Wien’s Law

Wein’s law is also known as the Wein’s displacement law, it is named after German Physicist Willhelm Wein in honor of his extraordinary contribution in explaining black body radiations. Wein’s law gives us a relationship between the wavelength of light that corresponds to the highest intensity and the absolute temperature of the object. 

In other words, Wein’s displacement law explains the fact that objects emit different wavelengths from the spectra at different temperatures. For example, Hotter objects emit shorter wavelengths hence they appear Reddish whereas colder objects emit long wavelengths hence they appear blue.

State Wien’s Displacement Law

At the beginning of quantum mechanics, the biggest challenge physicists faced was explaining the wave nature of atoms. Black body radiation plays an important role in quantum mechanics. Black bodies are those bodies that absorb all the radiations at absolute zero temperature i.e., there will be no transmission or emission of radiations. In explaining black body radiations many scientists have given their contributions. 

Max Planck explained black body radiation quantum mechanically, Rayleigh-Jeans and Wein’s gave special cases to Planck’s law. Wein’s law was developed for shorter wavelengths and Rayleigh-Jeans explained it for longer wavelengths.  But Wein’s law was developed way before Max Planck’s explanation. 

Wein’s explained the distribution of wavelengths of the black body with respect to the energies for shorter wavelengths, but it was not in good approximations for longer wavelengths. Later Planck’s law corrected this and gave a universal that was acceptable even for the longer wavelengths. Therefore, Wein’s displacement law is considered to be a special case of Planck’s law.

Now, to state Wien’s displacement law we must be aware of the concept of black body radiation and quantum mechanics. Wein’s law is explained for perfect black bodies and Wein’s observed that the intensity of energy radiated by a blackbody is not distributed uniformly over all the wavelengths but is maximum for a particular wavelength (λₘ). 

Now, Wien’s displacement law definition or Wein’s displacement law states that the product of maximum wavelength corresponds to maximum intensity and the absolute temperature is constant.

Mathematically we write,

⇒ λₘT = b…….(1)

Where,

λₘ - The maximum wavelength corresponding to maximum intensity

T - The absolute temperature

b - Wein’s Constant and the value of Wein’s constant is given by 2.88 x 10-3 m-K or 0.288 cm-K

(Image will be Updated soon)

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Equation (1) is known as the Wien's law formula or Wien’s displacement law formula. The value of maximum wavelength corresponding to maximum emission will be decreasing with increasing absolute temperature. The Wien’s constant (b) is a Physical constant determining the relationship between the thermodynamic absolute temperature of the black body and the maximum wavelength and it is denoted by b. It is a product of the temperature and wavelength of the black body which grows shorter as the wavelength reaches a maximum with temperature. Students will encounter this topic as Wien's displacement law class 11 in their academic syllabus.

Wien's Displacement Law Derivation

William Wiens used thermodynamics to explain the distribution of wavelengths according to energies emitted by the radiations and called it Wien’s law of distribution. Wien’s distribution says that energy distribution varies as a function of λ-5.

For the short values of λ, the exponential factor becomes large and contributes more which overcomes the other factor λ-5. This means that at shorter wavelengths E increases with λ. On the other hand at higher λ the exponential factor is very small. In this range, dominant mostly and hence E should be decreased at higher λ.

At first sight, we find Wien's law good to explain the blackbody radiation curve, But compare the curve plotted by Wien's distribution law with the experimental one. As we see that in the shorter A range Wien's law fits very well but we find a difference between these curves in the higher A range. This implies an error in the theoretical distribution law which is too large to ascribe to experimental uncertainties and indicates a flaw in the theory. Wien could neither explain the failure of his relation nor supply a better one.

Although Wien's law does not hold good for the complete explanation one can deduce the maximum spectral emissive power dependence on temperature by this as follows-

From Wien's displacement law we have at λ = λₘ,  λₘT = b

Where,

λₘ - The maximum wavelength corresponding to maximum intensity

T - The absolute temperature

b - Wein’s Constant and the value of Wein’s constant is given by 2.88 x 10-3 m-K or 0.288 cm-K

Importance of Wien’s Displacement

We can determine the temperature of astronomical objects using Wien’s displacement law. It is used in designing remote sensors. Other applications of Wien’s displacement law are given by:

• Incandescent Bulb Light: With the decrease in temperature of the filament, wavelengths are longer making light appear redder.

• The Temperature of the Sun: One can study the peak emission per nanometres of the sun with a wavelength of 500 nm in the green spectrum which is in the human eye sensitive range.

Wien's Law and the Stefan Boltzmann Law?

