Wien's Displacement Law: Understanding the Relationship Between Temperature and Emission

Introduction

Wien’s displacement law, formulated by Wilhelm Wien in 1893, is a fundamental principle in physics that describes the relationship between the wavelength of the peak emission of a black body and its temperature. According to this law, as the temperature of a black body increases, the wavelength at which it emits the most radiation shifts towards shorter wavelengths. This means that hotter objects emit more energy at shorter wavelengths, such as in the visible or ultraviolet range, while cooler objects emit more energy at longer wavelengths, such as in the infrared range.

Key Takeaways

Temperature (K)	Wavelength of Peak Emission (nm)
1000	2.897 × 10^6
2000	1.4485 × 10^6
3000	965,000
4000	724,000
5000	579,400

(Note: The values in the table are for illustrative purposes only and may not represent actual measurements.)

Understanding Wien’s Displacement Law

Wien displacement law — Image by Thomas-sk – Wikimedia Commons, Wikimedia Commons, Licensed under CC BY-SA 4.0.

Definition of Wien’s Displacement Law

Wien’s Displacement Law is a fundamental principle in physics that describes the relationship between the temperature of a black body radiator and the wavelength at which it emits the most intense radiation. It provides insights into the temperature dependence of the peak wavelength of electromagnetic radiation emitted by a black body.

Statement of Wien’s Displacement Law

Wien’s Displacement Law states that the peak wavelength of radiation emitted by a black body is inversely proportional to its temperature. Mathematically, it can be expressed as:

$lambda_{text{peak}} = frac{k}{T}$

Where:
– (lambda_{text{peak}}) is the peak wavelength of radiation emitted by the black body,
– (k) is Wien’s displacement constant ((2.898 times 10^{-3} , text{m} cdot text{K})),
– (T) is the temperature of the black body in Kelvin.

According to Wien’s Displacement Law, as the temperature of a black body increases, the peak wavelength of its emitted radiation shifts towards shorter wavelengths. This means that at higher temperatures, the black body emits more radiation in the ultraviolet and visible light regions, while at lower temperatures, it emits more in the infrared region.

The relationship described by Wien’s Displacement Law is a consequence of Planck’s law and the Stefan-Boltzmann law, which govern the behavior of black body radiation. Planck’s law describes the spectral distribution of radiation emitted by a black body, while the Stefan-Boltzmann law relates the total power radiated by a black body to its temperature.

Wien’s Displacement Law has important implications in various fields. For example, it helps explain the colors we perceive from different sources of light. The peak wavelength of radiation emitted by a light source determines its color. For instance, a light source with a peak wavelength in the red region of the spectrum appears reddish, while a light source with a peak wavelength in the blue region appears bluish.

Additionally, Wien’s Displacement Law is used in the parameterization of the solar spectrum. By analyzing the distribution of radiation wavelengths emitted by the Sun, scientists can determine its temperature and other important characteristics. This knowledge is crucial for understanding the behavior of our closest star and its impact on Earth.

Origin and History of Wien’s Displacement Law

Wien’s Displacement Law, also known as Wien’s Law, is a fundamental principle in physics that describes the relationship between the temperature of a black body radiator and the wavelength at which it emits the most radiation. This law was first formulated by the German physicist Wilhelm Wien in the late 19th century.

Why is Wien’s Displacement Law So Called?

Wien’s Displacement Law is named after Wilhelm Wien, who derived this relationship while studying the properties of thermal radiation emitted by heated objects. He observed that as the temperature of a black body radiator increases, the peak wavelength of its emitted radiation shifts to shorter wavelengths. This phenomenon is known as “displacement” because the peak of the radiation distribution moves towards the higher frequency end of the electromagnetic spectrum.

Wien’s Displacement Law from Planck’s Law

Wien’s Displacement Law can be derived from Planck’s Law, which describes the spectral distribution of electromagnetic radiation emitted by a black body at a given temperature. Planck’s Law states that the intensity of radiation at a specific wavelength is proportional to the reciprocal of the wavelength raised to the power of five, multiplied by a constant.

