DOING PHYSICS WITH MATLAB

 

THERMAL PHYSICS

 

BLACKBODY RADIATION

 

Ian Cooper

matlabvisualphysics@gmail.com

 

 

 

MATLAB SCRIPTS   (download files)

 

The continuous spectrum of a blackbody at different temperatures can be investigated.

 

tpSun.m

Simulation of the electromagnetic radiation emitted from the Sun. The Script can be used to create colour spectrums of the radiation emitted from the Sun by calling the Script Colorcode.m.

 

tpStar.m

Simulation of the blackbody curve of a star. You can change the temperature of the star and observe its blackbody temperature. The Script can be used to create colour spectrums of the radiation emitted from a star Sun by calling the Script Colorcode.m.

 

tpFilament.m

Simulation of the radiation emitted by a hot tungsten filament.

 

tpBlackbody.m

Simulation of the radiation emitted from a hot object at four temperatures.

 

simpson1d.m

Function to evaluate the area under a curve using Simpson’s 1/3 rule.

 

ColorCode.m

Function to return the appropriate colour for a wavelength in the visible range from 380 nm to 780 nm.

 

 

 

 

THERMAL RADIATION AND BLACKBODIES

   PARTICLE NATURE OF ELECTROMAGNETIC RADIATION

 

The wave nature of electromagnetic radiation is demonstrated by interference phenomena. However, electromagnetic radiation also has a particle nature. For example, to account for the observations of the radiation emitted from hot objects, it is necessary to use a particle model, where the radiation is considered to be a stream of particles called photons. The energy of a photon, E is

(1)     

 

The electromagnetic energy emitted from an object’s surface is called thermal radiation and is due a decrease in the internal energy of the object. This radiation consists of a continuous spectrum of frequencies extending over a wide range. Objects at room temperature emit mainly infrared and it is not until the temperature reaches about 800 K and above that objects glows visibly.

 

A blackbody is an object that completely absorbs all electromagnetic radiation falling on its surface at any temperature. It can be thought of as a perfect absorber and emitter of radiation. The power emitted from a blackbody, P is given by the Stefan-Boltzmann law and it depends only on the surface area of the emitter, A and its surface temperature, T

     (2)     

 

A more general form of equation (2) is

(2)     

 

where e is the emissivity of the object. For a blackbody, e = 1. When e  < 1 the object is called a graybody and the object is not a perfect emitter and absorber.

 

 

The amount of radiation emitted by a blackbody is given by Planck’s radiation law and is expressed in terms of the spectral exitance for wavelength or frequency Rl or Rf respectively

 

     (4)                        [W.m-2.m-1] 

 

or

 

     (5)                          [W.m-2.s-1]  

 

 

 

In the literature, many different terms and symbols are used for the spectral exitance. Sometimes the terms and the units given are wrong or misleading.

             

The power radiated per unit surface of a blackbody, PA within a wavelength interval or bandwidth, (l1, l2) or frequency interval or bandwidth (f1, f2) are given by equations 6 and 7

 

     (6)               [W.m-2]    

 

 

and

 

     (7)             [W.m-2]

 

 

 

The equations 6 and 7 give the Stefan-Boltzmann law (equation 2) when the bandwidths extend from 0 to ¥.

 

 

Wien’s Displacement law states that the wavelength lpeak corresponding to the peak of the spectral exitance given by equation 4 is inversely proportional to the temperature of the blackbody and the frequency fpeak for the spectral exitance peak frequency given by equation 5 is proportional to the temperature

     (8)      

                                                                                                          

The peaks in equations 4 and 5 occur in different parts of the electromagnetic spectrum and so

     (9)      

 

The Wien’s Displacement law explains why long wave radiation dominates more and more in the spectrum of the radiation emitted by an object as its temperature is lowered.

 

When classical theories were used to derive an expression for the spectral exitances Rl and Rf, the power emitted by a blackbody diverged to infinity as the wavelength became shorter and shorter. This is known as the ultraviolet catastrophe. In 1901 Max Planck proposed a new radical idea that was completely alien to classical notions, electromagnetic energy is quantized. Planck was able to derive the equations 4 and 5 for blackbody emission and these equations are in complete agreement with experimental measurements. The assumption that the energy of a system can vary in a continuous manner, i.e., it can take any arbitrary close consecutive values fails. Energy can only exist in integer multiples of the lowest amount or quantum, h f. This step marked the very beginning of modern quantum theory.

 

 

A summary of the physical quantities, units and values of constants used in the description of the radiation from a hot object.

