The energy that a body loses due to thermal radiation is characterized by the following quantities.
Radiation flux (F) - energy emitted per unit time from the entire surface of the body.
In fact, this is the power of thermal radiation. The dimension of the radiation flux is [J/s = W].
Energy luminosity (Re) - energy of thermal radiation emitted per unit time from a unit surface of a heated body:
In the SI system, energetic luminosity is measured - [W/m 2 ].
The radiation flux and energetic luminosity depend on the structure of the substance and its temperature: Ф = Ф(Т),
The distribution of energetic luminosity over the spectrum of thermal radiation characterizes it spectral density. Let us denote the energy of thermal radiation emitted by a single surface in 1 s in a narrow range of wavelengths from λ before λ + d λ, via dRe.
Spectral luminosity density (r) or emissivity The ratio of energetic luminosity in a narrow part of the spectrum (dRe) to the width of this part (dλ) is called:
Approximate form of spectral density and energetic luminosity (dRe) in the wavelength range from λ before λ + d λ, shown in Fig. 13.1.
Rice. 13.1. Spectral density of energetic luminosity
The dependence of the spectral density of energetic luminosity on wavelength is called body radiation spectrum. Knowledge of this dependence allows one to calculate the energetic luminosity of a body in any wavelength range. The formula for calculating the energetic luminosity of a body in a range of wavelengths is:
The total luminosity is:
Bodies not only emit, but also absorb thermal radiation. The ability of a body to absorb radiation energy depends on its substance, temperature and wavelength of the radiation. The absorption capacity of the body is characterized by monochromatic absorption coefficient α.
Let a stream fall on the surface of the body monochromatic radiation Φ λ with wavelength λ. Part of this flow is reflected, and part is absorbed by the body. Let us denote the magnitude of the absorbed flux Φ λ abs.
Monochromatic absorption coefficient α λ is the ratio of the radiation flux absorbed by a given body to the magnitude of the incident monochromatic flux:
Monochromatic absorption coefficient is a dimensionless quantity. Its values lie between zero and one: 0 ≤ α ≤ 1.
Function α = α(λ,Τ) , expressing the dependence of the monochromatic absorption coefficient on wavelength and temperature, is called absorption capacity bodies. Its appearance can be quite complex. The simplest types of absorption are discussed below.
Pure black body is a body whose absorption coefficient is equal to unity for all wavelengths: α = 1.
Gray body is a body for which the absorption coefficient does not depend on the wavelength: α = const< 1.
Absolutely white body is a body whose absorption coefficient is zero for all wavelengths: α = 0.
Kirchhoff's law
Kirchhoff's law- the ratio of the emissivity of a body to its absorption capacity is the same for all bodies and is equal to the spectral density of the energy luminosity of an absolutely black body:
= /
Corollary of the law:
1. If a body at a given temperature does not absorb any radiation, then it does not emit it. Indeed, if for a certain wavelength the absorption coefficient α = 0, then r = α∙ε(λT) = 0
1. At the same temperature black body radiates more than any other. Indeed, for all bodies except black,α < 1, поэтому для них r = α∙ε(λT) < ε
2. If for a certain body we experimentally determine the dependence of the monochromatic absorption coefficient on wavelength and temperature - α = r = α(λT), then we can calculate the spectrum of its radiation.