Wien’s Displacement Law provides a more specific relationship between the peak wavelength (λ_max) of the radiation distribution and the temperature (T) of the black body radiator. It can be mathematically expressed as:

$quicklatex.com addf04de85045a93ea9afcfee2d0f7a2 l3$ max = frac{b}{T}” title=”Rendered by QuickLaTeX.com” height=”26″ width=”250″ style=”vertical-align: -7px;”/>

Where λ_max is the peak wavelength, T is the temperature in Kelvin, and b is Wien’s constant.

Wien’s constant (b) is approximately equal to 2.898 × 10^-3 Kelvin-meter. It represents the proportionality between the peak wavelength and the temperature of the black body radiator.

By using Wien’s Displacement Law, we can determine the peak wavelength of the radiation emitted by a black body at a given temperature. This law is particularly useful in understanding the temperature dependence of the color of light emitted by various sources. For example, at lower temperatures, the peak wavelength is in the infrared region, resulting in a reddish glow. As the temperature increases, the peak wavelength shifts towards the visible light spectrum, giving rise to different colors. At even higher temperatures, the peak wavelength can reach the ultraviolet region.

Mathematical Representation of Wien’s Displacement Law

Wien’s Displacement Law Formula

Wien’s Displacement Law is a fundamental principle in physics that describes the relationship between the temperature of a black body radiator and the wavelength at which it emits the most radiation. The mathematical representation of Wien’s Displacement Law is expressed by the equation:

$lambda_{text{max}} = frac{k}{T}$

Where:
– ( lambda_{text{max}} ) represents the peak wavelength at which the black body radiation is emitted,
– ( k ) is Wien’s constant, and
– ( T ) is the temperature of the black body radiator.

Wien’s Displacement Law is Expressed by the Equation

Wien’s Displacement Law states that the peak wavelength of the electromagnetic radiation emitted by a black body radiator is inversely proportional to its temperature. As the temperature of the black body radiator increases, the peak wavelength decreases. This relationship is mathematically represented by the equation mentioned above.

To understand the significance of this equation, let’s break it down further:

( lambda_{text{max}} ) represents the peak wavelength of the radiation emitted by the black body. The peak wavelength is the wavelength at which the intensity of radiation is maximum. It determines the color of the light emitted by the black body.
( k ) is Wien’s constant, which has a value of approximately ( 2.898 times 10^{-3} , text{m} cdot text{K} ). It is a fundamental constant in physics that relates the peak wavelength to the temperature.
( T ) represents the temperature of the black body radiator in Kelvin (K). The temperature dependence in the equation shows that as the temperature increases, the peak wavelength decreases. This means that at higher temperatures, the black body emits radiation with shorter wavelengths, shifting towards the ultraviolet region. Conversely, at lower temperatures, the black body emits radiation with longer wavelengths, shifting towards the infrared region.

Wien’s Displacement Law is closely related to other principles in physics, such as the Stefan-Boltzmann law, Planck’s law, and the distribution of thermal radiation. It provides valuable insights into the behavior of electromagnetic radiation and helps us understand the relationship between temperature and the peak wavelength of black body radiation.

By applying Wien’s Displacement Law, we can determine the peak wavelength of radiation emitted by various sources. For example, the Sun has a peak wavelength in the visible light range, around 500 nm. This parameterization of the Sun’s radiation distribution gives it a yellowish color. On the other hand, a fluorescent light bulb has a peak wavelength in the ultraviolet range, around 365 nm. This parameterization of the fluorescent light’s radiation distribution gives it a bluish color.

Derivation and Proof of Wien’s Displacement Law

Wien Displacement — Image by Livinus – Wikimedia Commons, Wikimedia Commons, Licensed under CC BY-SA 3.0.

Wien’s Displacement Law Derivation

Wien’s Displacement Law is an important concept in the study of black body radiation. It describes the temperature dependence of the peak wavelength of electromagnetic radiation emitted by a black body. The law states that the product of the peak wavelength and the temperature of the black body is a constant.

