and emission of electromagnetic radiation energy by atoms and molecules is possible only in separate “portions”, which are called energy quanta. Magnitude of energy quantum ε proportional to the radiation frequency v(inversely proportional to wavelength λ ):
ε = hv = hc/λ
Proportionality factor h = 6.625·10 -34 J·s and is called Planck's constant. In the visible part of the spectrum for wavelength λ = 0.5 µm the value of the energy quantum is equal to:
ε = hc/λ= 3.79·10 -19 J·s = 2.4 eV
Based on this assumption, Planck obtained a formula for u λТ:
(2.1)
Where k– Boltzmann constant, With– speed of light in vacuum. l The curve corresponding to function (2.1) is also shown in Fig. 2.2.
From Planck's law (2.11) the Stefan-Boltzmann law and Wien's displacement law are obtained. Indeed, for the integral radiation density we obtain
Calculation using this formula gives a result that coincides with the empirical value of the Stefan-Boltzmann constant.
Wien's displacement law and its constant can be obtained from Planck's formula by finding the maximum of the function u λТ, why is the derivative of u λТ By λ , and is equal to zero. The calculation leads to the formula:
(2.2)
Calculation of constant b this formula also gives a result that coincides with the empirical value of the Wien constant.
Let us consider the most important applications of the laws of thermal radiation.
A. Thermal light sources. Most artificial light sources are thermal emitters (incandescent electric lamps, conventional arc lamps, etc.). However, these light sources are not very economical.
In § 1 it was said that the eye is sensitive only to a very narrow part of the spectrum (from 380 to 770 nm); all other waves do not produce a visual sensation. The maximum sensitivity of the eye corresponds to the wavelength λ = 0.555 µm. Based on this property of the eye, one should require from light sources such a distribution of energy in the spectrum at which the maximum spectral radiation density would fall on the wavelength λ = 0.555 µm or so. If we take an absolutely black body as such a source, then using Wien’s displacement law we can calculate its absolute temperature:
TO
Thus, the most advantageous thermal light source should have a temperature of 5200 K, which corresponds to the temperature of the solar surface. This coincidence is the result of the biological adaptation of human vision to the distribution of energy in the spectrum solar radiation. But even this light source efficiency(the ratio of the energy of visible radiation to the total energy of all radiation) will be small. Graphically in Fig. 2.3 this coefficient is expressed by the ratio of areas S 1 And S; square S 1 expresses the energy of radiation in the visible region of the spectrum, S- all radiation energy.
Calculations show that at a temperature of about 5000-6000 K, the light efficiency is only 14-15% (for an absolutely black body). At the temperature of existing artificial light sources (3000 K), this efficiency is only about 1-3%. Such a low “light output” of a thermal emitter is explained by the fact that during the chaotic movement of atoms and molecules, not only light (visible) waves are excited, but also other electromagnetic waves that do not have a light effect on the eye. Therefore, it is impossible to selectively force the body to emit only those waves to which the eye is sensitive: invisible waves are also emitted.
The most important of modern temperature light sources are incandescent electric lamps with tungsten filament. The melting point of tungsten is 3655 K. However, heating the filament to temperatures above 2500 K is dangerous, since tungsten at this temperature is very quickly atomized and the filament is destroyed. To reduce filament sputtering, it was proposed to fill the lamps with inert gases (argon, xenon, nitrogen) at a pressure of about 0.5 atm. This made it possible to raise the temperature of the filament to 3000-3200 K. At these temperatures, the maximum spectral density of radiation lies in the region of infrared waves (about 1.1 microns), therefore all modern incandescent lamps have an efficiency of slightly more than 1%.
B. Optical pyrometry. The laws of black body radiation outlined above make it possible to determine the temperature of this body if the wavelength is known λ 0 , corresponding to the maximum u λТ(according to Wien's law), or if the value of the integral radiation density is known (according to the Stefan-Boltzmann law). These methods of determining body temperature from its thermal radiation in the cabin optical pyrometry; they are especially useful when measuring very high temperatures. Since the mentioned laws apply only to an absolutely black body, optical pyrometry based on them gives good results only when measuring the temperatures of bodies close in their properties to absolutely black. In practice, these are factory furnaces, laboratory muffle furnaces, boiler furnaces, etc. Let's consider three ways to determine the temperature of heat emitters:
A. Method based on Wien's displacement law. If we know the wavelength at which the maximum spectral density of radiation falls, then the body temperature can be calculated using formula (2.2).