THERMAL RADIATION Stefan Boltzmann's law Relationship between the energy luminosity R e and the spectral density of the energy luminosity of a black body Energy luminosity of a gray body Wien's displacement law (1st law) Dependence of the maximum spectral density of the energy luminosity of a black body on temperature (2nd law) Planck's formula
THERMAL RADIATION 1. The maximum spectral density of the solar energy luminosity occurs at wavelength = 0.48 microns. Assuming that the Sun radiates as a black body, determine: 1) the temperature of its surface; 2) the power emitted by its surface. According to Wien's displacement law, Power emitted by the surface of the Sun According to Stefan Boltzmann's law,
THERMAL RADIATION 2. Determine the amount of heat lost by 50 cm 2 from the surface of molten platinum in 1 minute, if the absorption capacity of platinum A T = 0.8. The melting point of platinum is 1770 °C. The amount of heat lost by platinum is equal to the energy emitted by its hot surface. According to Stefan Boltzmann's law,
THERMAL RADIATION 3. An electric furnace consumes power P = 500 W. The temperature of its inner surface with an open small hole with a diameter of d = 5.0 cm is 700 °C. How much of the power consumption is dissipated by the walls? The total power is determined by the sum of the Power released through the hole Power dissipated by the walls According to Stefan Boltzmann's law,
THERMAL RADIATION 4 A tungsten filament is heated in a vacuum with a current of force I = 1 A to a temperature T 1 = 1000 K. At what current strength will the filament be heated to a temperature T 2 = 3000 K? The absorption coefficients of tungsten and its resistivity corresponding to temperatures T 1, T 2 are equal to: a 1 = 0.115 and a 2 = 0.334; 1 = 25, Ohm m, 2 = 96, Ohm m The power emitted is equal to the power consumed from the electrical circuit in steady state Electric power released in the conductor According to Stefan Boltzmann's law,
THERMAL RADIATION 5. In the spectrum of the Sun, the maximum spectral density of energy luminosity occurs at a wavelength of .0 = 0.47 microns. Assuming that the Sun emits as a completely black body, find the intensity of solar radiation (i.e., radiation flux density) near the Earth outside its atmosphere. Luminous intensity (radiation intensity) Luminous flux According to the laws of Stefan Boltzmann and Wien
THERMAL RADIATION 6. Wavelength 0, which accounts for the maximum energy in the black body radiation spectrum, is 0.58 microns. Determine the maximum spectral density of energy luminosity (r, T) max, calculated for the wavelength interval = 1 nm, near 0. The maximum spectral density of energy luminosity is proportional to the fifth power of temperature and is expressed by Wien’s 2nd law. Temperature T is expressed from Wien’s displacement law value C is given in SI units, in which the unit wavelength interval = 1 m. According to the conditions of the problem, it is necessary to calculate the spectral luminosity density calculated for the wavelength interval of 1 nm, so we write out the value of C in SI units and recalculate it for a given wavelength interval:
THERMAL RADIATION 7. A study of the solar radiation spectrum shows that the maximum spectral density of energy luminosity corresponds to a wavelength = 500 nm. Taking the Sun to be a black body, determine: 1) the energetic luminosity R e of the Sun; 2) energy flow F e emitted by the Sun; 3) the mass of electromagnetic waves (of all lengths) emitted by the Sun in 1 s. 1. According to the laws of Stefan Boltzmann and Wien 2. Luminous flux 3. The mass of electromagnetic waves (all lengths) emitted by the Sun during the time t = 1 s, we determine by applying the law of proportionality of mass and energy E = ms 2. The energy of electromagnetic waves emitted during time t, is equal to the product of energy flow Ф e ((radiation power) by time: E=Ф e t. Therefore, Ф e =ms 2, whence m=Ф e/s 2.
Thermal radiation of bodies is electromagnetic radiation arising from that part of the internal energy of the body, which is associated with the thermal motion of its particles.
The main characteristics of thermal radiation of bodies heated to a temperature T are:
1. Energy luminosityR (T ) -the amount of energy emitted per unit time from a unit surface of a body, over the entire wavelength range. Depends on the temperature, nature and condition of the surface of the radiating body. In the SI system R ( T ) has a dimension [W/m2].
2. Spectral density of energetic luminosityr ( ,T) =dW/ d - the amount of energy emitted by a unit surface of a body per unit time in a unit wavelength interval (near the wavelength in question ). Those. this quantity is numerically equal to the energy ratio dW, emitted from a unit area per unit time in a narrow range of wavelengths from before +d , to the width of this interval. It depends on the body temperature, wavelength, and also on the nature and condition of the surface of the emitting body. In the SI system r( , T) has a dimension [W/m 3 ].
Energetic luminosity R(T) related to the spectral density of energetic luminosity r( , T) in the following way:
(1) [W/m2]
3. All bodies not only emit, but also absorb electromagnetic waves incident on their surface. To determine the absorption capacity of bodies in relation to electromagnetic waves of a certain wavelength, the concept is introduced monochromatic absorption coefficient-the ratio of the magnitude of the energy of a monochromatic wave absorbed by the surface of a body to the magnitude of the energy of the incident monochromatic wave:
The monochromatic absorption coefficient is a dimensionless quantity that depends on temperature and wavelength. It shows what fraction of the energy of an incident monochromatic wave is absorbed by the surface of the body. Value ( , T) can take values from 0 to 1.