To derive Wien’s Displacement Law, we start with Planck’s law, which describes the spectral distribution of thermal radiation emitted by a black body. Planck’s law states that the spectral radiance of a black body is given by the equation:

$B(lambda, T) = frac{{2hc^2}}{{lambda^5}} cdot frac{1}{{e^{left(frac{{hc}}{{lambda kT}}right)} - 1}}$

In this equation, (B(lambda, T)) represents the spectral radiance at a given wavelength (lambda) and temperature (T), (h) is Planck’s constant, (c) is the speed of light, and (k) is the Boltzmann constant.

To find the peak wavelength, we need to determine the wavelength at which the spectral radiance is maximum. This can be done by taking the derivative of Planck’s law with respect to wavelength and setting it equal to zero:

$frac{{dB}}{{dlambda}} = 0$

Simplifying this equation, we get:

$frac{{d}}{{dlambda}}left(frac{{2hc^2}}{{lambda^5}} cdot frac{1}{{e^{left(frac{{hc}}{{lambda kT}}right)} - 1}}right) = 0$

After performing the differentiation and simplification, we arrive at the following equation:

$frac{{-10hc^2}}{{lambda^6}} cdot frac{1}{{e^{left(frac{{hc}}{{lambda kT}}right)} - 1}} + frac{{2hc^2}}{{lambda^5}} cdot frac{{hc}}{{lambda^2kT}} cdot frac{{e^{left(frac{{hc}}{{lambda kT}}right)}}}{{left(e^{left(frac{{hc}}{{lambda kT}}right)} - 1right)^2}} = 0$

Simplifying further, we obtain:

$-10 + 2left(frac{{hc}}{{lambda kT}}right) = 0$

Solving for (lambda), we find:

$lambda = frac{{hc}}{{5kT}}$

This equation represents Wien’s Displacement Equation, which relates the peak wavelength (lambda) to the temperature (T) of the black body.

Wien’s Displacement Law Proof

To prove Wien’s Displacement Law, we can use the Stefan-Boltzmann law, which states that the total power radiated by a black body is proportional to the fourth power of its temperature. The Stefan-Boltzmann law is given by the equation:

$P = sigma cdot A cdot T^4$

In this equation, (P) represents the power radiated, (sigma) is the Stefan-Boltzmann constant, (A) is the surface area of the black body, and (T) is its temperature.

By substituting the expression for the peak wavelength from Wien’s Displacement Equation into the Stefan-Boltzmann law, we can derive the relationship between the peak wavelength and the temperature.

$P = sigma cdot A cdot T^4$

$P = sigma cdot A cdot left(frac{{hc}}{{5klambda}}right)^4$

Simplifying further, we get:

$P = sigma cdot A cdot frac{{h^4c^4}}{{625k^4lambda^4}}$

Since the power radiated is directly proportional to the energy radiated, we can write:

$P propto frac{{h^4c^4}}{{lambda^4}}$

This equation shows that the power radiated is inversely proportional to the fourth power of the peak wavelength. Therefore, as the temperature of the black body increases, the peak wavelength decreases.

This relationship between the peak wavelength and temperature is known as Wien’s Displacement Law. It explains why objects at higher temperatures emit radiation with shorter wavelengths, such as infrared radiation, visible light, and even ultraviolet radiation.

Wien’s Displacement Law is a fundamental concept in understanding the distribution of electromagnetic radiation emitted by thermal sources. It provides a parameterization for the peak wavelength of radiation emitted by objects at different temperatures, allowing us to give a quantitative description of the color of light emitted by various sources. For example, a source with a peak wavelength in the infrared range would appear reddish, while a fluorescent light with a peak wavelength in the visible range would emit white light.

Wien’s Displacement Law Constants

Practicalplanckerp — Image by Anton~commonswiki – Wikimedia Commons, Wikimedia Commons, Licensed under CC BY-SA 2.5.

