Wien’s Displacement Law

Wein’s Displacement Law or Wein’s Law is named after the man who discovered it, Wilhelm Wien; a German physicist. Wilhelm Wein has done remarkable work in the field of Radiation, for which he was granted a noble price for physics in 1911. Max Planck who was a colleague of Wien, continued his work on the same topic and gave Wein-Plank Law and further a more general law Planck’s Law of Radiation. Wein’s Displacement Law is a fundamental concept in the study of radiation and describes the relationship between the temperature of an object and the wavelength of its maximum emission of radiation. This short article provides a thorough explanation of Wein’s law, including the mathematical formulation and various ways it can be expressed.

According to Wien’s Displacement Law also called Wein’s Law, the black body’s temperature has an inverse relationship with the wavelength with the highest emissive power. The relationship between the peak wavelength (wavelength with peak emissive power, m) and the temperature of the radiating black body is provided by this law.

In other words, this law implies that the hotter objects emit most of their radiation in the shorter wavelength region of the radiation, so they appear blue in color, and colder objects emit most of their radiation in the longer wavelength spectrum of the radiation, hence they appear redder in color. 

The law is expressed mathematically as follows:

λ_max ∝ 1/ T

λ_max = b / T

Where,

• λ_max is the Wavelength of maximum emission,
• b is th Wien’s displacement constant, and 
• T is the Temperature (in Kelvin).

Alternate Formulas for Wein’s Law

Some formulas which come under Wein’s law are as follows:

Formula for peak frequency of radiation is given by:

Mathematical formula for peak wavelength is given by:

\

Where,

• k_B is the Boltzmann constant which has value of 1.380649 × 10^-23 m² kg s^-2 K^-1
• h is the Planck’s constant which has value of 6.62607015 × 10^-34 m² kg / s,
• T is the Temperature of the surface of the balck body (in Kelvin).

Wein’s Displacement Constant is the physical constant that is represented by b in Wein’s Displacement Law. The value of Wein’s Displacement constant in SI units is 2.878 × 10⁻³ mK or if we change the unit of length to the micrometer, then b ≈ 2898 μm⋅K.

Black Body Radiation

Electromagnetic radiation emitted by the Black Body is called Black Body Radiation. Black Body is an ideal opaque matter which can absorb all the radiation falling onto it. It is also a perfect emitter of radiation which means it can emit radiation of all wavelengths.

Black Body Radiation with Planck’s Law

Planck’s Law or Planck’s Radiation law describes how the temperature of the surface of the black body and the emitted radiation by the black body are related, and the mathematical formula for this law is given as follows:

Where,

• B(ν, T) is the enegery radiated per unit area per unit time form the body,
• ν is Frequency,
• k_b is Boltzmann Constant,
• h is Planck’s Constant, and
• c is speed of light in vacuum.

The formula for Wien’s Displacement law can be derived by differentiating the formula for Planck’s radiation law. The mathematical procedure of the same is given as follows:

From the Planck’s Law of Radiation, we know

Differentiating the above equation, we get

For maximum frequency, 

Now, let u=\frac{h \nu}{k_B T}

Using Lambert’s W-functions to solve for u,

Putting the value of u in our assumption, we get

Comparing it to the Wien’s Law, λ_max = b/T

Thus, 

b =  

Examples of Wien’s Displacement Law include:

• Peak radiation from a wood fire that is around 1500K hot is produced at 2000 nm, as is evident. In other words, the majority of the radiation that the wood fire emits is not visible to the unaided eye. Because of this, a campfire is a terrible source of light while being a fantastic source of heat.
• When a piece of metal is heated, it first turns red hot. The longest wavelength in the visible spectrum is this one. The color shifts from red to orange to yellow with additional heating. At its hottest, the metal will shine white. The shorter wavelengths of light are dominant.
• In an Incandescent light bulb, the heat produced by the filament is directly proportional to the voltage, in the case of voltage drops temperature of the filament also drops with it. Which can be seen as the bulb looking red compared to its previous state.

