The term "black body" was introduced by Gustav Kirchhoff in The light emitted by a black body is called black-body radiation.

Transcription

1 Black body (Redirected from Black-body radiation) As the temperature decreases, the peak of the black body radiation curve moves to lower intensities and longer wavelengths. The black-body radiation graph is also compared with the classical model that preceded it. In physics a black body is an object that absorbs all light that falls onto it: no light passes through it nor is reflected. Despite the name, black bodies do produce thermal radiation such as light. The term "black body" was introduced by Gustav Kirchhoff in The light emitted by a black body is called black-body radiation. When used as a compound adjective, the term is typically hyphenated, as in "black-body radiation", or combined into one word, as in "blackbody radiation". The hyphenated and one-word forms should not generally be used as nouns, however. In the laboratory, the closest thing to a black body radiation is the radiation from a small hole in a cavity: it 'absorbs' little energy from the outside if the hole is small, and it 'radiates' all the energy from the inside which is black. However, the spectrum (i.e. the amount of light emitted at each wavelength) of its radiation will not be continuous, and only rays will appear whose wavelengths depend on the material in the cavity (see Emission spectrum). By extrapolating the spectrum curve for other frequencies, a general curve can be drawn, and any black-body radiation will follow it. This curve depends only on the temperature of the cavity walls. The observed spectrum of black-body radiation could not be explained with Classical electromagnetism and statistical mechanics: it predicted infinite brightness at low wavelength (i.e. high frequencies), a prediction often called the ultraviolet catastrophe. This theoretical problem was solved by Max Planck, who had to assume that electromagnetic radiation could propagate only in discrete packets, or quanta. This idea was later used by Einstein to explain the photoelectric effect. These theoretical advances eventually resulted in the replacement of classical electromagnetism by quantum mechanics. Today, the quanta are called photons. The intensity of radiation from a black body at temperature T is given by Planck's law of black body radiation: where is the amount of energy per unit surface per unit time per unit solid angle emitted in the frequency range between ν and ν+δν;

2 h is Planck's constant c is the speed of light k is Boltzmann's constant. The wavelength at which the radiation is strongest is given by Wien's law, and the overall power emitted per unit area is given by the Stefan-Boltzmann law. So, as temperature increases, the glow color changes from red to yellow to white to blue. Even as the peak wavelength moves into the ultra-violet enough radiation continues to be emitted in the blue wavelenths that the body will continue to appear blue. It will never become invisible indeed, the radiation of visible light increases monotonically with temperature. The radiance or observed intensity is not a function of direction. Therefore a black body is a perfect Lambertian radiator. Real objects never behave as full-ideal black bodies, and instead the emitted radiation at a given frequency is a fraction of what the ideal emission would be. The emissivity of a material specifies how well a real body radiates energy as compared with a black body. This emissivity depends on factors such as temperature, emission angle, and wavelength. However, a typical engineering assumption is to assume that a surface's spectral emissivity and absorptivity do not depend on wavelength, so that the emissivity is a constant. This is known as the grey body assumption. When dealing with non-black surfaces, the deviations from ideal black body behavior are determined by both the geometrical structure and the chemical composition, and follow Kirchhoff's Law: emissivity equals absorptivity, so that an object that does not absorb all incident light will also emit less radiation than an ideal black body. BLACK BODY RADIATION (i) Perfect black body: A body that absorbs all the radiation incident upon it and has an emissivity equal to 1 is called a perfectly black body. A black body is also an ideal radiator. It implies that if a black body and an identical another body are kept at the same temperature, then the black body will radiate maximum power as it is obvious from equation P = ea T 4 also. Because e = 1 for a perfectly black body while for any other body e < 1. Radiation entering the cavity has little chance of leaving before it is completely absorbed. (e 1) Materials like black velvet or lamp black come close to being ideal black bodies, but the best practical realization of an ideal black body is a small hole leading into a cavity, as this absorbs 98% of the radiation incident on them. Cavity approximating an ideal black body. (ii) Absorptive power a : It is defined as the ratio of the radiant energy absorbed by it in a given time to the total radiant energy incident on it in the same interval of time. a = energy absorbed/energy incident As a perfectly black body absorbs all radiations incident on it, the absorptive power of a perfectly black body is maximum and unity. (iii) Spectral absorptive power a l : The absorptive power a refers to radiations of all wavelengths (or the total energy) while the spectral absorptive power is the ratio of radiant energy absorbed by a surface to the radiant energy incident on it for a particular wavelength. It may have different values for different wavelengths for a given surface. Let us take an example, suppose a = 0.6, a = 0.4 for 1000 Å and a = 0.7 for 2000 o A for a given surface. Then it means that this surface will absorbs only 60% of the total radiant energy incident on it. Similarly it absorbs 40% of the energy incident on it corresponding to 1000 o A and 70% corresponding to 2000 o A. The spectral absorptive power a is related to absorptive power a through the relation

