What is Wien's displacement law?

Physicist Wilhelm Wien gave Wien's displacement law in 1893. This law states the relation between wavelength and temperature for a blackbody. The wavelength (maximum) of a blackbody has an inverse relationship with its absolute temperature. Also, there is no uniform distribution of energy radiated by the blackbody.

What is the formula of Wien's displacement law?

The formula for Wien's displacement law is:

${\lambda }_{max}=\frac{b}{T}$

Here, b is the Wien's constant, and T is the absolute temperature.

Also, ${\lambda }_{max}$ is the peak wavelength that gives the maximum radiation a blackbody emits.

The value of Wien's constant is $b=2.9\times {10}^{-3} \mathrm{mK}$.

Example of Wien's displacement law

An example of Wien's displacement law is an iron rod. When you heat the iron rod on a flame, it will first show dull red color. Later, it will change to reddish yellow and finally become white. This color change satisfies Wien's law. When you increase the temperature, the energy emission is the maximum, and the wavelength becomes smaller.

Importance of Wien's displacement law in Physics

In Physics, Wien's displacement law plays a vital role in the study of stars. This law helps determine the surface temperatures of the moon, the sun, and the other stars.

For example, moonlight has peak intensity at a wavelength of $14\times {10}^{-6} \mathrm{m}$. By Wien's displacement law, the temperature of the moon's surface can be found to be 200 K. When you consider the case of the sun, the intensity of solar radiation is the maximum at $48\times {10}^{-8} \mathrm{m}$. By Wien's displacement law, the temperature of the sun’s surface is 6,060 K.

Graph showing Wien's displacement law

How does blackbody radiation explain Wien's displacement law?

A blackbody is the one that absorbs almost all radiations incident on it. It can absorb and emit radiation at all frequencies. The radiation that is emitted by a blackbody is called blackbody radiation. The electromagnetic wave theory does not explain this radiation. However, the quantum theory of radiation has been able to provide a suitable explanation.

When you heat a piece of metal of a high melting point, its atoms vibrate and emit radiations with a certain frequency. The frequency of the emitted radiation increases on further heating the body. This change in frequency can be seen in terms of change in color from red to yellow, and finally white. This change in the color of the radiations emitted from the blackbody signifies the importance of Wien's law, in which there is a decrease in wavelength on the increase in temperature.

Sketch of blackbody radiation curve showing Wien's displacement law

Difference between Stefan-Boltzmann Law and Wien's displacement law

According to the Stefan-Boltzmann law, there is a direct relationship between the total energy emitted by a perfect blackbody per second per unit area and the fourth power of the absolute temperature.

The formula for Stefan-Boltzmann law is $H=EA\sigma T^4$.

Here, E is the rate of radiated energy, and A is the surface area, and T is the absolute temperature.

The Stefan-Boltzmann constant is a universal constant, and its value is $\sigma =5.67\times {10}^{-8} {\mathrm{Js}}^{-1}{\mathrm{m}}^{-2}{\mathrm{K}}^{-4}$.

Both Stefan-Boltzmann law and Wien's displacement law are defined for a blackbody. However, both are conceptually different. The former states the relationship between the total energy radiated by a blackbody and its temperature. On the other hand, the latter states the relationship between the wavelength of radiation emitted by a blackbody and its temperature.

Derivation of Wien's law from Planck's law

In 1901, the quantum theory of radiation was given by Max Planck. This theory successfully explains blackbody radiation and the photoelectric effect.

The key features of Planck quantum theory of radiation are:

• The energy radiated by a blackbody in the form of energy packets is called quanta.
• The energy of each quantum depends on the frequency of the incident of radiation.

This theory was further extended by Einstein in 1905. He explained the photoelectric effect, and got the Nobel Prize for the same. He considered light to be made of packets of particles called photons. The energy of each photon is directly proportional to the frequency of incident radiation.

The formula for quantum energy is E=hf.

Here, h is Planck's constant with the value h=6.626×10-34 J s, and f is the frequency of incident light.

However, the electromagnetic theory fails to explain Wein's displacement law. With the help of the quantum theory of radiation, Wilhelm Wien derived his displacement law.

From the concepts of quantum mechanics, the formula for optical frequency is

$f_{peak}=\frac{\alpha kT}{h}$

Here,

h = Planck's Constant

k = Boltzmann Constant

T is the absolute temperature in Kelvin

α is the maximization constant

From the above equation,

$h{f}_{peak}=\alpha kT$                                   ...(1)

From equation (1), frequency is found to be directly proportional to temperature.

From the relation between wavelength and frequency, we know:

$v=f\lambda$                                                ...(2)

From equation (2), frequency is found to be inversely proportional to wavelength.

From equations (1) and (2),

λT=constant                                     ...(3)

Equation (3) states Wien's displacement law.

Limitations of Wien's displacement law 

The limitation to Wien's displacement law indicates that it fails in the case of blackbody radiations of longer wavelengths. When the body's temperature is decreased, it is not possible to obtain a continuous Wein curve.

Common Mistakes

Students often get confused between Wien's displacement law and Planck's radiation law. Although both are historically related, yet they are conceptually different. Planck's law helps determine the spectrum of a blackbody, while Wien's law determines the relationship between peak wavelength and temperature.

Students also sometimes confuse between Wien's displacement law and the Stefan-Boltzmann law. The Stefan-Boltzmann law states the relationship between the total energy radiated by a blackbody and its temperature. On the other hand, Wien's law states the relationship between the wavelength of blackbody radiation and its temperature.

Context and applications

This topic is significant in the professional exams for both graduate and undergraduate courses such as:

• Bachelor of Technology in Chemical Engineering
• Bachelor of Science in Physics
• Master of Science in Chemistry
• Master of Science in Applied Chemistry
• Quantum Mechanics

Related Concepts

• Blackbody Radiation
• Stefan-Boltzmann Law
• Blackbody Spectrum
• Hydrogen Spectrum
• Quantum Theory of Radiation
• Planck's Quantum Theory

Practice Problems

Q.1 Which of the following statements is correct regarding Wien's displacement law?

a) The peak wavelength of a radiation is directly proportional to its absolute temperature.

b) The peak wavelength of a radiation is inversely proportional to its absolute temperature.

c) The peak wavelength of a radiation is independent of its absolute temperature.

d) None of these.

Correct option: (b)

Q.2 Name the scientist who first derived Wien's displacement law.

a) Max Planck

b) Albert Einstein

c) Stefan

d) Wilhelm Wien

Correct option: (d)

Q.3 According to Wien's displacement law:

a) ${\lambda }_{max}T=cons\tan t$

b) $\frac{{\lambda }_{max}}{T}=cons\tan t$

c) ${\lambda }_{max}{T}^2=cons\tan t$

d) ${\lambda }_{max}2T=cons\tan t$

Correct option: (a)

Q.4 What is the value of Wien's constant?

a) $2.9\times {10}^{-3} \mathrm{mK}$

b) $2.9\times {10}^{-6} \mathrm{mK}$

c) $4.9\times {10}^{-3} \mathrm{mK}$

d) $4.9\times {10}^{-6} \mathrm{mK}$

Correct option: (a)

Q.5 When a piece of iron is heated, it first becomes dull red. Later on, it changes to reddish yellow and finally to white hot. Which of the following laws correctly explains this?

a) Kirchhoff's Law

b) Newton's law of cooling

c) Wien's displacement law

d) Einstein's photoelectric effect

Correct option: (c)