Radiation from heated metal in the visible range
Absolutely black body - physical idealization used in thermodynamics, a body that absorbs everything falling on it electromagnetic radiation in all ranges and does not reflect anything. Despite the name, a completely black body itself can emit electromagnetic radiation of any frequency and visually have color.Emission spectrum 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 albedo, equal to 0.01) in the visible wavelength range, but infrared radiation is absorbed much worse by them. Among the bodies solar system has the properties of an absolutely black body to the greatest extent Sun.
The term was introduced by Gustav Kirchhoff in 1862. Practical model
Black body model
Absolutely black bodies do not exist in nature, so in physics they are used for experiments. model. It is a closed cavity with a small hole. Light entering through this hole will be completely absorbed after repeated reflections, and the hole will appear completely black from the outside. 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 exits (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 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 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 Classical approach
Initially, purely classical methods were applied to solve the problem, which gave a number of important and correct results, but they did not allow the problem to be completely solved, 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 appearance quantum mechanics.
Wien's first law of radiation
In 1893 Wilhelm Wien, using, in addition to classical thermodynamics, the electromagnetic theory of light, he derived the following formula:
uν - radiation energy density
ν - radiation frequency
T- temperature of the radiating body
f- a function that depends only on frequency and temperature. The form of this function cannot be established based only on thermodynamic considerations.
Wien's first formula is valid for all frequencies. Any more specific formula (for example, Planck's law) must satisfy Wien's first formula.
From the first formula of Wien we can deduce Wien's displacement law(maximum law) and Stefan-Boltzmann law, but it is impossible to 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 is called the law of maximum.
Absorbs 99.965% of radiation incident on it in the ranges of visible light, microwaves and radio waves.
The term “absolute black body” was introduced by Gustav Kirchhoff in 1862.
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✪ Physics for dummies. Lecture 59. Absolutely black body
✪ Absolutely black body
✪ Black body radiation
✪ Elementary particles | absolutely black body
✪ Absolutely black body
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Practical model
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 classical principles thermodynamics and electrodynamics leads to the Rayleigh-Jeans law:
u (ω , T) = k T ω 2 π 2 c 3 (\displaystyle u(\omega ,T)=kT(\frac (\omega ^(2))(\pi ^(2)c^(3) )))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 the 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 the Planck formula, which will coincide with the Rayleigh-Jeans formula for ℏ ω / k T ≪ 1 (\displaystyle \hbar \omega /kT\ll 1).
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 :
R (ν , T) = 2 π h ν 3 c 2 1 e h ν / k T − 1 , (\displaystyle R(\nu ,T)=(\frac (2\pi h\nu ^(3))( c^(2)))(\frac (1)(e^(h\nu /kT)-1)),)Where R (ν , T) (\displaystyle R(\nu ,T))- radiation power per unit area of the radiating surface in a unit frequency interval (dimension in SI: J s −1 m −2 Hz −1).
Which is equivalent,
R (λ , T) = 2 π h c 2 λ 5 1 e h c / λ k T − 1 , (\displaystyle R(\lambda ,T)=(2\pi h(c^(2)) \over \lambda ^ (5))(1 \over e^(hc/\lambda kT)-1),)Where R (λ , T) (\displaystyle R(\lambda ,T))- radiation power per unit area of the emitting surface in a unit wavelength interval (dimension in SI: J s −1 m −2 m −1).
Stefan-Boltzmann law
The total energy of thermal radiation is determined by the Stefan-Boltzmann law, which states:
j = σ T 4 , (\displaystyle j=\sigma T^(4),)Where j (\displaystyle j) is the power per unit area of the radiating surface, and
σ = 2 π 5 k 4 15 c 2 h 3 = π 2 k 4 60 ℏ 3 c 2 ≃ 5.670 400 (40) ⋅ 10 − 8 (\displaystyle \sigma =(\frac (2\pi ^(5)k ^(4))(15c^(2)h^(3)))=(\frac (\pi ^(2)k^(4))(60\hbar ^(3)c^(2))) \simeq 5(,)670400(40)\cdot 10^(-8)) W/(m²·K 4) - Stefan-Boltzmann constant.Thus, an absolutely black body at T (\displaystyle 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.