In particular, the temperature on the surface of the Sun, stars, etc. is determined in this way.
For non-black bodies, this method does not give the true body temperature; if there is one maximum in the emission spectrum and we calculate T according to formula (2.2), then the calculation gives us the temperature of an absolutely black body, which has almost the same energy distribution in the spectrum as the body under test. In this case, the color of the radiation of an absolutely black body will be the same as the color of the radiation under study. This body temperature is called its color temperature.
The color temperature of an incandescent lamp filament is 2700-3000 K, which is very close to its true temperature.
b. Radiation method of measuring temperatures based on measuring the integral radiation density of the body R and calculating its temperature using the Stefan-Boltzmann law. The corresponding devices are called radiation pyrometers.
Naturally, if the radiating body is not absolutely black, then the radiation pyrometer will not give the true temperature of the body, but will show the temperature of an absolutely black body at which the integral radiation density of the latter is equal to the integral radiation density of the test body. This body temperature is called radiation, or energy, temperature.
Among the disadvantages of a radiation pyrometer, we point out the impossibility of using it to determine the temperatures of small objects, as well as the influence of the medium located between the object and the pyrometer, which absorbs part of the radiation.
V. I brightness method for determining temperatures. Its operating principle is based on a visual comparison of the brightness of the hot filament of the pyrometer lamp with the brightness of the image of the heated test body. The device is a telescope with an electric lamp placed inside, powered by a battery. Equality, visually observed through a monochromatic filter, is determined by the disappearance of the image of the thread against the background of the image of the hot body. The filament is regulated by a rheostat, and the temperature is determined by the ammeter scale, graduated directly to the temperature.
Photo effect
The photoelectric effect was discovered in 1887 by the German physicist G. Hertz and experimentally studied by A. G. Stoletov in 1888–1890. Most full research the phenomenon of the photoelectric effect was carried out by F. Lenard in 1900. By this time, the electron had already been discovered (1897, J. Thomson), and it became clear that the photoelectric effect (or more precisely, the external photoelectric effect) consists in the ejection of electrons from a substance under the influence of an incident light on him.
The diagram of the experimental setup for studying the photoelectric effect is shown in Fig. 1.
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cleared. Some voltage was applied to the electrodes U, the polarity of which could be changed using double key. One of the electrodes (cathode K) was illuminated through a quartz window with monochromatic light of a certain wavelength λ. At a constant luminous flux, the dependence of the photocurrent strength was taken I from the applied voltage. In Fig. Figure 2 shows typical curves of such a dependence, obtained at two values of the intensity of the light flux incident on the cathode. The curves show that at sufficiently large positive voltages at anode A, the photocurrent reaches saturation, since all the electrons ejected from the cathode by light reach the anode. Careful measurements showed that the saturation current I n is directly proportional to the intensity of the incident light. When the anode voltage is negative, electric field electrons are inhibited between the cathode and anode. Only those electrons whose kinetic energy exceeds | eU|. If the voltage at the anode is less than - U h, the photocurrent stops. Measuring U h, we can determine the maximum kinetic energy of photoelectrons: ( mυ 2 / 2)max = eU h
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To the surprise of scientists, the value U h turned out to be independent of the intensity of the incident light flux. Careful measurements showed that the blocking potential increases linearly with increasing frequency ν of light (Fig. 3). Numerous experimenters have established the following basic principles of the photoelectric effect:
1. The maximum kinetic energy of photoelectrons increases linearly with increasing light frequency ν and does not depend on its intensity.
2. For each substance there is a so-called red limit of the photoelectric effect, i.e. the lowest frequency ν min at which the external photoelectric effect is still possible.
