How to Use Wien’s Displacement Law for Black Body Radiation Problems
Wien’s Displacement Law is a fundamental concept in thermal physics and modern Physics. It describes how the wavelength at which a blackbody emits radiation most strongly is inversely proportional to its absolute temperature. This concept helps explain why hotter objects shine with shorter wavelengths (bluer colors) and cooler objects emit at longer wavelengths (redder colors).
Blackbody radiation is the emission of electromagnetic waves from any object that absorbs all incident radiation without reflecting any. A black body is an ideal absorber and emitter, and its radiation spectrum depends only on temperature. The study of blackbody radiation provided key insights into quantum theory and underpins many principles in modern Physics.
Wien’s Displacement Law: Statement and Concept
Wien’s Displacement Law states that the peak wavelength of emission from a blackbody is inversely proportional to its absolute temperature. In simple terms, as an object gets hotter, the color of its emitted light shifts toward shorter (bluer) wavelengths. Cooler objects radiate longer (redder) wavelengths.
Wilhelm Wien formulated this law using thermodynamic arguments, demonstrating that as temperature increases, the peak of the blackbody radiation curve moves to shorter wavelengths.
Mathematical Expression and Formula
The law is mathematically written as:
λmax × T = b
Where:
λmax = Wavelength at which emission is maximum (in meters)
T = Absolute temperature of the blackbody (in Kelvin)
b = Wien’s constant = 2.897 × 10-3 m·K
This formula is a quick tool to estimate at which wavelength an object radiates most intensively if its temperature is known.
Derivation Outline (from Planck’s Law)
To derive Wien’s Law, start with Planck’s Law for blackbody spectral radiance:
B(λ, T) = (2hc2 / λ5) × [1 / (ehc/λkT – 1)]
The value of λ where B(λ, T) is maximum is found by differentiating with respect to λ and setting the derivative to zero. Solving the resulting equation gives λmax = b / T.
Key Constants and Practical Table
| Parameter | Symbol | Value/Formula | Unit |
|---|---|---|---|
| Peak wavelength | λmax | b/T | m |
| Wien’s constant | b | 2.897 × 10-3 | m·K |
| Temperature | T | Given/Calculated | K |
Physical Significance and Examples
Wien’s Law explains why the color of heated objects changes with temperature:
- As a metal is heated, it first glows red (longer wavelength). As temperature rises, it becomes orange, yellow, and finally white-hot.
- The surface of the sun (about 5700 K) has peak emission near 500 nm, in the green region visible to our eyes.
- A campfire at 1500 K emits most intensely at ~2000 nm (infrared), so we feel its heat more than see its light.
- Mammals (~300 K) emit in the infrared (around 10 micrometres), which is detected by thermal cameras and snakes.
By observing the peak emission wavelength from stars, astronomers can determine their surface temperatures.
Step-by-Step Problem Solving Using Wien’s Law
| Step | Description | Tip |
|---|---|---|
| 1 | Write down the formula: λmax × T = b | Remember λ in meters, T in Kelvin |
| 2 | Substitute the known value (temperature or wavelength) | Convert units if needed |
| 3 | Solve for the unknown parameter | Show all calculation steps clearly |
| 4 | Interpret result (check if wavelength is in visible, IR, or UV) | Relate answer to physical phenomena |
Solved Example
Example 1: The radiation from a star has a peak wavelength of 10-5 m. What is its temperature?
λmax × T = 2.897 × 10-3
T = 2.897 × 10-3 / 10-5 = 2.897 × 102 K = 289.7 K
Example 2: The Earth's temperature is 197 K. What is the wavelength where its emission peaks?
λmax = 2.897 × 10-3 / 197 = 0.0147 × 10-3 m = 1.47 × 10-5 m
Significance and Limitations
Wien’s Law makes it simple to estimate the temperature of hot objects based on the color they emit. It is crucial in fields like astrophysics, sensors, and thermal imaging.
However, the law is limited at low temperatures or longer wavelengths, where experimental results do not always match its predictions. At low temperatures, the true blackbody spectrum deviates from Wien’s Law, and Planck’s Law provides the accurate full-spectrum description.
Applications Summary
- Determining star and planet temperatures from radiation peaks.
- Explaining observed changes in color as objects are heated (e.g., iron turning red then white-hot).
- Thermal cameras detecting human or animal heat emission.