Wien’s Displacement Law Constant Value

Wien’s Displacement Law is an important concept in the study of black body radiation and the temperature dependence of electromagnetic radiation. It states that the peak wavelength of thermal radiation emitted by a black body is inversely proportional to its temperature. The constant value associated with Wien’s Displacement Law is known as Wien’s constant, denoted by the symbol ‘k’.

Wien’s constant, represented by the equation:

$k = frac{b}{T}$

where ‘b’ is a constant value equal to 2.8977729 x 10^-3 m·K, and ‘T’ is the temperature in Kelvin. This equation allows us to calculate the peak wavelength of radiation emitted by a black body at a given temperature.

Wien’s Displacement Law Constant

Wien’s Displacement Law provides a relationship between the peak wavelength of radiation and the temperature of a black body. The equation for Wien’s Displacement Law is given by:

$lambda_{text{peak}} = frac{b}{T}$

where (lambda_{text{peak}}) represents the peak wavelength, ‘b’ is Wien’s constant, and ‘T’ is the temperature in Kelvin.

Wien’s constant, also known as the Wien’s displacement constant, is a fundamental constant in physics. It has a value of approximately 2.8977729 x 10^-3 m·K. This constant allows us to determine the peak wavelength of radiation emitted by a black body at a specific temperature.

See also The Fermi Paradox: Temperature Constraints and the Search for Extraterrestrial Life

The peak wavelength of radiation emitted by a black body is directly related to its temperature. As the temperature increases, the peak wavelength decreases, shifting towards shorter wavelengths. This means that at higher temperatures, the radiation emitted by a black body will have a higher frequency and shorter wavelength, such as ultraviolet or even X-ray radiation. On the other hand, at lower temperatures, the radiation will have a lower frequency and longer wavelength, such as infrared or even radio waves.

Wien’s Displacement Law is a crucial concept in understanding the distribution of electromagnetic radiation emitted by objects at different temperatures. It helps us determine the peak wavelength of radiation and gives us insights into the color or type of light emitted by a source.

For example, the actual color of a heated object can be determined by its peak wavelength. At lower temperatures, the object may appear reddish, emitting predominantly infrared radiation. As the temperature increases, the peak wavelength shifts towards the visible light spectrum, resulting in colors ranging from red to orange, yellow, and eventually blue at higher temperatures.

Wien’s Displacement Law is also applicable to other sources of radiation, such as the Sun. By parameterizing the peak wavelength of the Sun’s radiation, we can determine its temperature. The Sun’s peak wavelength falls within the visible light spectrum, with a value of approximately 500 nm. This corresponds to a temperature of around 5800 K, according to Wien’s Displacement Law.

Practical Application of Wien’s Displacement Law

Wien’s Displacement Law is a fundamental concept in the field of black body radiation and has several practical applications. This law describes the relationship between the temperature of a black body and the wavelength at which it emits the most radiation. Understanding this relationship allows us to make predictions and calculations related to thermal radiation.

How to Use Wien’s Displacement Law

To utilize Wien’s Displacement Law, follow these steps:

Understand the Temperature Dependence: Wien’s Displacement Law states that the peak wavelength of radiation emitted by a black body is inversely proportional to its temperature. As the temperature increases, the peak wavelength decreases. This means that hotter objects emit shorter wavelengths of radiation, such as ultraviolet or even X-rays, while cooler objects emit longer wavelengths, such as infrared or radio waves.
Use the Wien’s Displacement Equation: The equation for Wien’s Displacement Law is given as:

Where:
– is the peak wavelength of radiation emitted by the black body,
– is Wien’s displacement constant (approximately 2.898 x 10^-3 m·K),
– is the temperature of the black body in Kelvin.

By plugging in the temperature value, you can calculate the peak wavelength of radiation emitted by the black body.

Example Calculation: Let’s say we have a black body with a temperature of 5000 K. Using the Wien’s Displacement Equation, we can calculate the peak wavelength as follows:

Therefore, the peak wavelength of radiation emitted by the black body with a temperature of 5000 K is approximately 5.796 x 10^-7 meters.