3. The number of photoelectrons emitted by light from the cathode in 1 s is directly proportional to the light intensity.
4. The photoelectric effect is practically inertialess, the photocurrent occurs instantly after the start of illumination of the cathode, provided that the light frequency ν > ν min.
All these laws of the photoelectric effect fundamentally contradicted the ideas of classical physics about the interaction of light with matter. According to wave concepts, when interacting with an electromagnetic light wave, an electron would gradually accumulate energy, and it would take a significant amount of time, depending on the intensity of the light, for the electron to accumulate enough energy to fly out of the cathode. As calculations show, this time should be calculated in minutes or hours. However, experience shows that photoelectrons appear immediately after the start of illumination of the cathode. In this model it was also impossible to understand the existence of the red boundary of the photoelectric effect. The wave theory of light could not explain the independence of photoelectron energy from the intensity of the light flux and the proportionality of the maximum kinetic energy frequency of light.
Thus, the electromagnetic theory of light was unable to explain these patterns.
The solution was found by A. Einstein in 1905. A theoretical explanation of the observed laws of the photoelectric effect was given by Einstein on the basis of M. Planck’s hypothesis that light is emitted and absorbed in certain portions, and the energy of each such portion is determined by the formula E = hν, where h– Planck’s constant. Einstein took the next step in the development of quantum concepts. He concluded that light has a discontinuous (discrete) structure. Electromagnetic wave consists of separate portions - quanta, later named photons. When interacting with matter, a photon completely transfers all its energy hν one electron. The electron can dissipate part of this energy during collisions with atoms of matter. In addition, part of the electron energy is spent on overcoming the potential barrier at the metal-vacuum interface. To do this, the electron must perform a work function A out, depending on the properties of the cathode material. The maximum kinetic energy that a photoelectron emitted from the cathode can have is determined by the law of conservation of energy:
This formula is usually called the Einstein equation for the photoelectric effect.
Using Einstein's equation, all the laws of the external photoelectric effect can be explained. Einstein's equation implies a linear dependence of the maximum kinetic energy on frequency and independence of light intensity, the existence of a red boundary, and the inertia-free photoelectric effect. Total number photoelectrons leaving the cathode surface in 1 s must be proportional to the number of photons incident on the surface during the same time. It follows from this that the saturation current must be directly proportional to the intensity of the light flux. This statement is called Stoletov's law.
As follows from Einstein’s equation, the tangent of the angle of inclination of the straight line expressing the dependence of the blocking potential U 3 from frequency ν (Fig. 3), equal to the ratio of Planck’s constant h to the electron charge e:
This allows us to experimentally determine the value of Planck's constant. Such measurements were carried out in 1914 by R. Millikan and gave good agreement with the value found by Planck. These measurements also made it possible to determine the work function A:
Where c– speed of light, λ cr – wavelength corresponding to the red boundary of the photoelectric effect.
Most metals have a work function A is several electron volts (1 eV = 1.602·10 -19 J). In quantum physics, the electron volt is often used as an energy unit. The value of Planck's constant, expressed in electron volts per second, is h=4.136·10 -15 eV·s.
Among the metals least amount of work alkaline elements have a yield. For example, sodium A= 1.9 eV, which corresponds to the red limit of the photoelectric effect λ cr ≈ 680 nm. Therefore, alkali metal compounds are used to create cathodes in photocells intended for recording visible light.
So, the laws of the photoelectric effect indicate that light, when emitted and absorbed, behaves like a stream of particles called photons or light quanta.
Thus, the doctrine of light, having completed a revolution lasting two centuries, again returned to the ideas of light particles - corpuscles.
But this was not a mechanical return to Newton's corpuscular theory. At the beginning of the 20th century, it became clear that light has a dual nature. As light spreads, it appears wave properties(interference, diffraction, polarization), and when interacting with matter - corpuscular (photoelectric effect). This dual nature of light is called wave-particle duality. Later, the dual nature of electrons and other elementary particles was discovered. Classical physics cannot give visual model combinations of wave and corpuscular properties of micro-objects. The movement of micro-objects is governed not by the laws of classical Newtonian mechanics, but by the laws of quantum mechanics. The theory of black body radiation developed by M. Planck and Einstein's quantum theory of the photoelectric effect lie at the basis of this modern science.