 

  Variable

Interpretation

Value

Unit

E

energy of photon

 

J

h

Planck’s constant

6.62608´10-34

J.s

c

speed of electromagnetic radiation

3.00x108

m.s-1

f

frequency of electromagnetic radiation

 

Hz

l

wavelength of electromagnetic radiation

 

 

T

surface temperature of object

 

K

A

surface area of object

 

m2

s

Stefan-Boltzmann constant

5.6696´10-8

W.m-2.K-4

P

power emitted from hot object

 

W

e

emissivity of object’s surface

 

 

Rl

spectral exitance: power radiated per unit area per unit wavelength interval

 

(W.m-2).m-1

Rf

spectral exitance: power radiated per unit area per unit frequency interval

 

(W.m-2).s-1

kB

Boltzmann constant

1.38066´10-23

J.K-1

bl

Wien constant: wavelength

2.898´10-3

m.K

bf

Wien constant: frequency

 

2.83 kB T / h

K-1.s-1

lpeak

wavelength of peak in solar spectrum

5.0225´10-7

m

RS

radius of the Sun

6.96´108

m

RE

radius of the Earth

6.96´106

m

RSE

Sun-Earth radius

6.96´1011

m

I0

Solar constant

1.36´103

W.m-2

a

Albedo of Earth’s surface

0.30

 

 

 

 

 

SIMULATION: THE SUN AND THE EARTH AS BLACKBODIES

 

Inspect and run the Script tpSun.m so that you are familiar with what the program and the code does. The Script calls the functions simpson1d.m and Colorcode.m.

 

The Sun can be considered as a blackbody, and the total power output of the Sun PS can be estimated by using the Sefan-Boltzmann law, equation 2, and by finding the area under the curves for Rl and Rf using equations 6 and 7. From observations on the Sun, the peak in the electromagnetic radiation emitted has a wavelength, lpeak = 502.25 nm (yellow). The temperature of the Sun’s surface (photosphere) can be estimated from the Wien displacement law, equation 8.

 

The distance from the Sun to the Earth, RSE can be used to estimate of the surface temperature of the Earth TE if there was no atmosphere. The intensity of the Sun’s radiation reaching the top of the atmosphere, I0 is known as the solar constant 

 

     (10)       

 

The power absorbed by the Earth, PEabs is

 

     (11)       

 

where a is the albedo (the reflectivity of the Earth’s surface). Assuming the Earth behaves as a blackbody then the power of the radiation emitted from the Earth, PErad is

 

     (12)       

 

It is known that the Earth’s surface temperature has remained relatively constant over many centuries, so that the power absorbed and the power emitted are equal, so the Earth’s equilibrium temperature TE is

 

     (13)      

 

 

 

Sample results using tpSun.m

 

Plots of the spectral exitance curves

 

 

 

Matlab screen output for sun.m

   

Sun: temperature of photosphere, T_S = 5770  K

   

Peak in Solar Spectrum

   Theory: Wavelength at peak in spectral exitance, wL = 5.02e-07  m 

   Graph:  Wavelength at peak in spectral exitance, wL = 5.04e-07  m 

   Corresponding frequency, f = 5.95e+14  Hz 

   

   Theory: Frequency at peak in spectral exitance, f = 3.39e+14  Hz 

   Graph:  Frequency at peak in spectral exitance, f = 3.40e+14  Hz 

   Corresponding wavelength, wL = 8.82e-07  m 

   

Total Solar Power Output

   P_Stefan_Boltzmann = 3.79e+26  W

   P(wL)_total        = 3.77e+26  W

   P(f)_total         = 3.79e+26  W

  

IR visible UV

   P_IR      = 1.92e+26  W 

   Percentage IR radiation      = 51.0   

   

   P_visible = 1.39e+26  W

   Percentage visible radiation = 36.8 

   

   P_UV      = 4.61e+25  W 

   Percentage UV radiation      = 12.2 

   

Sun - Earth   

   Theory: Solar constant I_O   = 1.360e+03  W/m^2 

   Computed: Solar constant I_E = 1.342e+03  W/m^2 

 

   Surface temperature of the Earth, T_E  = 254  K 

   Surface temperature of the Earth, T_E  = -19  deg C

 

 

 

Questions

1     How do the peaks in the plots Rl and Rf  compare with the predictions of the Wien displacement law and lpeak = 502.25 nm (yellow).

 

2     Compare the total solar power emitted by the Sun calculated from the Stefan-Boltzmann law and by the numerical integration to find the area under the spectral exitance (Rl and Rf) curves.

 

3     Compare the percentage the radiation in the ultraviolet, visible and infrared parts of the solar spectrum.

 

4     How does the computed value of the intensity of the radiation reaching the Earth’s surface, IE compare with the solar constant, I0?