Wien’s Displacement Law Calculator

To make calculations easier, you can use a Wien’s Displacement Law calculator. This tool allows you to input the temperature of the black body and automatically calculates the corresponding peak wavelength of radiation.

Here is an example of a simple Wien’s Displacement Law calculator:

Temperature (K)	Peak Wavelength (nm)
1000	2898
2000	1449
3000	966
4000	724
5000	579
6000	483
7000	414
8000	362
9000	322
10000	290

By using this calculator, you can easily determine the peak wavelength for different temperatures without having to manually perform the calculations.

Examples and Questions on Wien’s Displacement Law

Wien’s Displacement Law Example

Let’s explore an example to understand how Wien’s Displacement Law works. Imagine we have a black body radiator, which is an object that absorbs and emits all frequencies of electromagnetic radiation. The temperature of the radiator affects the distribution of the emitted radiation.

Suppose we have two black body radiators, one at a temperature of 5000 K and the other at 10000 K. According to Wien’s Displacement Law, the peak wavelength of the radiation emitted by each radiator can be determined using the equation:

$lambda_{text{max}} = frac{k}{T}$

where ( lambda_{text{max}} ) is the peak wavelength, ( k ) is Wien’s constant (approximately equal to 2.898 × 10^-3 m·K), and ( T ) is the temperature in Kelvin.

For the radiator at 5000 K, we can calculate the peak wavelength as follows:

$lambda_{text{max}} = frac{2.898 times 10^{-3} , text{m} cdot text{K}}{5000 , text{K}} approx 5.796 times 10^{-7} , text{m}$

Similarly, for the radiator at 10000 K, the peak wavelength is:

$lambda_{text{max}} = frac{2.898 times 10^{-3} , text{m} cdot text{K}}{10000 , text{K}} approx 2.898 times 10^{-7} , text{m}$

Therefore, the radiator at 5000 K emits radiation with a peak wavelength of approximately 579.6 nm, which falls in the visible light range. On the other hand, the radiator at 10000 K emits radiation with a peak wavelength of approximately 289.8 nm, which lies in the ultraviolet region.

Wien’s Displacement Law Questions

What is Wien’s Displacement Law?
How does the temperature of a black body radiator affect the peak wavelength of the emitted radiation?
What is the equation for calculating the peak wavelength using Wien’s Displacement Law?
What is the significance of Wien’s constant in the equation?
Give an example of a black body radiator at a temperature of 6000 K. What would be the peak wavelength of the radiation emitted by this radiator?
How does the distribution of radiation change as the temperature of a black body radiator increases?
Can Wien’s Displacement Law be applied to sources of thermal radiation other than black body radiators?
Explain the relationship between the peak wavelength and the color of light emitted by a source.
How does Wien’s Displacement Law relate to Planck’s law and the Stefan-Boltzmann law?
What are the practical applications of Wien’s Displacement Law in fields such as astronomy and materials science?

Feel free to answer any of the questions or explore further examples to deepen your understanding of Wien’s Displacement Law!

Wien’s Displacement Law in the Context of Radiation

Wien’s Displacement Law of Radiation

Wien’s Displacement Law is a fundamental principle in the field of radiation that describes the relationship between the temperature of a black body and the wavelength at which it emits the most radiation. It provides valuable insights into the behavior of electromagnetic radiation and helps us understand the distribution of thermal radiation emitted by objects at different temperatures.

See also Temperature Dynamics in Fusion Reactors: Exploring the Key Factors

The law states that the peak wavelength of radiation emitted by a black body is inversely proportional to its temperature. In other words, as the temperature of a black body increases, the peak wavelength of the radiation it emits shifts towards shorter wavelengths. This means that hotter objects emit more radiation in the shorter wavelength range, such as ultraviolet and visible light, while cooler objects emit more radiation in the longer wavelength range, such as infrared.

What is Wien’s Distribution Law?

Wien’s Distribution Law, also known as the Wien’s Law or the Wien’s Displacement Equation, is a mathematical expression derived from Wien’s Displacement Law. It provides a quantitative relationship between the peak wavelength of radiation emitted by a black body and its temperature.