In addition to the external photoelectric effect we considered (usually simply called the photoelectric effect), there is also an internal photoelectric effect observed in dielectrics and semiconductors. It consists in the redistribution of electrons due to the action of light energy levels. In this case, electrons are released throughout the entire volume.
The action of so-called photoresistors is based on the internal photoelectric effect. The number of current carriers generated is proportional to the incident luminous flux. Therefore, photoresistors are used for photometric purposes. The first semiconductor to be used for these purposes was selenium.
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In area р-n junction or at the interface of a metal with a semiconductor, a gate photoelectric effect can be observed. It consists in the appearance of an electromotive force (photo-emf) under the influence of light. In Fig. 173 shows the course of the potential energy of electrons (solid curve) and holes (dashed curve) in the region р-n transition. Minority carriers for this region (electrons in R-areas and holes in n-regions) formed under the influence of light pass through the transition. As a result, in p-area accumulates excess positive charge, in n-regions - excess negative charge. This results in a voltage applied to the junction, which is the photoelectromotive force. In particular, this effect is used in the creation of solar panels.
The concept of an “absolute black body” was introduced by the German physicist Gustav Kirchhoff in the mid-19th century. The need to introduce such a concept was associated with the development of the theory of thermal radiation.
An absolutely black body is an idealized body that absorbs all electromagnetic radiation incident on it in all wavelength ranges and does not reflect anything.
Thus, the energy of any incident radiation is completely transferred to the black body and converted into its internal energy. Simultaneously with absorption, the blackbody also emits electromagnetic radiation and loses energy. Moreover, the power of this radiation and its spectral range are determined only by the temperature of the black body. It is the temperature of the black body that determines how much radiation it emits in the infrared, visible, ultraviolet and other ranges. Therefore, the blackbody, despite its name, at sufficiently high temperature will emit in the visible range and visually have color. Our Sun is an example of an object heated to a temperature of 5800°C, with properties close to the black body.
Absolutely black bodies do not exist in nature, so in physics a model is used for experiments. Most often it is a closed cavity with a small entrance hole. The radiation entering through this hole is completely absorbed by the walls after repeated reflections. No part of the radiation entering the hole is reflected back from it - this corresponds to the definition of a blackbody (total absorption and no reflection). In this case, the cavity has its own radiation corresponding to its temperature. Since the own radiation of the inner walls of the cavity also performs a huge number of new absorptions and emissions, we can say that the radiation inside the cavity is in thermodynamic equilibrium with the walls. The characteristics of this equilibrium radiation are determined only by the cavity temperature (CBT): the total (at all wavelengths) radiation energy according to the Stefan-Boltzmann law, and the distribution of radiation energy over wavelengths is described by Planck’s formula.
There are no absolutely black bodies in nature. There are examples of bodies that are only closest in their characteristics to completely black. For example, soot can absorb up to 99% of the light falling on it. Obviously, the special surface roughness of the material makes it possible to reduce reflections to a minimum. It is thanks to multiple reflections followed by absorption that we see objects such as black velvet black.
I once met an object very close to the blackbody at the production of Gillette razor blades in St. Petersburg, where I had the opportunity to work even before taking up thermal imaging. Classic double-sided razor blades in technological process They are collected into “knives” with up to 3000 blades in a pack. Side surface, consisting of many sharpened blades pressed tightly together, is velvety black in color, although each individual steel blade has a shiny, sharpened steel edge. A block of blades left on a windowsill in sunny weather, could heat up to 80°C. At the same time, the individual blades practically did not heat up, since they reflected most of the radiation. Threads on bolts and studs have a similar surface shape; their emissivity is higher than on smooth surface. This property is often used in thermal imaging testing of electrical equipment.
