The following points highlight the five important laws of radiation. The laws are: 1. Wavelength Distribution of Black Body Radiation: Planck’s Law 2. Total Emissive Power; Stefan-Boltzman Law 3. Wien’s Displacement Law 4. Kirchoff’s Law 5. Intensity of Radiation and Lambert’s Cosine Law.
1. Wavelength Distribution of Black Body Radiation: Planck’s Law:
The energy emitted by a black surface varies in accordance with wavelength, temperature and surface characteristics of the body. For a prescribed wavelength, the body radiates much more energy at elevated temperatures. Likewise the amount of emitted radiation is strongly influenced by the wavelength even if temperature of the body remains at a constant fixed value.
The laws governing the distribution of radiant energy over wavelength for a black body at a fixed temperature were formulated by Planck.
Based upon extensive experimental evidence, Planck suggested the following law for the spectral distribution of emissive power:
The symbols used have the following meanings:
h = Planck constant, 6.6236 × 10-34 Js
c = Velocity of light in vacuum 2.998 × 108 m/s
k = Boltzman constant, 13.802 × 10-4 J/K
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λ = Wavelength of radiation waves, m
T = Absolute temperature of the black body, K
Quite often the above expression is written as-
The quantity denotes the monochromatic (single wavelength) emissive power, and is defined as the energy emitted by the black surface (in all directions) at a given wavelength λ per unit wavelength interval around λ. That is, the rate of energy emission in the interval dλ is equal to (Eλ)bdλ.
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The variation of distribution of the monochromatic emissive power with wavelength is called the spectral energy distribution, and this has been depicted in Fig. 7.5 for a number of selected temperatures.
The following important features can be noted from this plot:
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(i) The monochromatic emissive power varies across the wavelength spectrum; the distribution is continuous but non-uniform. The emitted radiation is practically zero at the zero wavelength. With increase in wavelength, the monochromatic emissive power increases and attains a certain maximum value. With further increase in wavelength, the emissive power diminishes and drops again to almost zero value at infinite wavelength.
(ii) At any wavelength the magnitude of the emitted radiation increases with increasing temperature.
(iii) The wavelength at which the monochromatic emissive power is maximum shifts in the direction of shorter wavelengths as the temperature increases. This shift signifies that at elevated temperature, much of the energy is emitted in a narrow band ranging on both sides of wavelength at which the monochromatic emissive power is maximum. For example, the sun with its surface temperature of about 5600°C emits 90% of its radiations between 0.1 and 3µ.
(iv) At any temperature, the area under the monochromatic emissive power versus wavelength gives the rate of radiant energy emitted within the wavelength interval dλ. Thus dEb = (Eλ)h dλ. Upon integration over the entire range of wavelength.
The integral measures the total area under the monochromatic emissive power versus wavelength curve for the black body, and it represents the total emissive power per unit area (radiant energy flux density) radiated from a black body.
(a) For shorter wavelengths, the factor C2/λT becomes very large, In that case-
Obviously the term (- 1) appearing in the denominator of the Planck’s distribution law can be neglected compared to this large value. The Planck’s law then reduces to-
Equation 7.5 is called Wien’s law, and it is accurate within 1 percent for λT less than 3000 µK.
(b) For longer wavelengths, the factor C2/λT is small. In that case exp (C2/λT) can be expanded in series to give,
The above identity is known as Rayleigh-Jean’s Law. It is accurate within 1 percent for λT > 8 × 105 µK. As black body emits over 99.9 percent of its energy at λT values below this limit; the Rayleigh-Jean’s formula is apparently well outside the range of thermal radiation. The relation, however, is quite useful for analysing long wave radiations such as radio waves.
2. Total Emissive Power; Stefan-Boltzman Law:
The total emissive power E of a surface is defined as the total radiant energy emitted by the surface in all directions over the entire wavelength range per unit surface area per unit time.
The basic rate equation for radiation transfer is based on Stefan-Boltzman law which states that the amount of radiant energy emitted per unit time from unit area of black surface is proportional to the fourth power of its absolute temperature.
