IB DP Physics: HL复习笔记8.2.2 Black-Body Radiation

Black-Body Radiation

  • Black body radiation is the name given to the thermal radiation emitted by all bodies (objects)
  • All objects, no matter what temperature, emit black body radiation in the form of electromagnetic waves
  • These electromagnetic waves usually lie in the infrared region of the spectrum
    • Black-body radiation can also be emitted in the form of visible light or other wavelengths, depending on the temperature
  • The hotter object, the more infrared radiation it radiates in a given time
  • A perfect black body is defined as:

An object that absorbs all of the radiation incident on it and does not reflect or transmit any radiation

  • Since a good absorber is also a good emitter, a perfect black body would be the best possible emitter too
  • As a result, an object which perfectly absorbs all radiation will be black
    • This is because the colour black is what is seen when all colours from the visible light spectrum are absorbed
  • The intensity and wavelength distribution of any emitted waves depends on the temperature of the body
  • This is represented on a black body radiation curve
    • As the temperature increases, the peak of the curve moves
    • This moves to a lower wavelength and a higher intensity

6.4.1-Black-Body-Curve

Black body spectrum for objects of different temperatures

  • From the electromagnetic spectrum, waves with a smaller wavelength have higher energy (e.g. UV rays, X-rays)
  • When an object gets hotter, the amount of thermal radiation it emits increases
  • This increases the thermal energy emitted and therefore the wavelength of the emitted radiation decreases
    • At room temperature objects emit thermal radiation in the infrared region of the spectrum
    • At around 1000 °C an object will emit a significant amount of red light
    • At 6000 °C an object will mainly emit white or blue light (and some ultraviolet)
    • At even higher temperatures objects will emit ultraviolet or even X-rays
  • A black body is an object which:
    • Absorbs all the radiation that falls on it, and is also a good emitter
    • Does not reflect or transmit any radiation
  • A black body is a theoretical object, however, stars are the best approximation there is
  • The radiation emitted from a black body has a characteristic spectrum that is determined by the temperature alone

25.1-Wiens-Law-Graph

The intensity-wavelength graph shows how thermodynamic temperature links to the peak wavelength for four different bodies

Stefan–Boltzmann Law & Wien’s Displacement Law

Wien’s Displacement Law

  • Wien’s displacement law relates the wavelength emitted by a body to its surface temperature, it states:

The black body radiation curve for different temperatures peaks at a wavelength which is inversely proportional to the temperature

  • This relation can be written as:

5.-Wiens-Displacement-Law-equation-1

  • The full equation for Wien's Law is given by

λmaxT = 2.9 × 10−3 m K

  • Where:
    • λmax = peak wavelength emitted by the body (m)
    • T = temperature of the body (K)
  • This equation shows:
    • The higher the temperature of a body, the shorter the wavelength it emits at the peak intensity
    • The higher the temperature of a body, the greater the intensity of the radiation at each wavelength

Stefan–Boltzmann Law

  • The power output of a black body depends on two factors:
    • Its surface temperature
    • Its radius
  • The relationship between these is known as the Stefan-Boltzmann Law, which states:

The total energy emitted by a black body per unit area per second is proportional to the fourth power of the absolute temperature of the body

  • It is equal to:

P = σAT4

  • Where:
    • P = total power emitted by the black body (W)
    • σ = the Stefan-Boltzmann constant
    • A = total surface area of the black body (m2)
    • T = absolute temperature of the body (K)
  • When considering a sphere (such as a star) the surface area is equal to 4πr2
  • In this case, the Stefan-Boltzmann law can be written as:

P = 4πr2σT4

Worked Example

Betelguese is our nearest red giant star. It has a power output of 4.49 × 1031 W and emits radiation with a peak wavelength of 850 nm.

Calculate the ratio of the radius of Betelgeuse rB to the radius of the Sun rs.

Radius of the sun rs = 6.95 × 108 m.

Step 1: Write down Wien’s displacement law

λmaxT = 2.9 × 10−3 m K

Step 2: Rearrange Wien’s displacement law to find the surface temperature of Betelguese

7.-Estimating-the-Radius-of-Stars-Worked-Example-equation-1

Step 3: Write down the Stefan-Boltzmann law

P = σAT4

P = 4πr2σT4

Step 4: Rearrange for r and calculate the stellar radius of Betelguese7.-Estimating-the-Radius-of-Stars-Worked-Example-equation-2

Step 5: Calculate the ratio rB / rs

7.-Estimating-the-Radius-of-Stars-Worked-Example-equation-3

    • Therefore, the radius of Betelguese is about 1000 times larger than the Sun’s radius

 

 

 

 

 

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