For non-black bodies we can approximately write:
j = ϵ σ T 4 , (\displaystyle j=\epsilon \sigma T^(4),\ )Where ϵ (\displaystyle \epsilon )- degree of blackness. For all substances ϵ < 1 {\displaystyle \epsilon <1} , for a completely black body ϵ = 1 (\displaystyle \epsilon =1), for other objects, by virtue of Kirchhoff’s law, the degree of emissivity is equal to the absorption coefficient: ϵ = α = 1 − ρ − τ (\displaystyle \epsilon =\alpha =1-\rho -\tau ), Where α (\displaystyle \alpha )- absorption coefficient, ρ (\displaystyle \rho )- reflection coefficient, and τ (\displaystyle \tau)- transmittance. That is why, to reduce thermal radiation, the surface is painted white or a shiny coating is applied, and to increase it, it is darkened.
Stefan-Boltzmann constant σ (\displaystyle \sigma ) can be theoretically calculated only from quantum considerations, using Planck's formula. At the same time, the general form of 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:
λ max = 0.002 8999 T (\displaystyle \lambda _(\max )=(\frac (0(,)0028999)(T)))Where T (\displaystyle T)- temperature in Kelvin, and λ max (\displaystyle \lambda _(\max )) - wavelength s 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).
| P = a 3 T 4 , (\displaystyle P=(\frac (a)(3))T^(4),) | (Thermal equation of state) |
| U = a V T 4 , (\displaystyle U=aVT^(4),) | (Caloric state equation for internal energy) |
| U = a V (3 S 4 a V) 4 3 , (\displaystyle U=aV\left((\frac (3S)(4aV))\right)^(\mathsf (\frac (4)(3)) ),) | (Canonical equation of state for internal energy) |
| H = (3 P a) 1 4 S , (\displaystyle H=\left((\frac (3P)(a))\right)^(\mathsf (\frac (1)(4)))S,) | enthalpy) |
| F = − 1 3 a V T 4 , (\displaystyle F=-(\frac (1)(3))aVT^(4),) | (Canonical equation of state for the Helmholtz potential) |
| Ω = − 1 3 α V T 4 , (\displaystyle \Omega =-(\frac (1)(3))\alpha VT^(4),) | (Canonical equation of state for the Landau potential) |
| S = 4 a 3 V T 3 , (\displaystyle S=(\frac (4a)(3))VT^(3),) | (Entropy) |
| C V = 4 a V T 3 , (\displaystyle C_(V)=4aVT^(3),) | (Heat capacity at constant volume) |
| γ = ∞ , (\displaystyle \gamma =\infty ,) | ( |
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 a 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. In the technological process, classic double-sided razor blades are assembled into “knives” of up to 3000 blades in a pack. The side surface, consisting of many sharpened blades pressed tightly together, is velvety black, although each individual steel blade has a shiny, sharpened steel edge. A block of blades left on a windowsill in sunny weather could reach temperatures of 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 a 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 with a high emissivity (low reflectance). 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
Light polarization is the process of ordering the oscillations of the electric field strength vector of a light wave when light passes through certain substances (during refraction) or when the light flux is reflected. There are several ways to produce polarized light.
1) Polarization using Polaroids. Polaroids are celluloid films coated with a thin layer of quinine sulfate crystals. The use of polaroids is currently the most common method of polarizing light.
2) Polarization by reflection. If a natural beam of light falls on a black polished surface, the reflected beam is partially polarized. Mirror or fairly well polished ordinary window glass, blackened on one side with asphalt varnish, can be used as a polarizer and analyzer.
The more correctly the angle of incidence is maintained, the greater the degree of polarization. For glass, the angle of incidence is 57°.
3) Polarization through refraction. A light beam is polarized not only during reflection, but also during refraction. In this case, a stack of 10-15 thin glass plates folded together, located at an angle of 57° to the light rays incident on them, is used as a polarizer and analyzer.
Wholesale And ical act And thoroughness, the ability of a medium to cause rotation of the plane of polarization of optical radiation (light) passing through it.
the angle j of rotation of the plane of polarization depends linearly on the thickness l layer of the active substance (or its solution) and concentration With of this substance - j = [a] lc(coefficient [a] is called specific O. a.); 2) rotation in a given environment occurs either clockwise (j > 0) or counterclockwise (j< 0), если смотреть навстречу ходу лучей света
43. Russ e veneration of St. e ta, change in the characteristics of the flow of optical radiation (light) during its interaction with matter. These characteristics can be the spatial distribution of intensity, frequency spectrum, and polarization of light. Often R. s. only a change in the direction of light propagation caused by the spatial heterogeneity of the medium is called, perceived as an improper glow of the medium.
