Black Body Radiation Notes

Black Body Radiation:

A black body is one that absorbs all electromagnetic radiations falling on it.

It is a perfect absorber & perfect emitter of radiation.

A black body in thermal equilibrium means constant temperature emits electromagnetic radiations. The radiation emitted by such a body is called black body radiation or full radiation or temperature radiation. This radiation is emitted according to Planck’s law & its spectrum is determined by the temperature alone, not by the shape or configuration of the body.  Again according to Kirchhoff’s law the emissive power of a black body is independent of its geometry, so universal in character.

A perfect black body is an idealization, it doesn’t really exist. Lamp black or platinum black is nearest approach to a perfectly black body. Lamp black can absorbs about 96% of radiant energy & platinum black absorbs about 98%. However, most glowing substances like a piece of red hot iron or the sun are good approximations.

A hollow closed cavity with blackened inner walls with a small opening serves as a very good approximation of a black body.

The spectrum of this radiation is not dependent on the chemical composition of the matter but it's only determined by its absolute temperature T. It turns out that all objects behave like black bodies, regardless if they are actually black or not.

A black body radiates more when it is hot than when it is cold, hence the spectrum of a hotter black body has its peak at higher frequency as compared to the cooler one.

Since hot bodies are very often used as light sources colour temperature is used to describe the colour of the light. For example the sun, a filament light bulb, a candle, and many other light sources can be considered as "black body" radiators.

Colour Perception:

1'000 K	Red
1'500 K	Reddish orange
2'000 K	Yellowish orange
2'800 K	Yellow
3'500 K	Yellowish white
4'500 K	Warm white
5'500 K	White
1'500 K	Candlelight
2'700 K	Incandescent lamp
3'200 K	Sunrise / sunset
3'400 K	Halogen incandescent lamp
5'500 K	Sunny daylight around noon
6'000 K	Electronic photo flash
7'000 K	Overcast sky
10'000 K	Blue sky

Idealization/ Ferry’s black body:

1. A widely used model of black body radiation which is a spherical cavity whose walls are opaque to radiations.
2. The sphere contains a small hole ‘O’ with a conical projection ‘P’ opposite to hole ‘O’.
3. The inner surface of the sphere is coated with lamp black. The space between the walls is evacuated to prevent loss of heat by conduction of convection.
4. When the radiant energy enters the hole, suffers multiple reflections inside the cavity & ultimately absorbed.
5. At each reflection about 96% of incident radiation energy is absorbed, hence after few reflections all the radiant energy is absorbed by the sphere.
6. The function of the conical projection is to prevent direct reflection of the radiant energy from the surface opposite to hole.
7.  Till thermal equilibrium, radiation is absorbed & emitted by the walls of the cavity. The energy is distributed until the bunch of photon achieves Planck’s distribution.
8. Thus sphere absorbs all radiant energy incidents on it. When the sphere is heated, blackbody radiations are emitted from the hole ‘O’.
9. To sphere is heated to & maintained at different temperature, wavelength & energy distribution is studied.
10. By an infrared spectrometer & Bolometer (instrument used to measure intensity), the intensities of different wavelengths are drawn.

Results:

1. The distribution of energy in the spectrum of black body radiations is not uniform over a wide range of wavelength.
2. As the temperature increases, E for every wavelength increases.
3. At constant temperature as λ increases till it becomes maximum at a certain wavelength _λmax & then with further increase of ˄, E decreases. At a higher temperature the wavelength  λmax at which E maximum, shifts towards a shorter wavelength.

Theoretical laws of Blackbody radiation:

1      1.Kirchhoff’s Law:

At any particular temperature, the ratio of emissive power (E_e) of a body in thermal equilibrium with its surroundings to its absorptive power (E_a) is a constant equal to the emissivity (ε) of a perfectly black body at that temperature.

Ε = E_e / E_a

           2. Stefan-Boltzmann Law:

This law states that the total energy radiated in unit time per unit surface area of a black body (I) is proportional to the fourth power of absolute temperature T of the black body.

I=    σT⁴

Where       σ =5.67x10^-8 W/m²K⁴ = Stefan-Boltzmann constant

3. Wien’s Displacement Law:

This law states that the wavelength corresponding to the maximum spectral intensity is inversely proportional to the absolute temperature of the body emitting radiation & expressed as

λmax=b/T

Where b=2.898x10^-3 mK= Wien’s displacement constant

          4.  Wien's radiation formula:

According to Wien, the energy density in the wavelength interval λ to λ+dλ, emitted by a blackbody at a temperature T is expressed as

   u_λ dλ=A/λ^⁵ exp(-B/kT) dλ

1    5. Rayleigh-Jeans Law:

According to this law

                u_λ dλ = 8πkT/ λ^⁴  dλ

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Problems on Black body Radiation

Objectives Blackbody Radiation:

1.    The emissivity & absorptivity of a real surface are equal for radiation with identical temperature & wavelength. This law is referred as

a)    Rayleigh Law

b)    Kirchhoff’s Law

c)    Planck’s Law

d)    Wien’s Law

2.    The energy radiated by a blackbody should not vary in accordance with

a)      Wavelength

b)      Temperature

c)       Surface Characteristics

d)      Time

3.    As the wavelength of the radiation decreases, the intensity of the black body radiations ____________
a) Increases
b) Decreases
c) First increases then decrease
d) First decreases then increase

4.     The radiations emitted by hot bodies are called as ________________
a) X-rays
b) Black-body radiation
c) Gamma radiations
d) Visible light

5.  What relation between emissivity, e, and Absorptive Power, a, is given by Kirchhoff’s law?

      a) e < a
b) e > a
c) e = a
d) no specific relation

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