Wien’s Law

According to Wien’s Law, objects with varying temperatures emit spectra with peak wavelengths at specific locations. It bears the name Wilhelm Wien after the German physicist. Shorter-wavelength light is emitted by hotter objects and gives them their blue color. Conversely, cooler things appear reddish because they emit longer-wavelength light. This short article provides a thorough explanation of Wein’s law, including the mathematical formulation and various ways it can be expressed.

Wien’s Law

According to Wien’s displacement 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.

With a rise in the black body’s absolute temperature (T), the wavelength at which it emits the most monochromatic emissive power drops.

Wien’s Displacement Constant

The 1893 invention of Wien’s law, also known as Wien’s displacement law, states that black body radiation exhibits a variety of temperature peaks at wavelengths that are inversely proportional to temperatures.

The law is expressed mathematically as follows:

λ_max ∝ b / T

OR

λ_max = b / T

Where,

• λ_max = Wavelength at which the blackbody dominantly radiates,
• b = Wien’s constant and its value is 2.897 × 10^-3 m K,
• T = Temperature (kelvin).

Wien’s Displacement Law Derivation

In order to explain the distribution of wavelengths in accordance with the energy released by the radiations, William Wiens employed thermodynamics. This explanation is known as Wien’s law of distribution. Energy distribution varies as a function of λ^-5, according to Wien’s distribution.

For the small values of, the exponential factor grows and makes a larger contribution, which outweighs the other component, λ^-5. As a result, E rises with at shorter wavelengths. However, the exponential factor becomes very little as the increases. E should be reduced at higher because dominant is predominant in this range.

At first glance, we think that Wien’s law explains the blackbody radiation curve rather well. But contrast the experimental curve with the one predicted by Wien’s distribution law. We can see that there is a difference between both curves in the larger A range because Wien’s rule fits rather well in the shorter A range. This indicates that the distribution rule has a theoretical flaw that is too substantial to be ascribed to experimental uncertainties. Wien was unable to provide a more suitable or appropriate reason for the breakdown of his relationship.

The maximum spectrum emissive power dependence on temperature can be determined using this in the following manner, even though Wien’s law does not apply to the entire explanation:

We have at from Wien’s displacement law that

λ = λ_m, λ_mT = b

Relevance of the Wien’s Law in Real Life

With the use of Wien’s displacement law, we may calculate the temperatures of celestial objects. When creating remote sensors, it is employed. Following are some further applications of how to use Wien’s displacement law:

• Incandescent Bulb Light: Since the filament’s temperature has dropped, the longer wavelengths make light look redder.
• Sun temperature: With a wavelength of 500 nanometers, which is in the green spectrum and within the range of human vision sensitivity, one can study the sun’s peak output per nanometer.

Significance of Wien’s Law

These are the significance of this law:

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

Solved Examples on Wien’s Law

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:

We have,

λ_mT = 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:

We have,

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.

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

FAQs on Wien’s Law

Question 1: What is the Wien’s 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.

Question 2: 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.

Question 3: 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.

Question 4: Write Examples of Wien’s Displacement Law.

Answer:

Examples of Wien’s Displacement Law:

1. 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.
2. When a piece of metal is heated, it first turns red hot. The longest wavelength in the visible spectrum is this one. The colour shifts from red to orange to yellow with additional heating. At its hottest, the metal will shine white. The shorter wavelengths of the light are dominant.

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