Radiation in an adiabatically closed system (not exchanging heat with the external environment) is called equilibrium. If you create a small hole in the wall of the cavity, the equilibrium state will change slightly and the radiation emerging from the cavity will correspond to the equilibrium radiation.
If a beam is directed into such a hole, then after repeated reflections and absorption on the walls of the cavity, it will not be able to come back out. This means that for such a hole the absorption coefficient ( , T) = 1.
The considered closed cavity with a small hole serves as one of the models absolutely black body.
Absolutely black bodyis a body that absorbs all radiation incident on it, regardless of the direction of the incident radiation, its spectral composition and polarization (without reflecting or transmitting anything).
For a completely black body, the spectral luminosity density is some universal function of wavelength and temperature f( , T) and does not depend on its nature.
All bodies in nature partially reflect radiation incident on their surface and therefore are not classified as absolute black bodies. If the monochromatic absorption coefficient of a body is the same for all wavelengths and lessunits(( , T) = Т =const<1),then such a body is called gray. The monochromatic absorption coefficient of a gray body depends only on the temperature of the body, its nature and the state of its surface.
Kirchhoff showed that for all bodies, regardless of their nature, the ratio of the spectral density of energy luminosity to the monochromatic absorption coefficient is the same universal function of wavelength and temperature f( , T) , the same as the spectral density of the energy luminosity of a completely black body :
Equation (3) represents Kirchhoff's law.
Kirchhoff's law can be formulated this way: for all bodies of the system that are in thermodynamic equilibrium, the ratio of the spectral density of energy luminosity to the coefficient monochromatic absorption does not depend on the nature of the body, is the same function for all bodies, depending on the wavelength and temperature T.
From the above and formula (3) it is clear that at a given temperature those gray bodies that have a large absorption coefficient emit more strongly, and absolutely black bodies emit the most strongly. Since for an absolutely black body( , T)=1, then from formula (3) it follows that the universal function f( , T) represents the spectral luminosity density of a black body
Energy luminosity of the body R T, is numerically equal to energy W, emitted by the body over the entire wavelength range (0
Body emissivity rl ,T numerically equal to the energy of the body dWl, emitted by a body from a unit of body surface, per unit of time at body temperature T, in the wavelength range from l to l +dl, 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 al ,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 l to l +dl, those.
The body for which al ,T =1 over the entire wavelength range is called an absolute black body (BLB).
The body for which al ,T =const<1 over the entire wavelength range is called gray.
Where- spectral density energetic luminosity, or body emissivity .
Experience shows that the emissivity of a body depends on the temperature of the body (for each temperature the maximum radiation lies in its own frequency range). Dimension .
Knowing the emissivity, we can calculate the energetic luminosity:
called absorption capacity of the body . It also depends greatly on temperature.
By definition, it cannot be greater than one. For a body that completely absorbs radiation of all frequencies, . Such a body is called absolutely black (this is an idealization).
A body for which and is less than unity for all frequencies,called gray body (this is also an idealization).
There is a certain connection between the emissive and absorptive capacity of a body. Let's mentally conduct the following experiment (Fig. 1.1).
Rice. 1.1
Let there be three bodies inside a closed shell. Bodies are in a vacuum, therefore energy exchange can only occur through radiation. Experience shows that such a system will, after some time, reach a state of thermal equilibrium (all bodies and the shell will have the same temperature).
In this state, a body with greater emissivity loses more energy per unit time, but, therefore, this body must also have greater absorption capacity:
Gustav Kirchhoff formulated in 1856 law and suggested black body model .
The ratio of emissivity to absorptivity does not depend on the nature of the body; it is the same for all bodies(universal)function of frequency and temperature.
| , | (1.2.3) |
Where - universal Kirchhoff function.
This function has a universal, or absolute, character.
The quantities themselves and, taken separately, can change extremely strongly when moving from one body to another, but their ratio constantly for all bodies (at a given frequency and temperature).
For an absolutely black body, therefore, for it, i.e. the universal Kirchhoff function is nothing more than the emissivity of a completely black body.
