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Chapter 5: Problem 41

Some of the hottest stars known have a blackbody temperature of \(100,000 \mathrm{K} .\) What is the peak wavelength of their radiation? What type of radiation is this?

Short Answer

Expert verified
The peak wavelength is about \( 29.977 \, \text{nm} \), which is ultraviolet radiation.

Step by step solution

01

Understand Wien's Law

Wien's Displacement Law relates the temperature of a blackbody to the wavelength at which it emits radiation most strongly. The law is given by the formula: \[ \lambda_{max} = \frac{b}{T} \] where \( \lambda_{max} \) is the peak wavelength, \( T \) is the absolute temperature in kelvins, and \( b \) is Wien's constant (\( b = 2.8977 \times 10^{-3} \, \text{m.K} \)).
02

Substitute the Values

Substitute the given temperature into Wien's Law: \[ \lambda_{max} = \frac{2.8977 \times 10^{-3} \, \text{m.K}}{100,000 \, \text{K}} \]
03

Calculate the Peak Wavelength

Perform the calculation: \[ \lambda_{max} = \frac{2.8977 \times 10^{-3} \, \text{m.K}}{100,000 \, \text{K}} = 2.8977 \times 10^{-8} \, \text{m} = 29.977 \, \text{nm} \]
04

Identify the Type of Radiation

The peak wavelength (\( 29.977 \, \text{nm} \)) falls within the ultraviolet range of the electromagnetic spectrum.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

blackbody radiation
Blackbody radiation is the thermal electromagnetic radiation emitted by an object that absorbs all incident radiation, regardless of wavelength or angle. This means it does not reflect or transmit any light. The concept is idealized in physics to study how objects emit radiation solely based on their temperature. The distribution and intensity of this radiation depend solely on the temperature of the blackbody.

A blackbody at a higher temperature emits more radiation across all wavelengths compared to one at a lower temperature. The color changes as the temperature increases, ranging from red to blue.
electromagnetic spectrum
The electromagnetic spectrum encompasses all types of electromagnetic radiation, which vary in wavelength and frequency. It includes visible light, radio waves, microwaves, infrared radiation, ultraviolet radiation, X-rays, and gamma rays.

The spectrum is usually divided into regions based on the type of radiation, which include:
  • Radio waves: Longest wavelength, used in communication.
  • Microwaves: Used in cooking and radar technology.
  • Infrared: Felt as heat.
  • Visible Light: The only part of the spectrum visible to the human eye.
  • Ultraviolet: Can cause sunburns.
  • X-rays: Used in medical imaging.
  • Gamma rays: Emitted by radioactive materials.
The type of electromagnetic radiation an object emits depends on its temperature.
ultraviolet radiation
Ultraviolet (UV) radiation is a type of electromagnetic radiation with wavelengths shorter than visible light, ranging from 10 nm to 400 nm. UV radiation is divided into three main types: UVA, UVB, and UVC.

  • UVA has the longest wavelengths and penetrates the atmosphere to reach the Earth's surface.
  • UVB causes sunburns and has slightly shorter wavelengths.
  • UVC has the shortest wavelengths and is mostly absorbed by the Earth's atmosphere.
UV radiation can be harmful, causing skin damage and increasing the risk of skin cancer. However, it also has beneficial uses, such as in sterilization and vitamin D synthesis.
peak wavelength
The peak wavelength is the wavelength at which the radiation emitted by a blackbody is most intense. According to Wien's Displacement Law, the peak wavelength is inversely proportional to the temperature. This means that as the temperature of a blackbody increases, the peak wavelength decreases, shifting to shorter wavelengths.

  • The formula for Wien's Law is \(\text{ } \lambda_{max} \text{ } = \text{ } \frac{b}{T} \text{ } \text{, where } \text{ } b \text{ } \text{ is a constant (2.8977 \times 10^{-3} \text{ } m \cdot \text{ } K).\text{ } \)
For example, stars with high temperatures have their peak wavelength in the ultraviolet range, as shown by the blackbody temperature of 100,000K stars with a peak wavelength of 29.977 nm.
temperature of stars
Stars radiate energy across the electromagnetic spectrum, but the peak wavelength of this radiation is determined by their surface temperature. Wien's Displacement Law helps us understand this relationship. Higher temperature stars emit radiation at shorter wavelengths.

  • Cooler stars emit light primarily in the red or infrared spectrum.
  • Hotter stars emit light in the blue or ultraviolet spectrum.
For instance, a star with a surface temperature of 100,000 K will have a peak wavelength in the ultraviolet range. This demonstrates that the color and type of radiation a star predominantly emits are key indicators of its temperature.

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