Optional Practical: Planck's law (GeoGebra)

Introduction

In this exercise you will use Planck's law to plot the black body spectra for different temperatures then show how Wien's law gives the peak wavelength and that the total power radiated per unit area is proportional to T4 (Stefan Boltzmann's law). This worksheet assumes that you know the basic functions of GeoGebra.

Note that Planck's law is not on the syllabus but it is interesting to see a bit of the theoretical background to black body radiation.

Planck's law

A black body is a body that absorbs all wavelengths of radiation. If light was shone through a small hole into into an empty box then it would reflect back and forth until absorbed, a "cavity" such as this would therefore be a black body.

If the box were heated then radiation would be emitted inside the cavity, this radiation would reflect back and forth forming standing waves. As with waves in a string, there are only certain possible wavelengths allowed, the radiation emitted from the hole would be made up of all of these wavelengths. The intensity of each wavelength would be related to the number of modes of vibration that could produce that wavelength,this is greater for shorter wavelengths as can be seen in the diagram below.

Classically, each wavelength is equally likely so the intensity of the shorter wavelengths would be greater, this would leads to an "ultra violet catastrophe" which is not observed.

According to the quantum model of light, the energy of a photon is given by E = hf. This means that the higher frequencies have higher energy so are less likely. This gives the following equation for the power per unit area for a given wavelength.

S subscript lambda equals fraction numerator 2 pi c squared h over denominator lambda to the power of 5 end fraction fraction numerator 1 over denominator e to the power of bevelled fraction numerator h c over denominator lambda k T end fraction end exponent minus 1 end fraction

c - speed of light
h - Planck constant
k - Boltzmann constant

Plotting the graph

With GeoGebra it is simply a matter of writing the equation and the graph will be plotted but you need to make the units fit. If power per unit area per unit wavelength is measured in 1013 Wm-3 and the wavelength in μm then the constants can be simplified to give the following equation:

y equals 38 over x to the power of 5. fraction numerator 1 over denominator e to the power of begin display style bevelled fraction numerator 14000 over denominator x T end fraction end style end exponent minus 1 end fraction

  • Make a slider for T from 0 to 8000 with increment 100.
  • Input the equation above.
  • add labels to the axis using the text tool.
  • Vary T and see how the curve changes.

To add a sequence of curves for different temperatures you can input the equation Sequence[38 / x⁵ 1 / (ℯ^(14000 / (x i)) - 1), i, 1000, 6000, 1000]. This will plot a line every 1000 K from 1000 to 6000 K.

Wien's Law

Wein's law gives the peak frequency for a given temperature


lambda subscript p e a k end subscript equals space fraction numerator 0.00289 over denominator T end fraction

This again needs to be adjusted for the units of the graph so use.

lambda subscript p e a k end subscript equals 2890 over T

  • Hide the sequence of lines.
  • Input the equation for a vertical line with x value = the peak wavelength (x=2890/T).
  • Place a point on the intersect of this vertical line and the spectrum graph.
  • Hide the line
  • Right click the point and tick "trace on".
  • Vary T and see how the peak wavelength changes.

Stefan-Boltzmann Law

The Stefan-Boltzmann law states that, the Power emitted per unit area is proportional to T4.

P equals sigma A T to the power of 4

The spectrum shows the power emitted per unit area wavelength against wavelength so the area under the curve will be the total power emitted per unit area. We can show that this is proportional to T4 by drawing another graph.

  • To find the area under the graph input "Integral[f, 0, 4]" this finds the area under the curve of f(x) from 0 to 4 where f(x) is the Planck's law function. The value will probably be assigned the letter a.
  • Open up second graph by choosing Graphics 2 from the view menu.

T^4 is going to be a very big number so to make it fit on the axis we need to scale it down by dividing by 1014.

  • place a point on the graph and then change its coordinates to (a, T^4/10^14).
  • Check "trace on" for this point
  • Vary T and see that the power per unit area is proportional to T4.

Excel version

This can also be done in Excel but you need to make a table of values before you plot the graph.

  • Make a column for wavelength from 0 to 3 in steps of 0.1
  • Make a cell containing the temperature 6000 (e.g. M3 in my example)
  • Write the equation =(38/$A1^5)*(1/((EXP(14000/$A1/M$3))-1)) in the first cell of the second column and copy it down.
  • Plot a scatter graph of Power per unit area per wavelength vs Wavelength
  • Try changing the value of T.
  • Multiple lines require multiple columns.
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