Calculations with Wien's Law & the Stefan-Boltzmann Law

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  • 0:01 Blackbody Radiation Laws
  • 1:05 Wien's Law
  • 2:53 The Stefan-Boltzmann Law
  • 6:16 Lesson Summary
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Lesson Transcript
Instructor: Artem Cheprasov

Artem has a doctor of veterinary medicine degree.

This lesson will go over the equations, as well as some calculations, for Wien's law and the Stefan-Boltzmann law. You will learn what factors these laws depend upon and what they reveal.

Blackbody Radiation Laws

In a couple other lessons, you might have learned about blackbodies and blackbody radiation. A blackbody is this theoretical object that is the best at emitting and absorbing all wavelengths of radiation that fall onto it. This radiation, a form of energy called electromagnetic radiation, is known as blackbody radiation when emitted by a blackbody.

The important properties of blackbody radiation can be described with two laws that can be put into quite simple mathematical equations. These laws are the Stefan-Boltzmann law and Wien's law. They boil down to this. The Stefan-Boltzmann law says that the total energy radiated from a blackbody is proportional to the fourth power of its temperature, while Wien's law is the relationship between the wavelength of maximum intensity a blackbody emits and its temperature. The wavelength of maximum intensity an object emits is known as lambda max.

This lesson will explain for you the equations involved in these laws and how they help astronomers understand the cosmos.

Wien's Law

Wien's law is written by the equation shown on your screen:

wiens law equation

Here, lambda max (in meters) is equal to a constant, b, divided by a temperature, T (in kelvin). The constant has a value of 2.9 * 10^-3 m K.

It is important to note that Wien's law gives you wavelength of maximum emission in meters. This means that if you want to get your answer converted to nanometers, you have to multiply your answer by 10^9 nm / 1 m.

If we know the temperature of a body, we can then use Wien's law to figure out lambda max, the wavelength at which it emits radiation most brightly. Conversely, Wien's law helps astronomers determine the surface temperature of a star without having to know its size, how much energy it radiates, and how far away it is. So, it's a pretty convenient law.

Let's do a practice calculation to see this law in action. The maximum intensity of sunlight is approximately 500 nm. How hot, then, is the sun's surface?

First, we need to convert 500 nm into meters. Dividing 500 nm by 10^9 nm / 1 m, we convert 500 nm into 5.0 * 10^-7 m.

After this, the rest is easy. Set 5.0 X 10^-7 m equal to (2.9 * 10^-3 m K) / T (temperature). This is the same as writing T = (2.9 * 10^-3 m K) / (5.0 * 10^-7 m). Our sun's surface temperature thus comes out to be 5,800 K.

The Stefan-Boltzmann Law

While Wien's law can be used to help us find the temperature or the wavelength of maximum intensity of a body, depending on which one of the two we have as a given, the Stefan-Boltzmann law is, in simple terms, used to find how much energy an object emits depending on temperature.

It's a bit more complex than that, though. Here's why. The total amount of energy a star radiates in one second is known as luminosity. Luminosity depends on the star's temperature and surface area (its 'size', if you will). Why?

Let's think of a familiar example. If you light a match, its temperature will be the same as that of a big log burning in a fireplace, but which one will radiate more heat? Which one can you stand closer to without being pushed back by the intensity of the heat? It's the tiny little match, isn't it? That means given the same temperature, larger stars emit more energy than smaller ones, like a log emits more energy than a tiny little match. That means surface area is kind of an important consideration in determining a star's total energy output.

Consequently, it's necessary to determine the amount of energy, in joules (J) per second, emitted from one square meter of an object's surface, something called energy flux, or F. As a result of this definition, energy flux is written as joules per square meter per second (J/m^2/s), or because one watt (W) equals one joule per second, as watts per square meter (W/m^2).

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