The document explains black body radiation, defining a black body as an ideal absorber and emitter of radiation. It discusses laws governing black body radiation, including Kirchhoff's law, Wien's displacement law, and Stefan’s law, and explores the energy density and distribution of emitted radiation based on temperature. The document also addresses historical theories like Rayleigh-Jeans law and the ultraviolet catastrophe, leading up to Planck's quantum theory of radiation.

1. Black Body Radiation
B y
Dr. C.D. Mungmode
M. G. College, Armori

2.  Black Body:
 A perfectly black body is one which absorbs
totally all the radiation of any wavelength which
fall on it. At room temperature, such an object
would appear to be perfectly black (hence the
term blackbody).
 When such a body heated to a suitable high
temperature, it emits full or total radiation. Hence
a perfect absorber is a perfect emitter.
 The radiations emitted by Black Body at higher
temperature are known as Black Body Radiations.

3. Fig. Ferry’s black body
 When this body is heated to high temperature, radiations come out
of the hole. The hole act as a black body radiator.
Black Body in Practice:
 It is possible to construct a nearly-perfect blackbody.
 A double walled copper sphere is taken and coated with lamp black
on its inner surface. It has a very narrow opening and a projection in
front of it. The radiations entering through narrow opening falls on
the projection and suffers multiple reflections and after some
reflections radiations are completely absorbs. It acts as a black body.

4. Black Body Radiation and its Temperature Dependence:
 When a Black Body placed inside a uniform temp. enclose, it will emit full radiation of encloser after
equilibrium.
 The radiations are independent of nature of substance, nature of walls of encloser and presence of any
other body.

5. Emissive Power:
The emissive power eλ of a body at a particular temp. (T) for a wavelength λ is defined as the energy emitted per
second per unit surface area of the body within a unit wavelength range.
Absorptive Power:
The absorptive power aλ of a body at a particular temp. (T) and for a particular wavelength λ is defined as the
ratio of the amount of energy absorbed in a given time by the surface to the amount of energy incident on the
surface in the same time.
Kirchhoff’s Law:
It states that the ration of the emissive power to the absorptive power for a given wavelength at a given
temperature is the same for all bodies and is equal to the emissive power of a perfectly black body at that
temperature.

6. Concept of Energy Density:
 Whenever a body gets hot, it starts emitting radiation.
 Max Planck gave his quantum theory of radiation to explain the distribution of energy in the
spectrum observed when a hot body emits heat radiations.
 Also the radiations of a hot body at a particular temperature are not of single frequency or
wavelength but contain many wavelengths.
 Energy density ρ(ν) is defined as the energy per unit volume per unit frequency.
 The energy per unit area of the body i.e. the energy density depends upon the temperature of
the hot body. Higher the temperature of the hot body, higher is the maximum energy density.

7.  Distribution of Black Body Radiation Spectrum:
 Planck was able to empirically describe the intensity of
light emitted by a blackbody as a function of wavelength.
Furthermore, he was able to describe how this spectrum
would change as the temperature changed.
 As the temperature of a blackbody increases, the total
amount of light emitted per second increases, and the
wavelength of the spectrum's peak (maximum intensity)
shifts to bluer colors (towards lower wavelength side.)

8. Wien's displacement law:
 For example, an iron bar becomes orange-red when heated to high temperatures and its color
progressively shifts toward blue and white as it is heated further.
 In 1893, German physicist Wilhelm Wien quantified the relationship between blackbody
temperature and the wavelength of the spectral peak with the following equation:
λmaxT = 0.298 cm k , where T is the temperature in Kelvin.
 Wien's law (also known as Wien's displacement law) states that the
wavelength of maximum emission from a blackbody is inversely
proportional to its temperature. This makes sense; shorter-
wavelength (higher-frequency) light corresponds to higher-energy
photons, which you would expect from a higher-temperature
object.

9.  Stefan’s law:
 It states that the rate of emission of radiant energy given out by unit area of a
perfectly black body is directly proportional to the fourth power of its absolute
temperature
E = σ T 4
Where, σ is Stefan’s constant = 5.67 x 10-8 Wm-2k-4
 If a black body at absolute temperature T is surrounded by another black body at
absolute temp T0, the net rate of loss of heat energy per unit area of the surface is given
by,
E = σ ( T4 – T0
4 )
 Josef Stefan deduced the rule in 1879 and Boltzmann provided a formal derivation in 1884.

10.  Wien’s law:
 The wavelength corresponding to maximum intensity is inversely proportional to the
absolute temperature of the black body.
lmaxT = 2.898 x 10-3 m.K -------- (1)
Where, l max is the wavelength at which the curve peaks
T is the absolute temperature
 As the temperature increases, the peak is “displaced” to shorter wavelengths.
Em T-5
= const ant ----------(2)
This is known as fifth power law.
Above relations can be combined into one and expressed as
El = C l- 5 f ( lT) -----------(3)
Where El is the emissive power of the black body at absolute temperature T.

11.  By making some arbitrary assumptions Wien concluded that f(lT) has the form A e-a/lT
El = C l- 5 A e - a / lT
= K l- 5 e - a / lT
where k is another constant having value 4.94 x 1015 units.
Eldl = K l- 5 e- a / lT dl
This is known as Wien’s law of energy distribution.
 This law is not accord with the experimental curves.
 It holds good only in the region of shorter wavelengths.
 It does not hold good at longer wavelengths.

12.  Rayleigh-Jeans Law:
 Tried to establish the relation, on the basis of classical theories, for distribution of energy
with wavelength.
 According to it, the electromagnetic radiation spectrum emitted by a black body
continuously vary in wavelength from zero to infinity.
 Energy distribution is given by,
E l d l = 8  k T l - 4 - - - - - - - - ( 1 )
Where, k is Boltzmann’s constant.
 This law holds good in the region of longer wavelengths but fails for shorter
wavelengths.
 Thus, Wien’s law and Rayleigh-Jeans law does not gives the perfect explanation of the
experimental results.