3 (iv) Emissive power e : (Don t confuse it with the emissivity e which is different from it, although both have the same symbol e). For a given surface it is defined as the radiant energy emitted per second per unit area of the surface. It has the units of W/m 2 or J/s m 2. For a black body e= T 4. (v) Spectral emissive power e : It is emissive power for a particular wavelength. Thus, Kirchoff s law: According to this law the ratio of emissive power to absorptive power is same for all surfaces at the same temperature. Hence, e 1 /a 1 = e 2 /a 2 = (e/a) but (a) black body = 1 and (e) black body = E (say) Then, (e/a) for any surface = constant = E, Similarly for a particular wavelength, (e /a ) = E Here E = emissive power of black body at temperature T = T 4 From the above expression, we can see that e a i.e., good absorbers for a particular wavelength are also good emitters of the same wavelength. COOLING BY RADIATION Consider a hot body at temperature T placed in an environment at a lower temperature T 0. The body emits more radiation than it absorbs and cools down while the surrounding absorb radiation from the body and warm up. The body is losing energy by emitting radiations at a rate. P 1 = ea T 4 and is receiving energy by absorbing radiations at a rate 4 P 2 = aa T 0 Here a is a pure number between 0 and 1 indicating the relative ability of the surface to absorb radiation from its surroundings. Note that this a is different from the absorptive power a. In thermal equilibrium, both the body and the surrounding have the same temperature (say T c ) and, P 1 = P 2 or ea T 4 4 c = aa T c or e = a Thus, when T > T 0, the net rate of heat transferred from the body to the surroundings is, dq/dt = ea (T 4 T 4 0 ) or mc = ( dt/dt) = ea (T 4 T 4 0 ) => Rate of cooling, ( dt/dt) = ea /mc (T 4 T 4 0 ) or dt/dt (T 4 T 4 0 ) Wien's displacement law Main article: Wien's displacement law Wien's displacement law shows how the spectrum of black body radiation at any temperature is related to the spectrum at any other temperature. If we know the shape of the spectrum at one temperature, we can calculate the shape at any other temperature. A consequence of Wien's displacement law is that the wavelength at which the intensity of the radiation produced by a black body is at a maximum, λ max, it is a function only of the temperature where the constant, b, known as Wien's displacement constant, is equal to (51) 10 3 m K.

4 Note that the peak intensity can be expressed in terms of intensity per unit wavelength or in terms of intensity per unit frequency. The expression for the peak wavelength given above refers to the intensity per unit wavelength; meanwhile the Planck's Law section above was in terms of intensity per unit frequency. Rayleigh Jeans Law: The Rayleigh-Jeans Radiation Law was a useful but not completely successful attempt at establishing the functional form of the spectra of thermal radiation. The energy density u ν per unit frequency interval at a frequency ν is, according to the the Rayleigh-Jeans Radiation, u ν = 8πν²kT/c² where k is Boltzmann's constant, T is the absolute temperature of the radiating body and c is the speed of light in a vacuum. This formula fits the empirical meansurements for low frequencies but fails increasingly for higher frequencies. The failure of the formula to match the new data was called the ultravioletl catastrophe. The significance of this inadequate so-called law is that it provides an asymptotic condition which other proposed formulas, such as Planck's, need to satisfy. It gives a value to an otherwise arbitrary constant in Planck's thermal radiation formula. Wein s distribution law: Wien derived his law from thermodynamic arguments, several years before Planck introduced the quantization of radiation. He showed that energy density u ν per unit frequency interval at a frequency ν is uν = 8πν3/c 3 e hυ/kt The Wien approximation was originally proposed as a description of the complete spectrum of thermal radiation, although it failed to accurately describe long wavelength (low frequency) emission. However, it was soon superseded by Planck's law, developed by Max Planck. Unlike the Wien approximation, Planck's law accurately describes the complete spectrum of thermal radiation. Derivation of Planck s radiation law Assumptions: (i) (ii) (iii) A cavity in a material that is maintained at constant temperature T The emission of radiation from the cavity walls is in equilibrium with the radiation that is absorbed by the walls The radiation field in an empty volume in thermal equilibrium with a container at T can be viewed as a superposition of standing harmonic Planck considered the black body radiations (in the hohlraum) to consist of linear oscillators of molecular dimensions and that the energy of a linear oscillator can assume only the discrete values 0,hv,2hv,3hv...nhv If N 0, N 1,N 2....are the number of oscillators per unit volume of the hologram possessing

5 energies0,hv,2hv...respectively, then the total number of oscillators N per unit volume will be N = N 0 +N 1+ N But the number of oscillators, N r having energy E r is given by (Maxwell s formula) N r = N 0 Putting these values in eqn., we get N = N0 +N0 +N0 +...N0 = N0( ) The total energy of N oscillators will be Hence the average energy per oscillator is (on dividing numerator and denominator bye-hv/kt) Thus we see that the average energy of the oscillator is not Kt (as given by classical theory)but equal to hv/(ehv/kt-1) according to Planck s quantum theory. Further, it can be deduced that he number of oscillators per unit volume having frequent in the range of v and v+dv is equal to Hence the average energy per unit value (i.e., energy density) inside the enclosure is obtained by multiplying with i.e. it is given by

6 Putting and,the average energy per unit volume in the enclosure of the wave lights between and +d will be Or the energy radiated by the black body corresponding to wavelength is Which is Planck s radiation law or Pluck s distribution law? The above equation is also quite often written in the form where and c 2 =hc/k are universal constants.