Black Body Radiation : Wien Displacement Law

Introduction

Thermal energy or heat energy is transferred from one object to another. This may happen due to three different possibilities. They are conduction, convection, and radiation. In the conduction process, the energy is transferred from one object to another without the movement of the particles. In the convection process, the energy transfer happens with the movement of particles. In the radiation process, the energy is transferred through space as a wave.

Radiation

Radiation transfer is a process of transmission of energy from one object to another through the medium or space. It can be ionizing or non-ionizing, which depends upon the energy of the radiation. Ionizing radiation has an energy of more than 10eV and is able to ionize the particles in their path.

X - rays, Gamma rays, alpha emissions, and beta emissions are some examples of ionizing radiation. But non-ionizing radiation has not much energy to ionize the particles in their path. Ultraviolet, infrared, visible, and microwave are some examples of non-ionizing radiation.

Black Body Radiation

Black body or an ideal body is the body that absorbs all electromagnetic radiation passing through it whatever the frequency. No radiation is reflected or transmitted through it. It can also emit electromagnetic radiation. Electromagnetic waves that are emitted by the black body are called black body radiation. For a perfect black body emissivity is ε=1.

Characteristics of Black Body Radiation

Characteristics of the black body are given below

• This body absorbs all the radiation that falls on it.

• The absorption coefficient of this body is equivalent to 1.

• Because of the black color of the body, its transmitting and reflecting power is zero.

Wien’s Displacement Law

The frequency spectrum of a black body depends only on the temperature of the body. Everybody emits electromagnetic radiation when its temperature becomes greater than its absolute temperature. As temperature increases the wavelength of the electromagnetic radiation decreases. That means the frequency of the radiation shifted higher for an increase in temperature. Wien’s displacement law shows the graphical representation of the frequency of the spectral line of a black body for different temperatures. It clearly tells us that the peak wavelength of the radiation spectra is inversely proportional to the temperature of the black body. According to this

$$\mathrm{\lambda_{peak}\propto \frac{1}{T}}$$

$$\mathrm{\lambda_{peak}=\frac{b}{T}}$$

$$\mathrm{b=proportionality\:constant}$$

$$\mathrm{b = 2.897 \times 10^{−3}\:m\: K}$$

This law is used to find the temperature of the celestial body. This law also explains the change in color of the solid for the given temperature.

Example for Wien’s Displacement Law

When a metal piece is heated to a particular temperature, at first it looks like a red hot whose wavelength is higher. As the temperature increases, it turns more orange-red, and then it will turn into white whose wavelength is low.

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Discovery of Wien’s Displacement Law

This law is named after Wilhelm Wien who defines this law by considering thermodynamics. His assumptions include attaining the thermal equilibrium when the adiabatic expansion of waves through the cavity take place. When an iron rod is heated it looks red hot at a particular temperature. If the temperature increases the color of the rod changes to orange and then yellow. Finally, it glows blue at a very high temperature.

Plank’s law Formula

When there is no net flow of energy between the black body and the environment, he describes the frequency of the radiation emitted by the black body as in thermal equilibrium. This law describes that the energy of the radiant energy increases with the increase in temperature thereby its peak value of the frequency-shifted to the shorter wavelength side.

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According to this frequency distribution of the black body, radiation is given by,

$$\mathrm{B_{
u(
u,T)}=2h
u^3/c^2\:e^{h
u/k}BT^{−1}}$$

Plank says that emission of electromagnetic radiation by the black body is not a continuous one. But it is emitted in a discrete manner.

Stefan Boltzmann Law

This law gives the relation between the power of the radiation emitted by a black body with the temperature of the body. The power of the radiant wave is directly proportional to the fourth power of the temperature of the black body and it is denoted as follows,

$$\mathrm{j\:\propto\: T^4}$$

T= absolute temperature.

Emissivity of anybody lies between 0<ε<1. It is 1 for a perfect black body.

Characteristics of Black Body Radiation Spectra

The emissive power of the black body is increased with respect to the increase in temperature for any wavelength.

• Each temperature has a certain range of maximum wavelength λpeak.

• The peak value of the wavelength for different temperature changes. If the temperature increases the peak value of the wavelength shifts towards the shorter wavelength side for decreasing temperature λpeak shifts to the longer wavelength side.

• The radiant power of the blackbody is calculated from the area covered under the graph.

Conclusion

Radiation is the process of energy transfer. If the radiation is done by a black body, it is denoted as black body radiation. Its characteristics are defined by three laws such as Wien’s displacement law, Plank’s law, and Stefan Boltzmann's law. Its spectral representations are also discussed.

FAQs

Q1. What is Kirchhoff's law of thermal radiation?

Ans. A black body that emits or absorbs radiation in thermal equilibrium, has equal emissivity and absorptivity. (i.e) a perfectly black body is a perfect absorber and a perfect emitter.

$$\mathrm{emissivity(\in)= absorptivity (\alpha)}$$

Q2. Give the relation for black body emission power.

Ans. The emission power of the perfect black body is directly proportional to the fourth power of its wavelength.

$$\mathrm{E = \sigma\:T^4}$$

Q3. Differentiate ionizing and non-ionizing radiation.

Ans.

Ionizing radiation	Non-ionizing radiation
High energy to ionize the atom in their path	As energy is low it cannot ionize the atom
When colliding with an atom it ejects the electron from the atom	When colliding with an atom it only makes the electron jump to the higher energy level
Gamma rays, X - rays	UV, IR, etc

Q4. Is the earth a black body?

Ans. Our earth is not a perfect black body. Because a perfect black body should absorb and emit radiation without considering the value of the wavelength. The earth absorbs all electromagnetic radiation but it does not emit all the radiation it absorbs.

Q5. What are all the applications of black body radiation?

Ans. Black body radiation is useful for lighting and heating. The body at high temperature gives radiation in the form of light. It is also used in thermal imaging and it is used in security systems such as alarms, photo sensors, etc.