13.  Ultraviolet Catastrophe:
 Rayleigh-Jeans law agrees well with the
experimental results at higher wavelength and at
lower wavelength it is in violent disagreement.
 According to it, energy density will be,
Eldl = 8 K T l- 4
 The energy density continuously increases with
decrease in wavelength (increase in frequency) and
approaches ∞ as l 0 .
 This is in contradiction with experimental results.
 This fatal objection to the law has been known as
Ultraviolet Catastrophe.

14.  Plank’s Quantum Postulates:
 A black body radiation chamber is filled up not only with radiation, but also with simple
harmonic oscillators of molecular dimension which can vibrate with all possible
frequencies. The vibration of a resonator is confine to one degree of freedom only.
 The oscillators can not radiate or absorb energy continuously, but energy is emitted or
absorbed in the form of packets or quanta called photons. Each photon has energy hƲ
where h is Plank’s constant having value 6.625 x 10-34 Joule-Sec and Ʋ is the frequency of
radiation. As the energy of the photon is hƲ, the energy emitted or absorbed is equal to 0,
hƲ, 2hƲ, 3hƲ …….., nhƲ.

15.  Derivation of Plank’s Radiation Law:
 Average energy of Plank’s oscillators:
If N is total no. of resonators and E is their total energy, average energy per oscillator is,
ε =
𝐸
𝑁
……… (1)
According to Maxwell’s law of molecular motion, the resonators having energies 0, ε, 2ε, 3ε,
……., nε are in the ratio
1 : exp(-𝜖/𝑘𝑇) : exp(-2𝜖/𝑘𝑇) : exp(-3𝜖/𝑘𝑇): …….
if N0 is the no. of resonators having energy zero, then no. of resonators N1 having energy 𝜖 will
be N0exp(-𝜖/𝑘𝑇), no. of resonators N2 having energy 2𝜖 will be N0exp(-2𝜖/𝑘𝑇) and so on.
∴ N = N0 + N1 + N2 +…………
= N0 + N0exp(-𝜖/𝑘𝑇) + N0exp(-2𝜖/𝑘𝑇) + …………

16. = N0 [ 1 + exp(-𝜖/𝑘𝑇) + exp(-2𝜖/𝑘𝑇) + exp(-3𝜖/𝑘𝑇) + …………..]
put, exp(-𝜖/𝑘𝑇) = y, we get
N = N0 [ 1+ y + y2 + y3 + ……………]
N =
N0
1 −𝑦
…….. (2)
The total energy of plank’s resonator will be
E = 0 x N0 + 𝜖 x N1 + 2𝜖 x N2 + ………..
= 𝜖N0exp(-𝜖/𝑘𝑇) + 2𝜖N0exp(-2𝜖/𝑘𝑇) + …………
= 𝜖N0 [exp(-𝜖/𝑘𝑇) + exp(-2𝜖/𝑘𝑇) + ………….]
= 𝜖N0
y
1−𝑦 2 ………(3)
Therefore, Avg. energy of a resonator will be
ε =
𝐸
𝑁
=
𝜖N0
y
1−𝑦 2
N0
1 −𝑦
=
𝜖exp(−𝜖/𝑘𝑇)
1−exp(−𝜖/𝑘𝑇)
=
𝜖
exp(𝜖/𝑘𝑇)− 1
………(4)

17. According to Plank’s hypothesis 𝜀 = h𝜈 , hence average energy becomes
ε =
h𝜈
exp(ℎ𝜈/𝑘𝑇)− 1
…………(5)
Plank’s derived an equation for no. of modes of vibration per unit volume with frequency
range 𝜈 and ( 𝜈 + d 𝜈 ), given by
=
8Π𝜈2d𝜈
c3 ……….(6)
Therefore, energy density belonging to the range d𝜈 can be obtained from equation (5) and
(6) as,
E𝜈 d𝜈 = (
8Π𝜈2d𝜈
c3 ) (
h𝜈
exp(ℎ𝜈/𝑘𝑇)− 1
)

18. E𝜈 d𝜈 =
8Πℎ𝜈3
c3
1
exp(ℎ𝜈/𝑘𝑇)− 1
d𝜈
This is known as Plank’s Radiation Law.
by using relation 𝜈 =
𝑐
λ
and d𝜈 = -
𝑐
λ2 dλ we get,
Eλ dλ =
8Πℎ𝑐
λ5
𝑑λ
exp(ℎ𝑐/λ𝑘𝑇)− 1

19. For shorter wavelengths, exp(
ℎ𝑐
λ𝑘𝑇
) becomes large as compared to unity and hence Plank’s law
reduces to,
Eλ dλ =
8Πℎ𝑐
λ5
𝑑λ
exp(ℎ𝑐/λ𝑘𝑇)
=
8Πℎ𝑐
λ5 exp(−
ℎ𝑐
λ𝑘𝑇
)𝑑λ
which is Wien’s law, fits for shorter wavelength region.
For longer wavelengths, exp(
ℎ𝑐
λ𝑘𝑇
) may be approximated to (1 +
ℎ𝑐
λ𝑘𝑇
) and hence Plan’s law
reduces to,
Eλ dλ =
8Πℎ𝑐
λ5
𝒅𝝀
(𝟏+ 𝒉𝒄
𝝀𝒌𝑻
−𝟏)
=
8Π𝑘𝑇
λ4 𝑑λ
which is Rayleigh-Jeans law, fits for longer wavelength region.