The peak wavelength of a blackbody is inversely proportional to its temperature, according to Wien's law. The power emitted by a blackbody is proportional to the fourth power of temperature, according to the Stefan-Boltzmann equation.

Planck’s Quantum Theory

Max Planck, a German physicist, proposed Planck's radiation law in 1900 to explain the spectral-energy distribution of radiation emitted by a blackbody. The quantum theory of Planck explains how radiation is emitted and absorbed. Planck believed that the sources of radiation are oscillating atoms and that each oscillator's vibrational energy can take any of a number of discrete values, but never any value in between. The following are the postulates of Planck's quantum theory.

Matter emits or absorbs energy in discrete quantities in the form of little packets or bundles in a discontinuous manner.

Quantum energy is the smallest bundle or packet of energy. A photon is a quantum of light in the case of light.

In whole number multiples of a quantum, a body or substance can emit or absorb energy as nhv. In this case, n is a positive number. As a result, energy can be absorbed or radiated in the form of hv, 2hv, 3hv, 4hv, etc., rather than 1.5hv, 2.5hv, etc.

The frequency of the radiation is exactly proportional to the energy of the quantum received or released.

Day-to-day Application of Wien’s Law

Incandescent bulb light - As the filament's temperature drops, wavelengths lengthen, making the light look redder.

The temperature of the sun - With a wavelength of 500 nm in the green spectrum, which is in the human eye's sensitive range, one may analyze the sun's peak emission per nanometres.

Examples

1. If Light From the Sun is Found to Have a Maximum Intensity Near the Wavelength of 500nm. Determine the Temperature of the surface of the Sun.

Given,

The maximum wavelength corresponding to the maximum intensity of light=λₘ=500 nm

Now, we are asked to estimate the temperature of the surface of the sun.

From Wien’s displacement law we have,

⇒ λₘT = b…….(1)

Where,

λₘ - The maximum wavelength corresponding to maximum intensity

T - The absolute temperature

b - Wein’s Constant and the value of Wein’s constant is given by 2.88 x 10-3 m-K or 0.288 cm-K

Substituting the corresponding values in the equation (1), we get:

⇒ (500 x 10-9)T = 2.88 x 10-3

\[\Rightarrow T = \frac{2.88\times 10-3}{500\times 10-9} = 5760 K\]

Therefore the temperature of the surface of the sun is 5760K. 

2. The Spectral Energy Distribution of the Sun (Temp = 6050 K) Has a Maximum at 4750A0. The Temperature of a Star for Which this Maximum is at 9500A0 is?

Given,

The maximum wavelength for the surface temperature of the sun = λ1 = 4750 A0

The Absolute temperature of the surface of the sun = T1 = 6050 K

The maximum wavelength of a star = λ2 = 9500 A0

We aim to calculate the temperature of the start at which the maximum wavelength is determined.

Now, from Wien’s law, we know that,

⇒ λₘT = b…….(1)

Where,

λₘ - The maximum wavelength corresponding to maximum intensity

T - The absolute temperature

b - Wein’s Constant and the value of Wein’s constant is given by 2.88 x 10-3 m-K or 0.288 cm-K

According to the given data, we write,

⇒ λ1T1 = b…..(2)

⇒ λ2T2 = b……(3)

From equation (2) and equation (3) we get,

⇒ λ1T1=λ2T2  

\[\Rightarrow T2 = \frac{\lambda_{1} T_{1}}{\lambda_{2}}\]......(4)

Substituting given values in equation (4) we get,

\[\Rightarrow T2 = \frac{(4750)(6050)}{9500} = 3025K\]

Therefore, the temperature of the given star for the maximum wavelength of 9500A0 is 3025 K.

3. Two Objects Emit Maximum Radiations at 4500A0and at 1500 A0 then Calculate the Ratio of the Temperature of the Stars.

Given,

The maximum wavelength of the first star=4500 A0

The maximum wavelength of the second star=1500 A0

We are asked to determine the ratio of temperatures. So according to Wien’s displacement law statement we have,

⇒ λₘT = b…….(1)

Where,

λₘ - The maximum wavelength corresponding to maximum intensity

T - The absolute temperature

b - Wein’s Constant and the value of Wein’s constant is given by 2.88 x 10-3 m-K or 0.288 cm-K

According to the given data, we write,

⇒ λ1T1 = b…..(2)

⇒ λ2T2 = b……(3)

From equation (2) and equation (3) we get,

⇒ λ1T1=λ2T2

⇒ T₂T₁=λ₁λ₂⇒T₂T₁=λ₁λ₂……….(4)

Substituting given values in the equation (4) we get,

⇒ T₂T₁=45001500=31⇒T₂T₁=45001500=31

Therefore the ratio of the temperature of the stars is 3:1.