Scientists are working to create materials with properties close to those of absolute black bodies. For example, significant results have been achieved in the optical range. In 2004, an alloy of nickel and phosphorus was developed in England, which was a microporous coating and had a reflectance of 0.16–0.18%. This material was listed in the Guinness Book of Records as the blackest material in the world. In 2008, American scientists set a new record - the thin film they grew, consisting of vertical carbon tubes, almost completely absorbs radiation, reflecting it by 0.045%. The diameter of such a tube is from ten nanometers and a length from ten to several hundred micrometers. The created material has a loose, velvety structure and a rough surface.
Each infrared device is calibrated according to the black body model(s). Temperature measurement accuracy can never be better than calibration accuracy. Therefore, the quality of calibration is very important. During calibration (or verification) using reference emitters, temperatures from the entire measurement range of the thermal imager or pyrometer are reproduced. In practice, reference thermal emitters are used in the form of a black body model of the following types:
Cavity models of the blackbody. They have a cavity with a small inlet hole. The temperature in the cavity is set, maintained and measured with high accuracy. Such emitters can produce high temperatures.
Extended or planar models of the black body. They have a platform painted with a composition containing high coefficient radiation (low reflectivity). The site temperature is set, maintained and measured with high accuracy. Low negative temperatures can be reproduced in such emitters.
When searching for information about imported black body models, use the term “black body”. It is also important to understand the difference between testing, calibrating and verifying a thermal imager. These procedures are described in detail on the website in the section on thermal imagers.
Materials used: Wikipedia; TSB; Infrared Training Center (ITC); Fluke Calibration
An absolutely black body is called such because it absorbs all radiation falling on it (or rather, into it) both in the visible spectrum and beyond. But if the body does not heat up, the energy is re-radiated back. This radiation emitted by a blackbody is of particular interest. The first attempts to study its properties were made even before the emergence of the model itself.
In the early 19th century, John Leslie experimented with various substances. As it turns out, black soot not only absorbs all visible light falling on it. It emitted much more strongly in the infrared than other lighter substances. It was thermal radiation, which differs from all other types in several properties. The radiation of an absolutely black body is equilibrium, homogeneous, occurs without energy transfer and depends only on
When the temperature of an object is high enough, thermal radiation becomes visible, and then any body, including a completely black one, acquires color.
Such a unique object, which radiates exclusively a certain thing, could not help but attract attention. Since we are talking about thermal radiation, the first formulas and theories regarding what the spectrum should look like were proposed within the framework of thermodynamics. Classical thermodynamics was able to determine where the maximum radiation should be at a given temperature, in which direction and how much it will shift when heated and cooled. However, it was not possible to predict what the energy distribution is in the black body spectrum at all wavelengths and, in particular, in the ultraviolet range.
According to the concepts of classical thermodynamics, energy can be emitted in any portions, including arbitrarily small ones. But in order for a completely black body to radiate at short wavelengths, the energy of some of its particles must be very large, and in the ultrashort wavelength region it would go to infinity. In reality, this is impossible, infinity appeared in the equations and was named. Only the fact that energy can be emitted in discrete portions - quanta - helped to resolve the difficulty. Today's thermodynamic equations are special cases of the equations
Initially, a completely black body was imagined as a cavity with a narrow opening. Radiation from the outside enters such a cavity and is absorbed by the walls. In this case, the spectrum of radiation from the entrance to a cave, the opening of a well, a window in dark room on a sunny day, etc. But most of all, the spectra of the Universe and stars, including the Sun, coincide with it.
It is safe to say that the more particles with different energies in an object, the more its radiation will resemble blackbody radiation. The energy distribution curve in the spectrum of an absolutely black body reflects the statistical patterns in the system of these particles, with the only correction that the energy transferred during interactions is discrete.
FEDERAL AGENCY FOR EDUCATION
state educational institution higher vocational education
"TYUMEN STATE OIL AND GAS UNIVERSITY"
Abstract on the discipline
"Technical Optics"
Topic: “Absolutely black body”
Completed by: student gr. OBDzs-07
Kobasnyan Stepan Sergeevich Checked by: teacher of the discipline
Sidorova Anastasia Eduardovna
Tyumen 2009
Absolutely black body- a physical abstraction used in thermodynamics, a body that absorbs all electromagnetic radiation incident on it in all ranges and does not reflect anything. Despite the name, a completely black body can itself emit electromagnetic radiation of any frequency and visually have color. The radiation spectrum of an absolutely black body is determined only by its temperature.
