Properties of the Planck Radiation Law

Explore properties of electromagnetic radiation emitted by bodies at given temperatures

Visualize the spectral radiance for the radiation emitted by a body at 5000 degrees Celsius, as a function of wavelength:

In[1]:=

plot=PlanckRadiationLaw[

5000

°C

,"SpectralPlot"]

Out[1]=

Determine the peak and mean wavelengths as well as the color of the body:

In[2]:=

{peak,mean,color}=N[PlanckRadiationLaw[

5000

°C

,#]]&/@{"MaxWavelength","MeanWavelength","Color"}

Out[2]=



549.533

1010.11



Find the spectral radiance for the peak and mean wavelengths:

In[3]:=

PlanckRadiationLaw[

5000

°C

,#]&/@{peak,mean}

Out[3]=



1.66984×

W/(

sr)

8.14981×

W/(

sr)



Compare the visible spectrum with the distribution of spectral radiance for the given temperature:

In[4]:=

ShowParametricPlot{x,y},{x,0,2*10^-6},{y,0,10^14},

,plot,



Out[4]=

Calculate the total radiance in the spectrum of green light:

In[5]:=

PlanckRadiationLaw

5000

°C

green light

LIGHT COLOR



minimum wavelength

maximum wavelength



Out[5]=

995946.

W/(

sr)

To compare Planck's radiation law to the Wien distribution law and Rayleigh–Jeans's law, first define the Wien distribution law and Rayleigh–Jeans's law:

In[6]:=

wiendistributionlaw[t_,ν_]:=Quantity[2,"PlanckConstant"/"SpeedOfLight"^2/"Steradians"]*ν^3*E^(-Quantity[1,"PlanckConstant"/"BoltzmannConstant"]*ν/t);rayleighjeanlaw[t_,ν_]:=Quantity[2,"BoltzmannConstant"/"SpeedOfLight"^2/"Steradians"]*ν^2*t;

Compare the three formulas symbolically:

In[7]:=

Grid{{"Wein distribution law",wiendistributionlaw[QuantityVariable["T","Temperature"],QuantityVariable["ν","Frequency"]]},{"Rayleigh–Jeans's law",rayleighjeanlaw[QuantityVariable["T","Temperature"],QuantityVariable["ν","Frequency"]]},{"Planck's radiation law",Last@FormulaData[{"PlanckRadiationLaw","Frequency"}]}},



Out[7]=

Wein distribution law	 -1 h / k  ν T   2 h /( 2 c sr)  3 ν
Rayleigh–Jeans's law	 2 k /( 2 c sr)  T 2 ν
Planck's radiation law	 2 h / 2 c  3 ν -1+  h / k  ν T 

Compute the values of the three formulas at 7000 K over the span of

Hz:

In[8]:=

wien=Table[{10^x,QuantityMagnitude[wiendistributionlaw[Quantity[7000,"Kelvins"],Quantity[10^x,"Hertz"]],"Watts"/("Hertz"*"Meters"^2*"Steradians")]},{x,13,15,0.1}];rayleighjean=Table[{10^x,QuantityMagnitude[rayleighjeanlaw[Quantity[7000,"Kelvins"],Quantity[10^x,"Hertz"]],"Watts"/("Hertz"*"Meters"^2*"Steradians")]},{x,13,15,0.1}];planck=Table[{10^x,QuantityMagnitude[PlanckRadiationLaw[Quantity[7000,"Kelvins"],Quantity[10^x,"Hertz"]]]},{x,13,15,0.1}];

Visualize and compare the values in a log-log plot:

In[9]:=