Absolutely black bodies do not exist in nature. Soot or platinum black has absorptive capacity, but only in a limited frequency range. However, a cavity with a small hole is very close in its properties to a completely black body. A beam that gets inside is necessarily absorbed after multiple reflections, and a beam of any frequency (Fig. 1.2).
Rice. 1.2
The emissivity of such a device (cavity) is very close to f(ν, ,T). Thus, if the cavity walls are maintained at a temperature T, then radiation comes out of the hole, very close in spectral composition to the radiation of an absolutely black body at the same temperature.
By decomposing this radiation into a spectrum, one can find the experimental form of the function f(ν, ,T)(Fig. 1.3), at different temperatures T 3 > T 2 > T 1 .
Rice. 1.3
The area covered by the curve gives the energetic luminosity of a black body at the corresponding temperature.
These curves are the same for all bodies.
The curves are similar to the molecular velocity distribution function. But there the areas covered by the curves are constant, but here with increasing temperature the area increases significantly. This suggests that energetic compatibility is highly dependent on temperature. Maximum radiation (emissivity) with increasing temperature shifts towards higher frequencies.
Laws of thermal radiation
Any heated body emits electromagnetic waves. The higher the body temperature, the shorter the waves it emits. A body in thermodynamic equilibrium with its radiation is called absolutely black (ACHT). The radiation of a completely black body depends only on its temperature. In 1900, Max Planck derived a formula by which, at a given temperature of an absolutely black body, one can calculate the intensity of its radiation.
The Austrian physicists Stefan and Boltzmann established a law expressing the quantitative relationship between the total emissivity and the temperature of a black body:
This law is called Stefan–Boltzmann law . The constant σ = 5.67∙10 –8 W/(m 2 ∙K 4) is called Stefan–Boltzmann constant .
All Planck curves have a noticeably pronounced maximum at the wavelength
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This law was called Wien's law . Thus, for the Sun T 0 = 5,800 K, and the maximum occurs at the wavelength λ max ≈ 500 nm, which corresponds to the green color in the optical range.
With increasing temperature, the maximum radiation of a completely black body shifts to the shorter wavelength part of the spectrum. A hotter star emits most of its energy in the ultraviolet, while a cooler star emits most of its energy in the infrared.
Photo effect. Photons
Photoelectric effect was discovered in 1887 by the German physicist G. Hertz and experimentally studied by A. G. Stoletov in 1888–1890. The most complete study of 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 photoeffect) consists of the ejection of electrons from a substance under the influence of light falling on it.
The diagram of the experimental setup for studying the photoelectric effect is shown in Fig. 5.2.1.
The experiments used a glass vacuum bottle with two metal electrodes, the surface of which was thoroughly cleaned. Some voltage was applied to the electrodes U, the polarity of which could be changed using a 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 5.2.2 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 voltage at the anode is negative, the electric field between the cathode and anode inhibits the electrons. 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:
Numerous experimenters have established the following basic principles of the photoelectric effect:
- The maximum kinetic energy of photoelectrons increases linearly with increasing light frequency ν and does not depend on its intensity.
- For each substance there is a so-called red photo effect border , i.e. the lowest frequency ν min at which the external photoelectric effect is still possible.
- The number of photoelectrons emitted by light from the cathode in 1 s is directly proportional to the light intensity.
- 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 the energy of photoelectrons from the intensity of the light flux and the proportionality of the maximum kinetic energy to the 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. An electromagnetic wave consists of separate portions - quanta, later named photons. When interacting with matter, a photon completely transfers all its energy hνone 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, 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:
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This formula is usually called Einstein's 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. The total number of 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.
As follows from Einstein’s equation, the tangent of the angle of inclination of the straight line expressing the dependence of the blocking potential Uз from frequency ν (Fig. 5.2.3), equal to the ratio of Planck’s constant h to the electron charge e:
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
Among metals, alkali elements have the lowest work function. 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 , designed 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 .
The photon energy is
it follows that the photon has momentum
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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. When light propagates, its wave properties appear (interference, diffraction, polarization), and when it interacts with matter, its corpuscular properties appear (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 provide a visual model of the combination 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.
d Φ e (\displaystyle d\Phi _(e)), emitted by a small area of the surface of the radiation source, to its area d S (\displaystyle dS) : M e = d Φ e d S . (\displaystyle M_(e)=(\frac (d\Phi _(e))(dS)).)It is also said that energetic luminosity is the surface density of the emitted radiation flux.