Black body model
Laws of black body radiation
Classic approach
The study of the laws of black body radiation was one of the prerequisites for the emergence of quantum mechanics.
Wien's first law of radiation
In 1893, Wilhelm Wien, based on the concepts of classical thermodynamics, derived the following formula:
From the first Wien formula one can derive the Wien displacement law (maximum law) and the Stefan-Boltzmann law, but one cannot find the values of the constants included in these laws.
Historically, it was Wien’s first law that was called the displacement law, but currently the term “Wien’s displacement law” refers to the maximum law.
Wien's second law of radiation
Experience shows that Wien's second formula is valid only in the limit high frequencies(short wavelengths). It is a special case of Wien's first law.
Later, Max Planck showed that Wien's second law follows from Planck's law for high quantum energies, and also found the constants C 1 and C 2. Taking this into account, Wien's second law can be written as:
Rayleigh-Jeans law
This formula assumes a quadratic increase in the spectral density of radiation depending on its frequency. In practice, such a law would mean the impossibility of thermodynamic equilibrium between matter and radiation, since according to it all thermal energy would have to transform into radiation energy in the short-wave region of the spectrum. This hypothetical phenomenon was called an ultraviolet catastrophe.
Nevertheless, the Rayleigh-Jeans radiation law is valid for the long-wave region of the spectrum and adequately describes the nature of the radiation. The fact of such correspondence can be explained only by using a quantum mechanical approach, according to which radiation occurs discretely. Based on quantum laws, we can obtain Planck's formula, which will coincide with the Rayleigh-Jeans formula for .
This fact is an excellent illustration of the principle of correspondence, according to which a new physical theory must explain everything that the old one was able to explain.
Planck's law
Dependence of black body radiation power on wavelength
The radiation intensity of an absolutely black body, depending on temperature and frequency, is determined by Planck's law :
Where I (ν) dν - radiation power per unit area of the radiating surface in the frequency range from ν to ν + d ν.
Equivalently,
,
Where u (λ) dλ - radiation power per unit area of the emitting surface in the wavelength range from λ to λ + d λ.
Stefan-Boltzmann law
The total energy of thermal radiation is determined Stefan-Boltzmann law :
Where j is the power per unit area of the radiating surface, and
W/(m²·K 4) - Stefan-Boltzmann constant .
Thus, an absolutely black body at T= 100 K emits 5.67 watts square meter its surface. At a temperature of 1000 K, the radiation power increases to 56.7 kilowatts per square meter.
Wien's displacement law
The wavelength at which the radiation energy of a completely black body is maximum is determined by Wien's displacement law :
Where T is the temperature in Kelvin, and λ max is the wavelength with maximum intensity in meters.
Visible color absolutely black bodies with different temperatures presented in the diagram.
Blackbody radiation
Electromagnetic radiation that is in thermodynamic equilibrium with a blackbody at a given temperature (for example, radiation inside a cavity in a blackbody) is called blackbody (or thermal equilibrium) radiation. Equilibrium thermal radiation is homogeneous, isotropic and non-polarized, there is no energy transfer in it, all its characteristics depend only on the temperature of the absolutely blackbody emitter (and, since blackbody radiation is in thermal equilibrium with this body, this temperature can be attributed to radiation). The volumetric energy density of blackbody radiation is equal to , its pressure is equal to 
. The so-called cosmic microwave background, or cosmic microwave background, is very close in its properties to black-body radiation, a radiation that fills the Universe with a temperature of about 3 K.
Blackbody chromaticity
Note: Colors are given in comparison with diffused daylight(D 65). The actual perceived color may be distorted by the eye's adaptation to lighting conditions.