This law holds a very high significance in the study of radiation, some of its useful aspects are as follows:

• The law explains the relationship between the blackbody’s temperature and the peak wavelength that is emitted.
• Knowing the peak wavelength that a body emits makes it simple to estimate the approximate temperature of hotter bodies.
• It implies that black bodies emitting peak emissive power at lower wavelengths are hotter than black bodies emitting peak emissive power at higher wavelengths.

Read More,

• Radiation
• Radiation Pressure
• Types of Radioactivity

Example 1: The North star emits energy with a wavelength of 410 nm, while the sun emits light at a maximum intensity of 621 nm. What is the ratio of the surface temperatures of the sun and the north star, if these stars behave like black bodies?

Answer:

According to Wein’s Displacement Law, we know 

λ_maxT = constant

⇒ T(S) / T(N) = λ_m(N) / λ_m(S)

⇒ T(S) / T(N) = 410 / 621

⇒ T(S) / T(N) = 0.66

Example 2: Determine the maximum amount of solar radiation using the assumption that the sun’s surface temperature is 5800 K. Where does this value fall on the electromagnetic spectrum? (b = 2.897 × 10^-3 m K).

Answer:

We have,

λ_max = b / T

⇒ λ_max = 2.897 × 10^-3 / 5800

⇒ λ_max = 4.995 × 10^-7 m 

⇒ λ_max = 4995 Ă

Example 3: A black body has a wavelength when it is 3510 K in temperature. Its comparable wavelength will be at a temperature of 4100 K.

Answer:

According to Wien’s displacement law, the black body radiation curve peaks for various temperatures at a wavelength that is inversely proportional to the temperature. The typical wavelength is shown to be the wavelength with the highest intensity. i.e.,

λT = constant

⇒  λ × 3510 = λ’ × 4100

⇒  λ’ = 3510 / 4100 λ

⇒ λ’ = 0.85 λ

Example 4: Radiation from stars has a maximal wavelength of 10^-5 m. Identify the star’s rough temperature.

Answer:

We have,

λ_mT = 2.897 × 10^-3

⇒ T = 2.897 × 10^-3 / 10^-5

⇒ T = 2.897 × 10² K

Example 5: Consider that the temperature of the earth is 197 K. Analyze the energy that the planet is emitting at its peak wavelength. 

Answer:

We have,

λ_mT = 2.897 × 10^-3

⇒ λ_m = 2.897 × 10-3 / 197

⇒ λ_m = 0.014 × 10^-3 m

Q1: What is the Wien’s Displacement Law?

Answer:

According to the Wiens Displacement Law, a black body’s temperature has an inverse relationship with the wavelength at which it produces its peak energy.

Q2: What is Wien’s Constant?

Answer:

The physical constant known as the Wien’s constant (b) controls the relationship between the black body’s maximum wavelength and its absolute thermodynamic temperature. It is a result of the black body’s temperature and wavelength, which gets shorter as the temperature rises to its maximum.

Q3: What makes Wien’s Law significant?

Answer:

The relationship between the black body’s temperature and the wavelength at which it releases its peak energy is given by Wien’s law.

Q4: Who discovered Wein’s Displacement Law?

Answer:

Wilhelm Wien, a German physicist, discovered this law in 1893.

Q5: What is a Black Body?

Answer:

A black body is an idealized object that absorbs all radiation incident on it and emits radiation according to its temperature. It is an idealized object because no real object can absorb or emit all radiation perfectly.

Q6: What is the Formula for Wein’s Displacement Law?

Answer:

The formula for Wein’s Displacement Law is: 

λ_maxT = b, 

Where, 

• λ_max is the Wavelength of maximum emission,
• b is th Wien’s displacement constant, and 
• T is the Temperature (in Kelvin).

Q7: What are the Applications of Wein’s Displacement Law?

Answer:

Wein’s Displacement Law is very useful in infrared spectroscopy and thermal imaging.

Q8: Does Wein’s Displacement Law apply only to black bodies?

Answer:

No, Wein’s Displacement is applied to all the real object which emits radiation.

Q9: What is Relation between the Peak Wavelength and Temperature of an Object?

Answer:

The relationship between the peak wavelength and the temperature of an object is inversely proportional, according to Wein’s displacement law.