 

5     From our simple model, the surface temperature of the Earth was estimated to be -19 oC. Is this sensible? What is the surface temperature on the moon? The average the temperature of the Earth is much higher than this, about +15 oC. Explain the difference. 

 

6     What changes occur in the calculations if the Sun was hotter (peak in the blue part of the spectrum) or cooler (peak in the red) part of the spectrum?

 

7     What would be wavelength lpeak and the temperature of the Sun’s surface if the Earth’s equilibrium temperature was -15 oC instead -19 oC? (In the m-script, increase the value of lpeak  until you reach the required equilibrium temperature of the Earth.)

 

 

 

M-script highlights

 

1         Suitable values for the wavelength and frequency integration limits for equations (6) and (7) are determined so that the spectral exitances at the limits are small compared to the peak values.

 

2         The Matlab function area is used to plot the spectral exitance curves, for example, in plotting the Rl curve:

               h_area1 = area(wL,R_wL);

               set(h_area1,'FaceColor',[0 0 0]);

               set(h_area1,'EdgeColor','none');

 

3         The color for the shading of the curve matches that of the wavelength in the visible part of the spectrum. A call is made to the function ColorCode.m to assign a color for a given wavelength band. For the shading of the Rl curve:

 

           thisColorMap = hsv(128);

           for cn = 1 : num_wL-1

           thisColor = ColorCode(wL_vis(cn));

           h_area = area(wL_vis(cn:cn+1),R_wL_vis(cn:cn+1));

           set(h_area,'FaceColor',thisColor);

           set(h_area,'EdgeColor',thisColor);

 

4         Simpson’s 1/3 rule is used for the numerical integration (simpson1d.m) to find the area under the spectral intensity curves. For the Rl curve, the total power radiated by the Sun:

               P_total = A_sun * simpson1d(R_wL,wL1,wL2);

 

5         The peaks in spectral intensities are calculated using Matlab logical functions:

             wL_peak_graph = wL(R_wL == max(R_wL)); 

             f_peak_graph = f(R_f == max(R_f));  

  

 

SIMULATION:   HOW EFFICIENT IS A HOT TUNGSTEN FILAMENT ?

Description: Description: Description: Image result for image hot tungsten filament

 

Inspect and run the Script tpFilament.m so that you are familiar with what the program and the code does. The Script calls the functions simpson1d.

 

Some car headlights use a hot tungsten filament to emit electromagnetic radiation. We can estimate the percentage of this radiation in the visible part of the electromagnetic spectrum for a hot tungsten filament that has a surface temperature of 2400 K and an electrical power of 55 W (the thermal power radiated is also 55 W). The first step is to calculate the thermal power radiated P by a hot object using equation 14

     (14)                                 [W]  

 

 

where N is a normalizing constant and includes a factor for the surface area such that P = 55 W.

 

The area under the power per unit wavelength curve is shaded yellow to show the visible part of the spectrum. Lastly, the function Rl is numerical integrated for the limits corresponding to only the visible part of the electromagnetic spectrum

 

              l1 = 700 nm (red)  and  l2 = 400 nm (blue)

 

This gives only the total power radiated in the visible part of the electromagnetic spectrum, Pvisible.

 

The filament efficiency, h given as percentage of visible radiation emitted by the hot tungsten filament to the power consumed by the filament

     (16)      

 

 

Sample Results for tpFilament.m

 

 

 

Matlab screen output      

 

 

   wavelength at peak  = 1.21e-006  m 

   wavelength at peak  = 1.21  um 

   

   P_total    = 55.0  W 

   

   P_visible  = 1.9  W 

   

   efficiency (percentage) =   3.5 

   

  Check normalization

   P_check    = 55.0  W 

 

 

 

Questions

1     Are your surprised by the efficiency of the tungsten filament used in a light globe? 

 

2     What part of the electromagnetic spectrum does the peak in the spectral intensity curve occur?

 

3     Most of the energy emitted from the light globe is not emitted in the visible part of the electromagnetic spectrum. What happens to most of the electrical energy supplied to the light globe?

 

4     What temperature would the filament have to be at so that the peak is in the visible part of the spectrum? Is this possible?

 

5     What is the minimum temperature of the filament so that the globe just starts to glow?

 

6     How do the results change if the power emitted by the hot tungsten filament was 75 W?

 

 

 

SIMULATION: THERMAL RADIATION EMITTED FROM A HOT OBJECT

 

Inspect and run the Script tpBlackbody.m so that you are familiar with what the program and the code does. The Script calls the function simpson1d.m for the numerical integration to compute the power emitted from the spectral exitance function.

