Wien’s Displacement Law: Formula and Applications

Wien's Law states that objects of different temperatures emit spectra that peak at distinct wavelengths. It was named after the German physicist Wilhelm Wien. Hotter objects emit shorter-wavelength light, which makes them appear blue. Cooler objects, on the other hand, produce longer-wavelength light, making them seem reddish. Learn about Wein's law in depth, including the mathematical representation and numerous possible methods to state the formula, in this brief article.

Key Terms- Wien’s Law, black body radiation,wavelength, temperature emit spectra, thermodynamics, electromagnetic radiation.

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What is Wien's Law Formula?

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Wien's law, also known as Wien's displacement law, was developed in 1893 and asserts that black body radiation has various temperature peaks at wavelengths that are inversely proportional to temperatures.

The following is a mathematical version of the law:

\(\lambda _{max} = \frac{b}{T}\)

where b = 2.8977 x 10³ m.K is the Wien's displacement constant

The temperature is expressed in kelvins (T).

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Wien’s Constant: b (Wien’s displacement constant)

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Wien's constant is a physical constant that defines the relationship between the black body's thermodynamic temperature and wavelength. It's a combination of temperature and the black body's wavelength, which gets shorter as the temperature rises and the wavelength approaches a maximum.

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Wien Displacement Law and Black Body Radiation

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A black body is a physics idealized version of a body that absorbs all electromagnetic radiation that strikes it, regardless of frequency or angle.

What is Black Body Radiation, and how does it work?

A black body must emit radiation at the same rate that it absorbs to maintain thermal equilibrium, and hence it must be a good emitter of radiation, emitting electromagnetic waves of as many frequencies as it can absorb, i.e. all frequencies. Blackbody radiation is the radiation emitted by the blackbody.

A blackbody absorbs all incident radiation and enables all incident radiation to flow through it (no reflected energy) (no energy transmitted through the body). This is true for radiation of all wavelengths and incidence angles. As a result, the blackbody absorbs all incident radiation perfectly.

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Blackbody Radiation Characteristics

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The following laws can be used to explain the characteristics of blackbody radiation:

Wien’s displacement law

Planck’s law

Stefan-Boltzmann law

Wien’s Displacement Law

According to Wien's displacement law,

For varying temperatures, the blackbody radiation curve peaks at a wavelength that is inversely proportional to the temperature.

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Alternate Formulas

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Frequency dependent formula:

\(\upsilon _{max} = \frac{\alpha }{h}kT\approx (5.879 \times 10^{10}\frac{Hz}{k})T\)

where,

k is the Boltzmann constant

h is the Planck’s constant

T is the temperature in kelvin

α is the equivalent value = 2.821

From Planck’s law:

\(\lambda _{max} = \frac{h_{c}}{x}\frac{1}{kT}=\frac{2.8977 \times 10^{6}nm.K}{T}\)

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Regular Application

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Here is the Regular application of The Wein’s Law-

• Light from incandescent bulbs has longer wavelengths when the filament temperature drops, making the light appear redder.
• The temperature of the sun: With a wavelength of 500 nm in the green spectrum, which is within the human eye's sensitive range, one can study the sun's peak emission per nanometre.

List of Important Formula:

1. Wien’s Law Formula

Λmax = b/T

b is the Wien’s displacement constant = 2.8977 x 10³ m.K

T is the temperature in kelvins

2. Planck’s Law

The spectral density of the emission is determined for each wavelength at a given temperature using Planck's law of blackbody radiation.

Planck’s Law Formula:

\(E_\lambda = \frac{8\pi h c}{\lambda^5} \times \frac{1}{exp (\frac{hc}{kT\lambda}) - 1}\)

Eλ is the wavelength

T is the absolute temperature

3. Stefan-Boltzmann Law

The correlation between total energy emitted and absolute temperature is explained by the Stefan-Boltzmann law.

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Example of Wien's Displacement Law

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• A wood fire that is around 1500K hot emits peak radiation at 2000 nm, which is easily deduced. This means that the bulk of the radiation emitted by the wood fire is invisible to the naked eye. This is why, while a bonfire is a great source of warmth, it is a terrible source of light.
• The surface of the sun has a temperature of 5700 K. We can determine the peak radiation output at a wavelength of 500 nm using the Wien displacement law. This colour belongs to the visible light spectrum's green region. Our eyes, it turns out, are extremely sensitive to this wavelength of visible light. We should be grateful for the fact that a disproportionately big amount of the sun's radiation falls within a relatively narrow visible spectrum.
• When a piece of metal is heated, it becomes red hot at first. This is the visible wavelength that is the longest. After further heating, the colour changes from red to orange, then yellow. The metal will glow white when it is at its hottest. The radiation is dominated by shorter wavelengths.

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Things to Remember

• Wein’s Displacement Law helps us to understand the “relationship between the temperature of a blackbody and the wavelength at which it limits the most light.”
• The term Black body can be explained as an ideal substance which absorbes all the light frequancies.
• It can be denoted as- \(\lambda _{max} = \frac{b}{T}\)
• This law helps the space scientists to calculate the temperature of a certain star.
• With the help of this formula we can determine the connection between colour and temperature as well.

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