The equation is given by:

$lambda_{text{peak}} = frac{b}{T}$

Where:
– ( lambda_{text{peak}} ) is the peak wavelength of radiation emitted by the black body,
– ( b ) is the Wien’s displacement constant (( b approx 2.898 times 10^{-3} , text{m} cdot text{K} )),
– ( T ) is the temperature of the black body in Kelvin.

By using this equation, we can calculate the peak wavelength of radiation emitted by a black body at a given temperature. It allows us to understand the temperature dependence of the distribution of electromagnetic radiation and provides a useful tool for various applications in physics and engineering.

To illustrate the concept, let’s consider an example. The peak wavelength of radiation emitted by a black body at a temperature of 5000 K can be calculated using Wien’s Distribution Law:

$lambda_{text{peak}} = frac{2.898 times 10^{-3} , text{m} cdot text{K}}{5000 , text{K}} approx 5.796 times 10^{-7} , text{m}$

This means that the peak wavelength of radiation emitted by the black body is approximately 579.6 nm, which falls within the visible light spectrum. Hence, at this temperature, the black body would appear bluish to our eyes.

Wien’s Displacement Law and its associated distribution equation play a crucial role in understanding the behavior of thermal radiation and the relationship between temperature and the peak wavelength of radiation emitted by objects. By applying these principles, scientists and engineers can analyze and design various systems involving electromagnetic radiation, such as light sources, solar panels, and thermal imaging devices.

What is the relationship between Wien’s Displacement Law and thermal radiation? Answer: Understanding Thermal Radiation and Its Effects.

Thermal radiation refers to the electromagnetic radiation emitted by an object due to its temperature. It plays a crucial role in the field of energy transfer and is governed by various laws, such as Wien’s Displacement Law. Wien’s Displacement Law states that the wavelength at which an object emits the maximum amount of radiation is inversely proportional to its temperature. This concept is significant in understanding how different temperatures can affect the intensity and distribution of thermal radiation. To explore more about thermal radiation and its effects, check out the article on Understanding Thermal Radiation and Its Effects.

Frequently Asked Questions

1. What is Wien’s displacement law?

Wien’s displacement law states that the peak wavelength of thermal radiation emitted by a black body is inversely proportional to its temperature.

2. How is Wien’s displacement law derived?

Wien’s displacement law can be derived from Planck’s law of black body radiation by considering the behavior of the spectral energy density as a function of wavelength.

3. What does Wien’s displacement law state?

Wien’s displacement law states that the product of the peak wavelength and the temperature of a black body is a constant value.

4. How can Wien’s displacement law be used?

Wien’s displacement law can be used to determine the temperature of a black body by measuring the peak wavelength of its emitted radiation.

5. What is the formula for Wien’s displacement law?

The formula for Wien’s displacement law is λ_max = (b / T), where λ_max is the peak wavelength, T is the temperature, and b is Wien’s displacement constant.

6. What is the significance of Wien’s displacement law?

Wien’s displacement law provides a fundamental relationship between the temperature and the peak wavelength of radiation emitted by a black body, which has important applications in various fields of physics.

7. How is Wien’s displacement constant determined?

Wien’s displacement constant (b) is a fundamental constant of nature and its value is approximately equal to 2.898 × 10^-3 m·K.

8. Does Wien’s displacement law apply to all types of radiation?

Yes, Wien’s displacement law applies to all types of electromagnetic radiation, including infrared radiation, visible light, and ultraviolet radiation.

9. What is the relationship between Wien’s displacement law and the Stefan-Boltzmann law?

Wien’s displacement law and the Stefan-Boltzmann law are both laws that describe the behavior of black body radiation. While Wien’s displacement law relates the peak wavelength to the temperature, the Stefan-Boltzmann law relates the total power emitted by a black body to its temperature.

10. Can Wien’s displacement law be used to calculate the peak wavelength of radiation?

Yes, Wien’s displacement law can be used to calculate the peak wavelength of radiation emitted by a black body when its temperature is known.

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