Numerically, the energetic luminosity is equal to the time-average modulus of the Poynting vector component perpendicular to the surface. In this case, averaging is carried out over a time significantly exceeding the period of electromagnetic oscillations.
The emitted radiation can arise in the surface itself, then they speak of a self-luminous surface. Another option is observed when the surface is illuminated from the outside. In such cases, some part of the incident flux necessarily returns back as a result of scattering and reflection. Then the expression for energetic luminosity has the form:
M e = (ρ + σ) ⋅ E e , (\displaystyle M_(e)=(\rho +\sigma)\cdot E_(e),)Where ρ (\displaystyle \rho ) And σ (\displaystyle \sigma )- reflection coefficient and scattering coefficient of the surface, respectively, and - its irradiance.
Other names of energetic luminosity, sometimes used in the literature, but not provided for by GOST: - emissivity And integral emissivity.
Spectral density of energetic luminosity
Spectral density of energetic luminosity M e , λ (λ) (\displaystyle M_(e,\lambda )(\lambda))- ratio of the magnitude of energetic luminosity d M e (λ) , (\displaystyle dM_(e)(\lambda),) falling on a small spectral interval d λ , (\displaystyle d\lambda ,), concluded between λ (\displaystyle \lambda) And λ + d λ (\displaystyle \lambda +d\lambda ), to the width of this interval:
M e , λ (λ) = d M e (λ) d λ . (\displaystyle M_(e,\lambda )(\lambda)=(\frac (dM_(e)(\lambda))(d\lambda )).)The SI unit is W m−3. Since wavelengths of optical radiation are usually measured in nanometers, in practice W m −2 nm −1 is often used.
Sometimes in literature M e , λ (\displaystyle M_(e,\lambda )) are called spectral emissivity.
Light analogue
M v = K m ⋅ ∫ 380 n m 780 n m M e , λ (λ) V (λ) d λ , (\displaystyle M_(v)=K_(m)\cdot \int \limits _(380~nm)^ (780~nm)M_(e,\lambda )(\lambda)V(\lambda)d\lambda ,)Where K m (\displaystyle K_(m))- maximum luminous radiation efficiency equal to 683 lm / W in the SI system. Its numerical value follows directly from the definition of candela.
Information about other basic energy photometric quantities and their light analogues is given in the table. Designations of quantities are given according to GOST 26148-84.
Energy photometric SI quantities| Name (synonym) | Quantity designation | Definition | SI units notation | Luminous magnitude |
|---|---|---|---|---|
| Radiation energy (radiant energy) | Q e (\displaystyle Q_(e)) or W (\displaystyle W) | Energy transferred by radiation | J | Light energy |
| Radiation flux (radiant flux) | Φ (\displaystyle \Phi ) e or P (\displaystyle P) | Φ e = d Q e d t (\displaystyle \Phi _(e)=(\frac (dQ_(e))(dt))) | W | Light flow |
| Radiation intensity (light energy intensity) | I e (\displaystyle I_(e)) | I e = d Φ e d Ω (\displaystyle I_(e)=(\frac (d\Phi _(e))(d\Omega ))) | W sr −1 | The power of light |
| Volumetric radiation energy density | U e (\displaystyle U_(e)) | U e = d Q e d V (\displaystyle U_(e)=(\frac (dQ_(e))(dV))) | J m −3 | Volumetric density of light energy |
| Energy brightness | L e (\displaystyle L_(e)) | L e = d 2 Φ e d Ω d S 1 cos ε (\displaystyle L_(e)=(\frac (d^(2)\Phi _(e))(d\Omega \,dS_(1)\, \cos \varepsilon ))) | W m−2 sr−1 | Brightness |
| Integral energy brightness | Λ e (\displaystyle \Lambda _(e)) | Λ e = ∫ 0 t L e (t ′) d t ′ (\displaystyle \Lambda _(e)=\int _(0)^(t)L_(e)(t")dt") | J m −2 sr −1 | Integral brightness |
| Irradiance (irradiance) | E e (\displaystyle E_(e)) | E e = d Φ e d S 2 (\displaystyle E_(e)=(\frac (d\Phi _(e))(dS_(2)))) | W m−2 |