## Wien’s Displacement Law

Wein’s law is also known as the Wein’s displacement law, it is named after German Physicist Willhelm Wein in honour 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 radiations play 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 radiations 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 it 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 corresponding 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

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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 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, dominate 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.

### 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.

Ans:

Given,

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

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}

\[\Rightarrow T=\frac{2.88\times10^{-3}}{500\times10^{-9}}=5760K\]

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 4750A^{0}. The Temperature of a Star for Which this Maximum is at 9500A^{0} is?

Ans:

Given,

The maximum wavelength for the surface temperature of the sun = λ_{1 }= 4750 A^{0}

The Absolute temperature of the surface of the sun = T_{1 }= 6050 K

The maximum wavelength of a star = λ_{2 }= 9500 A^{0}

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,

⇒ λ_{1}T_{1 }= b…..(2)

⇒ λ_{2}T_{2 }= b……(3)

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

⇒ λ_{1}T_{1}=λ_{2}T_{2}

\[\Rightarrow T_{2}=\frac{\lambda _{1}T_{1}}{\lambda _{2}}\]……….(4)

Substituting given values in equation (4) we get,

\[\Rightarrow T_{2}=\frac{(4750)(6050)}{9500}=3025K\]

Therefore, the temperature of the given star for the maximum wavelength of 9500A^{0 }is 3025 K.

3. Two Objects Emit Maximum Radiations at 4500A^{0}and at 1500 A^{0 }then Calculate the Ratio of the Temperature of the Stars.

Sol:

Given,

The maximum wavelength of the first star=4500 A^{0}

The maximum wavelength of the second star=1500 A^{0}

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,

⇒ λ_{1}T_{1 }= b…..(2)

⇒ λ_{2}T_{2 }= b……(3)

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

⇒ λ_{1}T_{1}=λ_{2}T_{2}

\[\Rightarrow \frac{T_{2}}{T_{1}}=\frac{\lambda _{1}}{\lambda _{2}}\]……….(4)

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

\[\Rightarrow \frac{T_{2}}{T_{1}}=\frac{4500}{1500}=\frac{3}{1}\]

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

1. What is Wien's Displacement Law? What is the Use of Wien’s Law?

Ans: Wein’s displacement law states that the product of maximum wavelength corresponding to maximum intensity and the absolute temperature is constant. The important application of Wien's law is to determine the temperature of astronomical objects. Wien’s law is used in designing remote sensors, incandescent light etc...

2. What is Wien’s Constant?

Ans: 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.