Eh = σb T4 … (7.7)
Where, σb is the radiation coefficient of a black body. This rate equation can be set-up by the integration of monochromatic emissive power over the entire band width of wavelength for λ = 0 to λ = ¥.
Undoubtedly, the Stefan Boltzman law helps us to determine the amount of radiations emitted in all the directions and over the entire wavelength spectrum from a simple knowledge of the temperature of the black body.
Normally a body radiating heat is simultaneously receiving heat from other bodies as radiation. Consider that surface 1 at temperature T1 is completely enclosed by another black surface at temperature T2. The net radiant heat flux is then given by-
Example 1:
List the salient features of a black body radiation.
Calculate the radiant flux density from a black body at 400°C? If the emitted radiant energy is to be doubled, to what temperature surface of the black body needs to be raised?
Solution:
A black body is an ideal or hypothetical surface having the following radiation heat transfer characteristics:
(i) A black body absorbs all the incident radiation regardless of wavelength and direction.
(ii) A black body neither reflects nor transmits any amount of incident radiation.
(iii) For a prescribed wavelength a black body radiates the maximum energy possible at the temperature of the body.
(iv) The black body is a diffused emitter. This implies that the radiation emitted by a black surface is a function of wavelength and temperature but is independent of direction.
From Stefan-Boltzman law, the rate of energy transmission from a black body is-
3. Wien’s Displacement Law:
From the spectral distribution of black body emissive power, it is apparent that the wavelength associated with maximum rate of emission depends upon the absolute temperature of the radiating surface. The nature of this dependence can be obtained by differentiating the Planck’s expression,
where, λmax denotes the wavelength at which emissive power is maximum. The Wien’s displacement law may be stated as “the product of absolute temperature and the wavelength at which the emissive power is maximum, is constant”. The law suggests that λmax is inversely proportional to the absolute temperature and accordingly the maximum spectral intensity of radiation shifts towards the shorter wavelength with rising temperature. The locus of points described by-
Wien’s law has been plotted as the dashed curve in Fig. 7.6.
A combination of Planck’s law and the Wien’s displacement law yields the correlation for maximum monochromatic emissive power for a black body.
Thus the magnitude of the maximum monochromatic emissive power varies proportionally with the fifth power of the absolute temperature of the black surface.
Wien’s displacement law holds true for more real substances; there is however some deviation in the case of a metallic radiator where the product (λmax T) is found to vary with absolute temperature. The law finds application in the prediction of a very high temperature through measurement of wavelength.
Example 2:
What are the ranges of wavelength of electromagnetic waves covering ultra-violets visible, infracted and thermal radiation.
A small black body has a total emissive power of 4.5 kW/m2. Determine its surface temperature and the wavelength of emission maximum. In which range of the spectrum does this wavelength fall?
Solution:
From Stefan Boltzman law, the rate of energy transmission from a black body is-
4. Kirchoff’s Law:
Consider two surfaces, one absolutely black at temperature Tb and the other non-black at temperature T (Fig. 7.7). The surfaces are arranged parallel to each other and so close that the radiation of one falls totally on the other. The radiant energy E emitted by the non-black surface impinges on the black surface and gets fully absorbed. Likewise the radiant energy Eh emitted by the black surface strikes the non-black surface.
If the non-black surface has absorptivity α, it will absorb αEb radiations and the remainder (1 – α) Eb will be reflected back for full absorption at the black surface. Radiant interchange for the non-black surface equals (E – αEb). If both the surfaces are at the same temperature, T = Tb, then the conditions correspond to mobile thermal equilibrium for which the resultant interchange of heat is zero.
This is because the absorptivity αb of a black body equals unity.
Equations 7.14 shows that the ratio of the emissive power E to absorptivity α is same for all bodies, and is equal to the emissive power of a black body at the same temperature. This relationship is known as the Kirchoff s law.
The ratio of the emissive power of a certain non-black body E to the emissive power of a black body Eb, both bodies being at the same temperature, is called the emissivity of the body. Emissivity of a body is a function of its physical and chemical properties and the state of its surface-whether rough or smooth.