An absolutely black body is a physical abstraction used in thermodynamics, a body that absorbs all electromagnetic radiation incident on it in all ranges and does not reflect anything. Despite the name, a completely black body can itself emit electromagnetic radiation of any frequency and visually have color. The radiation spectrum of an absolutely black body is determined only by its temperature.
The blackest real substances, for example, soot, absorb up to 99% of incident radiation (i.e., have an albedo of 0.01) in the visible wavelength range, however infrared radiation absorbed by them much worse. Among the bodies solar system The Sun possesses the properties of an absolutely black body to the greatest extent. The term was introduced by Gustav Kirchhoff in 1862.
******draw a body model.******
Black body model
Absolutely black bodies do not exist in nature, so in physics a model is used for experiments. It is a closed cavity with a small hole. Light entering through this hole will, after repeated reflections, be completely absorbed, and the outside of the hole will appear completely black. But when this cavity is heated, it will develop its own visible radiation.
Wien's first law of radiation
In 1893, Wilhelm Wien.
Wien's first formula is valid for all frequencies. Any more specific formula (for example, Planck's law) must satisfy Wien's first formula.
Wien's second law of radiation
In 1896, Wien derived the second law based on additional assumptions:
Wien's second formula is valid only in the limit of high frequencies (short wavelengths). It is a special case of Wien's first law.
Rayleigh-Jeans law
An attempt to describe the radiation of a completely black body based on classical principles thermodynamics and electrodynamics leads to the Rayleigh-Jeans law:
In practice, such a law would mean the impossibility of thermodynamic equilibrium between matter and radiation, since according to it all thermal energy would have to be converted into radiation energy in the short-wave region of the spectrum. This hypothetical phenomenon was called an ultraviolet catastrophe.
Planck's law determines the intensity of radiation of a completely black body depending on temperature and frequency
The Stefan-Boltzmann law determines the total energy of thermal radiation determined by the law
The wavelength at which the radiation energy of an absolutely black body is maximum is determined by Wien's displacement law:
So, if we assume as a first approximation that human skin is close in properties to an absolutely black body, then the maximum of the radiation spectrum at a temperature of 36°C (309 K) lies at a wavelength of 9400 nm (in the infrared region of the spectrum).
In all ranges and not reflecting anything. Despite the name, a completely black body itself can emit electromagnetic radiation of any frequency and visually have a . The radiation spectrum of an absolutely black body is determined only by its temperature.
The importance of an absolutely black body in the question of the spectrum of thermal radiation of any (gray and colored) bodies in general, in addition to the fact that it represents the simplest non-trivial case, also lies in the fact that the question of the spectrum of equilibrium thermal radiation of bodies of any color and reflection coefficient is reduced by the methods of classical thermodynamics to the question of the radiation of an absolutely black body (and historically this was already done by the end of the 19th century, when the problem of radiation of an absolutely black body came to the fore).
The blackest real substances, for example, soot, absorb up to 99% of incident radiation (that is, they have an albedo of 0.01) in the visible wavelength range, but they absorb infrared radiation much worse. Among the bodies of the Solar System, the Sun has the properties of an absolutely black body to the greatest extent.
Practical model
Black body model
Absolutely black bodies do not exist in nature (except for black holes), so in physics a model is used for experiments. It is a closed cavity with a small hole. Light entering through this hole will, after repeated reflections, be completely absorbed, and the outside of the hole will appear completely black. But when this cavity is heated, it will develop its own visible radiation. Since the radiation emitted by the inner walls of the cavity, before it leaves (after all, the hole is very small), in the overwhelming majority of cases will undergo a huge amount of new absorption and radiation, we can say with confidence that the radiation inside the cavity is in thermodynamic equilibrium with the walls. (In fact, the hole is not important for this model at all, it is only needed to emphasize the fundamental observability of the radiation located inside; the hole can, for example, be completely closed, and quickly opened only when equilibrium has already been established and the measurement is being carried out).
Laws of black body radiation
Classic approach
Initially, purely classical methods, which gave a number of important and correct results, but did not completely solve the problem, ultimately leading not only to a sharp discrepancy with experiment, but also to an internal contradiction - the so-called ultraviolet disaster.