SCATTERINGINDEX, the reciprocal of the distance at which the flux of radiation forming a parallel light beam is attenuated as a result scattering in the environment by 10 times or e times.
Rel e I'm zach O n, states that the intensity I light scattered by the medium is inversely proportional to the 4th power of the wavelength l of the incident light ( I~ l -4) in the case when the medium consists of dielectric particles whose dimensions are much smaller than l . I rass ~1/ 4
44. Absorbing e tion of St. e ta, a decrease in the intensity of optical radiation (light) passing through a material medium due to the processes of its interaction with the medium. Light energy at P. s. goes into various shapes internal energy of the medium or optical radiation composition; it can be completely or partially re-emitted by the medium at frequencies different from the frequency of the absorbed radiation.
Bouguer's law. The physical meaning is that the process of loss of beam photons in the medium does not depend on their density in the light beam, i.e. on light intensity and half-length I.
I=I 0 exp(λl ); l – wavelength, λ - absorption rate, I 0– intensity of the absorbing beam.
Bug e ra - L A Mberta - B e rak O n, determines the gradual attenuation of a parallel monochromatic (one-color) beam of light as it propagates in an absorbing substance. If the power of the beam entering a layer of substance thick l, equal to I o, then, according to B.-L.-B. h., beam power at exit from the layer
I(l)= I o e- c cl,
where c is the specific indicator of light absorption, calculated per unit of concentration With absorption determining substance;
Absorption rate (k l), the reciprocal of the distance at which the monochromatic radiation flux frequency n, forming a parallel beam, is attenuated due to absorption in matter in e times or 10 times. Measured in cm -1 or m -1 . In spectroscopy and some other branches of applied optics, the term “PP.” traditionally used to denote the absorption coefficient.
Molar absorption rate
Transmittance is the ratio of the radiation flux passing through a medium to the flux incident on its surface. t = F/F 0
Optical density is a measure of the opacity of a layer of substance for light rays D = log(-F 0 /F)
Transparency of the environment- the ratio of the magnitude of the radiation flux that passed without changing direction through a layer of medium of unit thickness to the magnitude of the incident flux (that is, without taking into account the effects of scattering and the influence of effects on interfaces).
45. Thermal radiation- electromagnetic radiation with a continuous spectrum, emitted by heated bodies due to their thermal energy.
Absolutely black body- a physical idealization 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.
Gray body- this is a body whose absorption coefficient does not depend on frequency, but depends only on temperature
For gray body
GRAY BODY- body, absorption coefficient which is less than 1 and does not depend on the radiation wavelength and abs. temperatures T. Coef. absorption (also called blackness coefficient S. t.) of all real bodies depends on (selective absorption) and T, therefore they can be considered gray only in the intervals and T, where coefficient approx. permanent. In the visible region of the spectrum, solar radiation properties have coal( = 0.80 at 400-900 K), soot ( = 0.94-0.96 at 370-470 K); platinum and bismuth black absorb and emit as light in the widest range - from visible light to 25-30 microns (= 0.93-0.99).
Basic laws of radiation:
Stefan-Boltzmann law- the law of black body radiation. Determines the dependence of the radiation power of an absolutely black body on its temperature. Statement of the law:
where is the degree of blackness (for all substances, for an absolutely black body). Using Planck's law for radiation, the constant σ can be defined as
where is Planck's constant, k- Boltzmann constant, c- speed of light.
Numerical value J s −1 m −2 K −4.
Kirchhoff's radiation law- a physical law established by the German physicist Kirchhoff in 1859.
In its modern formulation, the law reads as follows:
The ratio of the emissivity of any body to its absorption capacity is the same for all bodies at a given temperature for a given frequency and does not depend on their shape and chemical nature.
It is known that when electromagnetic radiation falls on a certain body, part of it is reflected, part is absorbed, and part can be transmitted. The fraction of radiation absorbed at a given frequency is called absorption capacity body. On the other hand, every heated body emits energy according to some law called emissivity of the body.
The values of and can vary greatly when moving from one body to another, however, according to Kirchhoff’s law of radiation, the ratio of emissive and absorption abilities does not depend on the nature of the body and is a universal function of frequency (wavelength) and temperature:
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.