From Equation 7.14:
E/Eb = α or ɛ = α … (7.15)
where, ɛ = E/Eb is emissivity.
Kirchoff’s law can also be stated as:
“The emissivity e and absorptivity of a real surface are equal for radiation with identical temperature and wavelength”. The equivalence of e and a does suggest that a perfect absorber (the perfect black body) is also a perfect radiator.
5. Intensity of Radiation and Lambert’s Cosine Law:
Subtended Plane and Solid Angles:
The plane angle a is defined by a region by the rays of a circle, and is measured as the ratio of the element of arc of length I on the circle to the radius r of the circle- α = l/r
The solid angle ω is defined by a region by the rays of a sphere, and is measured as,
Where,
An = projection of the incident surface normal to the line of propagation
A = area of incident surface
ϴ = angle between the normal to the incident surface and the line of propagation
r = length of the line of propagation between the radiating and the incident surfaces
The relationship between A, An and ϴ has been illustrated in Fig. 7.10. When the incident surface is a sphere, the projection of surface normal to the line of propagation is the silhouette disk of the sphere, which is a circle of the diameter of the sphere. The unit of measure of solid angle is the steradian (sr).
Intensity of Radiation:
Consider a small black surface dA (emitter) arbitrarily located at a point in the space under consideration and emitting radiations in different directions. A black body radiation collector through which the radiations pass is located at an angular position characterised by zenith angle ϴ towards the surface normal and the azimuth angle ɸ of a spherical coordinate system. Further, the collector subtends a solid angle dω when viewed from a point on the emitter.
The intensity of radiation I is the energy emitted (of all wavelengths) in a particular direction per unit surface area and through a unit solid angle. The area is the projected area of the surface on a plane perpendicular to the direction of radiation.
The collector or the incidence surface measures a variation of the emitted radiations depending upon its angular position. Maximum amount of radiation is measured (received) by the collector when it is at the position normal to the emitter, and the intensity in a direction ϴ from normal to the emitter follows the Lambert’s cosine law,
“The intensity of radiation in a direction ϴ from the normal to a black emitter is proportional to cosine of the angle ϴ.”
If In denotes the normal intensity and I0 represents the intensity at angle ϴ from the normal, then-
Iϴ = In cos ϴ … (7.20)
Apparently the energy radiated out decreases with increase in ϴ and becomes zero at ϴ = 90°.
When the collector is oriented at an angle ϴ1 from the normal to the emitter, then the radiations striking and being absorbed by the collector can be expressed as-
Where, dω1, is the solid angle subtended by the collector at the surface of the emitter.
The collector could be located at different angular positions and still maintain the same radial distance from the emitter. Let it subtend a solid angle dω2 at the emitter surface when located in a direction ϴ2 from the normal.
Then the rate of flow of energy through it will be:
It follows from equations 7.21 and 7.22 that for any surface located at an angle ϴ from the normal and subtending a solid angle dω at the emitter dA,
(dEb)ϴ2 = In cos ϴ dω dA … (7.23)
Relation between the normal intensity and emissive power:
To establish a relation between the normal intensity and the emissive power, we relate the differential solid angle dω to the zenith and azimuth angles by noting that for a spherical surface:
Then the radiations leaving the emitter and striking the collector is,
dEb = In cos ϴ sin ϴ dϴ cɸ dA …(7.24)
The total energy Eb radiated by the emitter and passing through a spherical region can be worked out be integrating equation 7.23 over the limits.
But the total emissive power of the emitter with area dA and temperature T is also given by:
Thus for a unit surface, the intensity of normal radiation In is the 1/π times the emissive power Eb.
Example 3:
Define intensity of radiation and prove that the intensity of normal radiation is 1/π times the total emissive power.
A black body of 0.2 m2 area has an effective temperature of 800 K. Calculate (a) the total rate of energy emission (b) the intensity of normal radiation (c) the intensity of radiation along a direction 60° to the normal, and (d) the wavelength of maximum monochromatic emissive power.