The study of the laws of black body radiation was one of the prerequisites for the emergence of quantum mechanics.
Wien's first law of radiation
k- Boltzmann constant, c- speed of light in vacuum.Rayleigh-Jeans law
An attempt to describe the radiation of a completely black body based on the classical principles of thermodynamics and electrodynamics leads to the Rayleigh-Jeans law:
This formula assumes a quadratic increase in the spectral density of radiation depending on its frequency. In practice, such a law would mean the impossibility of thermodynamic equilibrium between matter and radiation, since according to it all thermal energy would have to be converted into radiation energy in the short-wave region of the spectrum. Such a hypothetical phenomenon was called an ultraviolet catastrophe.
Nevertheless, the Rayleigh-Jeans radiation law is valid for the long-wave region of the spectrum and adequately describes the nature of the radiation. The fact of such correspondence can be explained only by using a quantum mechanical approach, according to which radiation occurs discretely. Based on quantum laws, one can obtain Planck's formula, which will coincide with the Rayleigh-Jeans formula at .
This fact is an excellent illustration of the principle of correspondence, according to which a new physical theory must explain everything that the old one was able to explain.
Planck's law
The radiation intensity of an absolutely black body, depending on temperature and frequency, is determined by Planck's law:
where is the radiation power per unit area of the radiating surface in a unit frequency interval in the perpendicular direction per unit solid angle (dimension in SI: J s −1 m −2 Hz −1 sr −1).
Equivalently,
where is the radiation power per unit area of the emitting surface in a unit wavelength interval in the perpendicular direction per unit solid angle (SI dimension: J s −1 m −2 m −1 sr −1).
The total (i.e. emitted in all directions) spectral radiation power per unit surface of an absolutely black body is described by the same formulas accurate to the coefficient π: ε(ν, T) = π I(ν, T) , ε(λ, T) = π u(λ, T) .
Stefan-Boltzmann law
The total energy of thermal radiation is determined by the Stefan-Boltzmann law, which states:
The radiation power of an absolutely black body (integrated power over the entire spectrum) per unit surface area is directly proportional to the fourth power of the body temperature:
Where j is the power per unit area of the radiating surface, and
W/(m²·K 4) - Stefan-Boltzmann constant.Thus, an absolutely black body at T= 100 K emits 5.67 watts per square meter of its surface. At a temperature of 1000 K, the radiation power increases to 56.7 kilowatts per square meter.
For non-black bodies we can approximately write:
where is the degree of blackness (for all substances, for an absolutely black body).
The Stefan-Boltzmann constant can be theoretically calculated only from quantum considerations, using Planck's formula. In the same time general form the formula can be obtained from classical considerations (which does not eliminate the problem of the ultraviolet catastrophe).
Wien's displacement law
The wavelength at which the radiation energy of a completely black body is maximum is determined by Wien's displacement law:
Where T is the temperature in Kelvin, and is the wavelength with maximum intensity in meters.
So, if we assume as a first approximation that human skin is close in properties to an absolutely black body, then the maximum of the radiation spectrum at a temperature of 36 °C (309 K) lies at a wavelength of 9400 nm (in the infrared region of the spectrum).
The apparent color of completely black bodies at different temperatures is shown in the diagram.
Blackbody radiation
Electromagnetic radiation that is in thermodynamic equilibrium with a blackbody at a given temperature (for example, radiation inside a cavity in a blackbody) is called blackbody (or thermal equilibrium) radiation. Equilibrium thermal radiation is homogeneous, isotropic and non-polarized, there is no energy transfer in it, all its characteristics depend only on the temperature of the absolutely blackbody emitter (and, since blackbody radiation is in thermal equilibrium with this body, this temperature can be attributed to radiation). The volumetric energy density of blackbody radiation is equal to its pressure is equal to. Very close in its properties to blackbody radiation is the so-called relict radiation, or cosmic microwave background - radiation filling the Universe with a temperature of about 3 K.
Blackbody chromaticity
Colors are given in comparison with diffuse daylight (