 

Description: Description: Description: Image result for image glow from a hot iron bar

What part of the rod is at the highest temperature?

 

The thermal radiation emitted by a blackbody at four different temperatures is modeled. The spectral exitance curves for each temperature are plotted. Notice that even a temperatures as high as 2000 K only a small amount of radiation is emitted in the visible part of the electromagnetic spectrum. From the graphical output and the numerical values displayed in the Command Window, it is easy to verify that the Stefan-Boltzmann Law and the Wien’s Displacement Law are satisfied.

 

A table can be displayed in Command Window using the Matlab table function. The centre column is the relative power emitted by the hot object. You can see that when the temperature is doubled, the power emitted increases by a factor of 16 (24) and the peak wavelength in the blackbody curve is inversely proportional to the surface temperature of the object.

  disp('    ');

  CWT = table(T,round(P,2),round(wL_Peak,0), 'VariableNames', {'T [K]', 'Prel', 'wL_Peak [nm]'});

  disp(CWT)

  disp('  ')

 

      T [K]    Prel     wL_Peak [nm]

    _____    _____    ___________

 

    1000       1             2898   

    1500      5.08        1932   

    2000     16.07       1449   

    2500     39.25       1159    

 

 

 

 

SIMULATION:  STAR TEMPERATURES

 

Inspect and run the Script tpStar.m so that you are familiar with what the program and the code does. The Script calls the function simpson1d.m for the numerical integration to compute the power emitted from the spectral exitance function. The star temperature is entered in the INPUT section of the Script.

 

Stars approximate blackbody radiators and their visible color depends upon the temperature of the radiator. The curves below are for a blue, a yellow-white, and a red star. The yellow-white star has a colour is like our Sun.

 

Blue Star

Star: temperature of photosphere, T_S = 7000  K

   

Peak in Star Spectrum

   Theory: Wavelength at peak in spectral exitance, wL = 414  nm 

   Graph:  Wavelength at peak in spectral exitance, wL = 415   nm 

   Correspondending frequency, f = 7.22e+14  Hz 

   

   Theory: Frequency at peak in spectral exitance, f = 4.11e+14  Hz 

   Graph:  Frequency at peak in spectral exitance, f = 4.13e+14  Hz 

   Correspondending wavelength, wL = 7.27e-07  m 

   

Total Solar Power Output

   P(wL)_total        = 1.35e+08  a.u.

  

IR visible UV

   P_IR      = 5.09e+07  W 

   Percentage IR radiation      = 37.6   

   

   P_visible = 5.35e+07  W

   Percentage visible radiation = 39.5 

   

   P_UV      = 3.10e+07  W 

   Percentage UV radiation      = 22.9 

 

 

Yellow-White Star   6000 K

Star: temperature of photosphere, T_S = 6000  K

   

Peak in Star Spectrum

   Theory: Wavelength at peak in spectral exitance, wL = 483  nm 

   Graph:  Wavelength at peak in spectral exitance, wL = 485   nm 

   Correspondending frequency, f = 6.19e+14  Hz 

   

   Theory: Frequency at peak in spectral exitance, f = 3.53e+14  Hz 

   Graph:  Frequency at peak in spectral exitance, f = 3.54e+14  Hz 

   Correspondending wavelength, wL = 8.48e-07  m 

   

Total Solar Power Output

   P(wL)_total        = 7.31e+07  a.u.

  

IR visible UV

   P_IR      = 3.52e+07  W 

   Percentage IR radiation      = 48.1   

   

   P_visible = 2.76e+07  W

   Percentage visible radiation = 37.8 

   

   P_UV      = 1.03e+07  W 

   Percentage UV radiation      = 14.1 

 

 

Red Star   4000 K

Star: temperature of photosphere, T_S = 4000  K

   

Peak in Star Spectrum

   Theory: Wavelength at peak in spectral exitance, wL = 725  nm 

   Graph:  Wavelength at peak in spectral exitance, wL = 727   nm 

   Correspondending frequency, f = 4.12e+14  Hz 

   

   Theory: Frequency at peak in spectral exitance, f = 2.35e+14  Hz 

   Graph:  Frequency at peak in spectral exitance, f = 2.36e+14  Hz 

   Correspondending wavelength, wL = 1.27e-06  m 

   

Total Solar Power Output

   P(wL)_total        = 1.44e+07  a.u.

  

IR visible UV

   P_IR      = 1.11e+07  W 

   Percentage IR radiation      = 77.1   

   

   P_visible = 3.02e+06  W

   Percentage visible radiation = 20.9 

   

   P_UV      = 2.86e+05  W 

   Percentage UV radiation      = 2.0