Characteristics of thermal radiation
Bodies heated to temperatures as high as 424e43ie glow. The glow of bodies caused by heating is called thermal (temperature) radiation. Thermal radiation, being the most common in nature, occurs due to the energy of thermal movement of atoms and molecules of a substance (i.e., due to its internal energy) and is characteristic of all bodies at temperatures above 0 K. Thermal radiation is characterized by continuous spectrum, the position of the maximum of which depends on temperature. At high temperatures short (visible and ultraviolet) electromagnetic waves are emitted; at low ones, predominantly long (infrared) waves are emitted.
Thermal radiation is practically the only type of radiation that can be equilibrium. Let us assume that a heated (radiating) body is placed in a cavity bounded by an ideally reflective shell. Over time, as a result of the continuous exchange of energy between the body and radiation, equilibrium will occur, i.e., the body will absorb as much energy per unit time as it emits. Let us assume that the balance between the body and radiation is disturbed for some reason and the body emits more energy than it absorbs. If per unit time a body emits more than it absorbs (or vice versa), then the body temperature will begin to decrease (or increase). As a result, the amount of energy emitted by the body will be weakened (or age) until, finally, equilibrium is established. All other types of radiation are nonequilibrium.
A quantitative characteristic of thermal radiation is spectral density of energy luminosity (emissivity) of a body≈ radiation power per unit surface area of a body in a frequency range of unit width:
where d ≈ energy of electromagnetic radiation emitted per unit time (radiation power) per unit surface area of the body in the frequency range from n before n+d n.
Unit of spectral density of energetic luminosity ( Rn,T) ≈joule per meter squared(J/m2).
The written formula can be represented as a function of wavelength:
Because c=ln, That
where the minus sign indicates that from age 424e43ie ;the absence of one of the quantities ( n or l) another quantity decreases. Therefore, in what follows we will omit the minus sign. Thus,
Using formula (197.1) you can go from R n,T ═ To R l,T and vice versa.
Knowing spectral density energetic luminosity, can be calculated integral energy luminosity (integral emissivity)(it is simply called the energetic luminosity of the body), summed over all frequencies:
The ability of bodies to absorb radiation incident on them is characterized by spectral absorption capacity
showing what fraction of the energy brought per unit time per unit surface area of a body by incident objects electromagnetic waves frequencies from n before n+d n, is absorbed by the body. Spectral absorption capacity is a dimensionless quantity. Quantities Rn,T═and A n,T depend on the nature of the body, its thermodynamic temperature and at the same time differ for radiation with different frequencies. Therefore, these values are referred to as certain T And n(or rather, to enough 424e43ie; exactly a narrow frequency range from n before n+d n).
A body capable of completely absorbing at any temperature all radiation of any frequency incident on it is called black. Consequently, the spectral absorption capacity of a black body for all frequencies and temperatures is identically equal to unity ( ). There are no absolutely black bodies in nature, but bodies such as soot, platinum black, black velvet and some others, in a certain frequency range, are close to them in their properties.
The ideal model black body is a closed cavity with a small hole ABOUT, the inner surface of which is blackened (Fig. 286). A ray of light entering such a cavity experiences multiple reflections from the walls, as a result of which the intensity of the emitted radiation is practically equal to zero. Experience shows that when the hole size is less than 0.1 of the cavity diameter, incident radiation of all frequencies is completely absorbed. Consequently open windows the houses from the street appear black, although inside the rooms there are enough 424e43ie; exactly light due to the reflection of light from the walls.