- Designing more efficient lighting and heating elements.
Practice and Further Learning
- Review Black Body Radiation & Wien’s Law for a connected understanding.
- Practice problems using b = 2.897 × 10-3 m·K, with all values converted to SI units.
- For self-assessment, solve more examples using different temperatures or wavelengths.
Explore more Physics topics and resources to master concepts for deeper understanding and exam readiness. For structured lessons and doubt resolution on topics like Wien’s Law, visit relevant lessons on Wien’s Displacement Law.
FAQs on Wien’s Displacement Law in Physics – Formula, Derivation, & Examples
1. What is Wien’s Displacement Law in simple terms?
Wien’s Displacement Law states that the wavelength at which a black body emits radiation most strongly (peak wavelength) is inversely proportional to its absolute temperature.
In other words: as the temperature of an object increases, its peak emission shifts to shorter (bluer) wavelengths; as the temperature decreases, it shifts to longer (redder) wavelengths.
2. What is the formula for Wien’s Displacement Law?
The formula is:
λmax = b / T
where:
• λmax = wavelength of maximum emission (in meters)
• b = Wien’s displacement constant = 2.898 × 10-3 m·K
• T = absolute temperature (in Kelvin)
3. What is the value and unit of Wien’s displacement constant?
Wien’s displacement constant (b) has the official value:
2.898 × 10-3 m·K (meter–Kelvin).
This means the product of peak wavelength (in meters) and absolute temperature (in Kelvin) is always equal to b for black body radiation.
4. How is Wien’s Displacement Law derived?
To derive Wien’s Displacement Law:
• Start with Planck’s Law for black body radiation, which gives spectral intensity as a function of wavelength and temperature.
• Differentiate Planck’s equation with respect to wavelength and set the derivative to zero to find the maximum (peak).
• The result shows λmax × T = constant, leading to the law’s formula.
Derivation involves calculus and understanding Planck’s spectral distribution.
5. What are the main applications of Wien’s Law?
Key applications include:
• Determining surface temperatures of stars (Astrophysics)
• Understanding color changes in heated metals and objects
• Designing thermal sensors and infrared cameras
• Explaining why humans and animals emit infrared radiation
• Solving thermal radiation numericals in competitive exams
6. Why did Wien’s Law fail at long wavelengths?
Wien’s Law fails at long wavelengths (low frequencies) because it only predicts the position of peak emission, not the entire spectral distribution.
At longer wavelengths, Rayleigh-Jeans Law is more accurate, and for all wavelengths, Planck’s Law is required.
7. What is the difference between Wien’s Law, Planck’s Law, and Rayleigh-Jeans Law?
• Wien’s Law: Gives the peak wavelength for a given temperature (λmax = b/T).
• Planck’s Law: Describes the full spectral energy distribution for any wavelength and temperature.
• Rayleigh-Jeans Law: Approximates black body radiation at long wavelengths, but fails at short wavelengths (ultraviolet catastrophe).
8. Can Wien’s Law be used for non-black bodies?
Wien’s Law is strictly accurate only for ideal black bodies. Real objects may deviate due to less than perfect emissivity, but the law provides a useful estimation for objects that approximate black body behavior.
9. How do you use Wien’s Law in numerical problems?
To solve numericals:
1. Write the formula: λmax = b/T
2. Check that T is in Kelvin and b has correct units
3. Substitute values and calculate λmax (in meters or nanometers)
4. Interpret the spectral region (e.g., visible, infrared)
10. What region of the spectrum does the human body emit most radiation in?
The human body, at about 310 K (37°C), emits most radiation in the infrared (IR) region. According to Wien’s Law, this explains why humans can be detected by thermal cameras, as our peak emission is not visible but infrared.
11. What are common mistakes when applying Wien’s Displacement Law?
Common mistakes include:
• Using temperature in Celsius instead of Kelvin
• Confusing Wien’s Law with Stefan-Boltzmann Law
• Forgetting to convert units (e.g., meters to nanometers)
• Applying Wien's Law to non-black body objects without considering emissivity
12. Why is Wien’s Law important for understanding stars?
Wien’s Law allows scientists to estimate a star’s surface temperature by observing the color (peak wavelength) of its light. Hotter stars appear blue due to shorter peak wavelengths, while cooler stars appear red due to longer peak wavelengths. This is key in astrophysics and the classification of stars.