Along with the concept of a black body, the concept is used gray body≈ a body whose absorption capacity is less than unity, but is the same for all frequencies and depends only on the temperature, material and state of the surface of the body. Thus, for a gray body = A T= const The study of thermal radiation played an important role in the creation of the quantum theory of light, so it is necessary to consider the laws that it obeys. Energy luminosity of the bodyR T, is numerically equal to energy W, emitted by the body over the entire wavelength range (0<<)
per unit body surface, per unit time, at body temperature T, i.e. Body emissivityr ,T numerically equal to the energy of the body dW, emitted by a body from a unit of body surface, per unit time at body temperature T, in the wavelength range from to +d, those. This quantity is also called the spectral density of the body's energy luminosity. Energetic luminosity is related to emissivity by the formula Absorbency body ,T- a number showing what fraction of the radiation energy incident on the surface of a body is absorbed by it in the wavelength range from to +d, those. A body for which ,T =1 over the entire wavelength range is called an absolute black body (BLB). A body for which ,T =const<1
over the entire wavelength range is called gray. 46. Special physical instruments called actinometers can measure the amount of solar energy received on the earth's surface per unit area per unit time. Before rays of the suns When they reach the Earth’s surface and enter the actinometer, they must pass through the entire thickness of our atmosphere, as a result of which part of the energy will be absorbed by the atmosphere. The magnitude of this absorption varies greatly depending on the state of the atmosphere, so that the amount of solar energy received on the earth's surface at different times is very different. The solar constant is the amount of energy received by one square centimeter of area exposed at the boundary of the earth's atmosphere perpendicular to the rays of the Sun, in one minute in small calories. From a large series of actinometric observations from many geophysical observatories, the following value was obtained for the solar constant: A = 1.94 cal/cm2 min. On 1 square meter of the surface of the site in the vicinity of the Earth facing the Sun, 1400 J of energy transferred by solar electromagnetic radiation are received every second. This value is called the solar constant. In other words, the energy flux density of solar radiation is 1.4 kW/m2. SOLAR SPECTRUM - distribution of energy of electromagnetic radiation from the Sun in the wavelength range from several fractions of nm (gamma radiation) to meter radio waves. In the visible region, the solar spectrum is close to the spectrum of a completely black body at a temperature of about 5800 K; has an energy maximum in the region of 430-500 nm. The solar spectrum is a continuous spectrum on which more than 20 thousand absorption lines (Fraunhofer lines) of various chemical elements are superimposed. Actin O meter- a device for measuring the intensity of direct solar radiation. The principle of operation of alumina is based on the absorption of incident radiation by a blackened surface and the conversion of its energy into heat. A. is a relative device, because The intensity of radiation is judged by various phenomena accompanying heating, in contrast to pyrheliometers - absolute instruments. For example, the principle of operation of the Michelson actinometer is based on heating a bimetallic plate blackened with soot by the sun's rays 1
, pressed from iron and invar. When heated, iron elongates, and invar experiences almost no thermal expansion, so the plate bends. The amount of bending serves as a measure of the intensity of solar radiation. The movement of a quartz filament is observed using a microscope. ,
located at the end of the plate. Pure black body- this is a body for which the absorption capacity is identically equal to unity for all frequencies or wavelengths and for any temperature, i.e.: From the definition of an absolutely black body it follows that it must absorb all radiation incident on it. The concept of "absolutely black body" is a model concept. Absolute black bodies do not exist in nature, but it is possible to create a device that is a good approximation to an absolutely black body - black body model
.
Black body model- this is a closed cavity with a small hole compared to its size (Fig. 1.2). The cavity is made of a material that absorbs radiation quite well. The radiation entering the hole is reflected many times from the inner surface of the cavity before leaving the hole. With each reflection, part of the energy is absorbed, as a result, the reflected flux dФ comes out of the hole, which is a very small part of the radiation flux dФ that entered it. As a result, the absorption capacity holes in the cavity
will be close to unity. If the inner walls of the cavity are maintained at temperature T, then radiation will emerge from the hole, the properties of which will be very close to the properties of black body radiation. Inside the cavity, this radiation will be in thermodynamic equilibrium with the cavity matter. By definition of energy density, the volumetric energy density w(T) of equilibrium radiation in a cavity is: where dE is the radiation energy in the volume dV. Spectral distribution of volume density is given by the functions u(λ,T) (or u(ω,T)), which are introduced similarly to the spectral density of energetic luminosity ((1.6) and (1.9)), i.e.: Here dw λ and dw ω are the volumetric energy density in the corresponding interval of wavelengths dλ or frequencies dω. Kirchhoff's law states that the relationship emissivity
body ((1.6) and (1.9)) to its absorption capacity
(1.14) is the same for all bodies and is a universal function of frequency ω (or wavelength λ) and temperature T, i.e.: It is obvious that the absorption capacity aω (or a λ) is different for different bodies, then from Kirchhoff’s law it follows that the stronger a body absorbs radiation, the stronger it should emit this radiation. Since for an absolute black body aω ≡ 1 (or aλ ≡ 1), then it follows that in the case of a completely black body:
In other words, f(ω,T) or φ(λ,T) ,
is nothing more than the spectral energy luminosity density (or emissivity) of a completely black body.
The function φ(λ,T) and f(ω,T) are related to the spectral energy density of black body radiation by the following relations: where c is the speed of light in vacuum. Installation diagram for experimental determination of the dependence φ(λ,T) is shown in Figure 1.3. Radiation is emitted from the opening of a closed cavity, heated to a temperature T, then hits a spectral device (prism or grating monochromator), which emits radiation in the frequency range from λ to λ + dλ. This radiation hits a receiver, which allows the radiation power incident on it to be measured. By dividing this power per interval from λ to λ + dλ by the area of the emitter (the area of the hole in the cavity!), we obtain the value of the function φ(λ,T) for a given wavelength λ and temperature T. The experimental results obtained are reproduced in Figure 1.4. Results of lecture No. 1 1. German physicist Max Planck in 1900 put forward a hypothesis according to which electromagnetic energy is emitted in portions, energy quanta. The magnitude of the energy quantum (see (1.2): ε = h v, where h=6.6261·10 -34 J·s is Planck’s constant, v- frequency of oscillations of an electromagnetic wave emitted by a body. This hypothesis allowed Planck to solve the problem of black body radiation. 2. And Einstein, developing Planck’s concept of energy quanta, introduced in 1905 the concept of “quantum of light” or photon. According to Einstein, quantum of electromagnetic energy ε = h v moves in the form of a photon localized in a small region of space. The idea of photons allowed Einstein to solve the problem of the photoelectric effect. 3. English physicist E. Rutherford, based on experimental studies conducted in 1909-1910, built a planetary model of the atom. According to this model, at the center of the atom there is a very small nucleus (r I ~ 10 -15 m), in which almost the entire mass of the atom is concentrated. The nuclear charge is positive. Negatively charged electrons move around the nucleus like the planets of the solar system in orbits whose size is ~ 10 -10 m. 4. The atom in Rutherford’s model turned out to be unstable: according to Maxwell’s electrodynamics, electrons, moving in circular orbits, should continuously emit energy, as a result of which they should fall onto the nucleus in ~ 10 -8 s. But all our experience testifies to the stability of the atom. This is how the problem of atomic stability arose. 5. The problem of atomic stability was solved in 1913 by the Danish physicist Niels Bohr on the basis of two postulates he put forward. In the theory of the hydrogen atom, developed by N. Bohr, Planck's constant plays a significant role. 6. Thermal radiation is electromagnetic radiation emitted by a substance due to its internal energy. Thermal radiation can be in thermodynamic equilibrium with surrounding bodies. 7. The energetic luminosity of a body R is the ratio of the energy dE emitted during a time dt by the surface dS in all directions to dt and dS (see (1.5)): 8. Spectral density of energy luminosity r λ (or emissivity of a body) is the ratio of energy luminosity dR, taken in an infinitesimal wavelength interval dλ, to the value dλ (see (1.6)): 9. Radiation flux Ф is the ratio of the energy dE transferred by electromagnetic radiation through any surface to the transfer time dt, which significantly exceeds the period of electromagnetic oscillations (see (1.13)): 10. Body absorption capacity a λ is the ratio of the radiation flux dФ λ "absorbed by a body in the wavelength interval dλ to the flux dФ λ incident on it in the same interval dλ, (see (1.14): 11. An absolutely black body is a body for which the absorption capacity is identically equal to unity for all wavelengths and for any temperature, i.e. A completely black body is a model concept. 12. Kirchhoff’s law states that the ratio of the emissivity of a body r λ to its absorption capacity a λ is the same for all bodies
and is a universal function of wavelength λ (or frequency ω) and temperature T (see (1.17)): LECTURE N 2 The problem of blackbody radiation. Planck's formula. Stefan-Boltzmann law, Wien's law § 1. The problem of black body radiation. Planck's formula The problem with black body radiation was to theoretically get addictedφ(λ,T)- the spectral density of the energy luminosity of an absolutely black body. It seemed that the situation was clear: at a given temperature T, the molecules of the substance of the radiating cavity have a Maxwellian velocity distribution and emit electromagnetic waves in accordance with the laws of classical electrodynamics. Radiation is in thermodynamic equilibrium with matter, which means that the laws of thermodynamics and classical statistics can be used to find the spectral radiation energy density u(λ,T) and the associated function φ(λ,T). However, all attempts by theorists to obtain the law of black body radiation based on classical physics have failed. Partial contributions to the solution of this problem were made by Gustav Kirchhoff, Wilhelm Wien, Joseph Stefan, Ludwig Boltzmann, John William Rayleigh, James Honwood Jeans. The problem of blackbody radiation was solved by Max Planck. To do this, he had to abandon classical concepts and make the assumption that a charge oscillating with a frequency v, can receive or give energy in portions, or quanta. The magnitude of the energy quantum in accordance with (1.2) and (1.4): where h is Planck's constant; Based on the concept of energy quanta, M. Planck, using the methods of statistical thermodynamics, obtained an expression for the function u(ω,T), giving distribution of energy density in the radiation spectrum of an absolute black body: The derivation of this formula will be given in Lecture No. 12, § 3 after we become acquainted with the basics of quantum statistics. To go to the spectral density of energy luminosity f(ω,T), we write the second formula (1.19): Using this relation and Planck’s formula (2.1) for u(ω,T), we obtain that: This is Planck's formula for spectral density of energetic luminosity f(ω ,T). Now we get Planck's formula for φ(λ,T). As we know from (1.18), in the case of a completely black body f(ω,T) = r ω, and φ(λ,T) = r λ. The relationship between r λ and r ω is given by formula (1.12), applying it we get: Here we expressed the argument ω of the function f(ω,T) in terms of the wavelength λ. Substituting here Planck’s formula for f(ω,T) from (2.2), we obtain Planck’s formula for φ(λ,T) - the spectral density of energy luminosity depending on the wavelength λ: The graph of this function coincides well with the experimental graphs of φ(λ,T) for all wavelengths and temperatures. This means that the problem of black body radiation has been solved. § 2. Stefan-Boltzmann law and Wien's law From (1.11) for an absolutely black body, when r ω = f(λ,T), we obtain the energy luminosity R(T) ,
integrating the function f(ω,Т) (2.2) over the entire frequency range. Integration gives: Let us introduce the notation: then the expression for the energetic luminosity R will take the following form: That's what it is Stefan-Boltzmann law
. M. Stefan, based on an analysis of experimental data, came to the conclusion in 1879 that the energetic luminosity of any body is proportional to the fourth power of temperature. L. Boltzmann in 1884 found from thermodynamic considerations that such a dependence of energetic luminosity on temperature is valid only for an absolutely black body. The constant σ is called Stefan-Boltzmann constant
. Its experimental significance: Calculations using the theoretical formula give a result for σ that is in very good agreement with the experimental one. Note that graphically the energetic luminosity is equal to the area limited by the graph of the function f(ω,T), this is illustrated in Figure 2.1. The maximum of the graph of the spectral density of energy luminosity φ(λ,T) shifts to the region of shorter waves with increasing temperature (Fig. 2.2). To find the law according to which the maximum φ(λ,T) shifts depending on temperature, it is necessary to study the function φ(λ,T) to the maximum. Having determined the position of this maximum, we obtain the law of its movement with temperature change. As is known from mathematics, to study a function to its maximum, you need to find its derivative and equate it to zero: Substituting here φ(λ,Т) from (1.23) and taking the derivative, we obtain three roots of the algebraic equation with respect to the variable λ. Two of them (λ = 0 and λ = ∞) correspond to zero minima of the function φ(λ,Т). For the third root, an approximate expression is obtained: Let us introduce the notation: then the position of the maximum of the function φ(λ,T) will be determined by a simple formula: That's what it is Wien's displacement law
. It is named after V. Wien, who theoretically obtained this ratio in 1894. The constant in Wien's displacement law has the following numerical value: Results of lecture No. 2 1. The problem of black body radiation was that all attempts to obtain, on the basis of classical physics, the dependence φ(λ,T) - the spectral density of the energy luminosity of a black body failed. 2. This problem was solved in 1900 by M. Planck on the basis of his quantum hypothesis: a charge oscillating with a frequency v, can receive or give out energy in portions or quanta. Energy quantum value: here h = 6.626 10 -34 is Planck’s constant, the value 3. Planck’s formula for the spectral density of the energy luminosity of an absolutely black body has the following form (see (2.4): here λ is the wavelength of electromagnetic radiation, T is the absolute temperature, h is Planck’s constant, c is the speed of light in vacuum, k is Boltzmann’s constant. 4. From Planck’s formula follows the expression for the energy luminosity R of an absolutely black body: which allows us to theoretically calculate the Stefan-Boltzmann constant (see (2.5)): the theoretical value of which coincides well with its experimental value: in the Stefan-Boltzmann law (see (2.6)): 5. From Planck’s formula follows Wien’s displacement law, which determines λ max - the position of the maximum of the function φ(λ,T) depending on the absolute temperature (see (2.9): For b - the Wien constant - the following expression is obtained from Planck’s formula (see (2.8)): Wien's constant has the following value b = 2.90 ·10 -3 m·K. LECTURE N 3 Photoelectric effect problem
. Einstein's equation for the photoelectric effect § 1. The photoelectric effect problem A The photoelectric effect is the emission of electrons by a substance under the influence of electromagnetic radiation.
This photoelectric effect is called external. This is what we will talk about in this chapter. There is also internal photoelectric effect
. (see lecture 13, § 2). In 1887, German physicist Heinrich Hertz discovered that ultraviolet light shining on the negative electrode in a spark gap facilitated the passage of the discharge. In 1888-89 Russian physicist A. G. Stoletov is engaged in a systematic study of the photoelectric effect (a diagram of its installation is shown in the figure). The research was carried out in a gas atmosphere, which greatly complicated the processes taking place. Stoletov discovered that: 1) ultraviolet rays have the greatest impact; 2) the current increases with increasing intensity of light illuminating the photocathode; 3) charges emitted under the influence of light have a negative sign. Further studies of the photoelectric effect were carried out in 1900-1904. German physicist F. Lenard in the highest vacuum achieved at that time. Lenard was able to establish that the speed of electrons escaping from the photocathode does not depend
on light intensity and directly proportional to its frequency
. Thus was born photoelectric effect problem
. It was impossible to explain the results of Lenard's experiments on the basis of Maxwell's electrodynamics! Figure 3.2 shows a setup that allows you to study the photoelectric effect in detail. Electrodes, photocathode
And anode
, placed in balloon,
from which the air has been pumped out. Light is supplied to the photocathode through quartz window
. Quartz, unlike glass, transmits ultraviolet rays well. The potential difference (voltage) between the photocathode and anode measures voltmeter
. The current in the anode circuit is measured by a sensitive microammeter
. To regulate voltage power battery
connected to rheostat
with a midpoint. If the rheostat motor is opposite the midpoint connected through a microammeter to the anode, then the potential difference between the photocathode and the anode is zero. When the slider is shifted to the left, the anode potential becomes negative relative to the cathode. If the rheostat slider is moved to the right from the midpoint, then the anode potential becomes positive. The current-voltage characteristic of the installation for studying the photoelectric effect allows one to obtain information about the energy of electrons emitted by the photocathode. The current-voltage characteristic is the dependence of the photocurrent i on the voltage between the cathode and anode U. When illuminated with light, the frequency v which is sufficient for the photoelectric effect to occur, the current-voltage characteristic has the form of the graph shown in Fig. 3.3: From this characteristic it follows that at a certain positive voltage at the anode, the photocurrent i reaches saturation. In this case, all electrons emitted by the photocathode per unit time fall on the anode during the same time. At U = 0, some electrons reach the anode and create a photocurrent i 0 . At some negative voltage at the anode - U back - the photocurrent stops. At this voltage value, the maximum kinetic energy of the photoelectron at the photocathode (mv 2 max)/2 is completely spent on doing work against the forces of the electric field: In this formula, m e is the mass of the electron; v max - its maximum speed at the photocathode; e is the absolute value of the electron charge. Thus, by measuring the retarding voltage U back, you can find the kinetic energy (and speed of the electron) immediately after its departure from the photocathode. Experience has shown that 1)the energy of the electrons emitted from the photocathode (and their speed) did not depend on the light intensity!
When the frequency of light changes v U back also changes, i.e. maximum kinetic energy of electrons leaving the photocathode; 2)maximum kinetic energy of electrons, at the photocathode,(mv 2 max)/2 , is directly proportional to the frequency v of the light illuminating the photocathode.
Problem, as in the case of black body radiation, was that theoretical predictions made for the photoelectric effect based on classical physics (Maxwellian electrodynamics) contradicted the experimental results.
Light intensity I in classical electrodynamics is the energy flux density of a light wave. Firstly, from this point of view, the energy transferred by a light wave to an electron must be proportional to the intensity of the light. Experience does not confirm this prediction.
Secondly, in classical electrodynamics there are no explanations for the direct proportionality of the kinetic energy of electrons,(mv 2 max)/2 , light frequency v.
(1)
(2)
(3)
. (4)
v- frequency of oscillations of an electromagnetic wave emitted by an oscillating charge; ω = 2π v- circular frequency.
J s is also called Planck's constant ["ash" with a bar], ω is the circular (cyclic) frequency.
