Particle Nature of Electromagnetic Radiation
PARTICLE NATURE OF ELECTROMAGNETIC RADIATION
THERMAL RADIATION AND BLACKBODIES
Outline
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
E = hf(1)
where
h = 6.6260810-34 J.s is the Planck constant
frequency of the radiation, f in Hz.
The electromagnetic energy emitted from the surface due to the internal energy of the object is called thermal radiation. 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, Pis given by the Stefan-Boltzmann law and it depends only on the surface area of the emitter, A and its surface temperature, T
(2)
where
= 5.669610-8 W.m-2.K-4 is the Stefan-Boltzmann constant
power, P in W
surface area, A in m2
temperature, T in K.
A more general form of equation (2) is
(3)
where is the emissivity of the object. For a blackbody, = 1. When < 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 intensity (radiant emittance)Ror Rf
(4)
or
(5)
The power radiatedper unit surface of a blackbody, R(T) within in a bandwidth are given by the equations
(6)
and
(7)
where
c = 2.99792458108 m.s-1 is the speed of light
kB = 1.3806610-23 J.K-1 is the Boltzmann constant
wavelength, in m
R in W.m-3 or (W.m-2).m-1 is the power radiated per unit area per unit wavelength interval
Rf in W.m-2.s or (W.m-2).Hz-1 is the power radiated per unit area per unit frequency interval
R in W.m-2 is the intensity of the emitted radiation or the power per unit area
(1, 2) wavelength interval or bandwidth
(f1, f2) frequency interval or bandwidth
The values of the integrals in equations (6) and (7) give the Stefan-Boltzmann law
(2)
when the bandwidths extend from 0 to .
Wien’s displacement law states that the wavelength, peak corresponding to the peak of the spectral intensity given by equation (4) is inversely proportional to the temperature of the blackbody and the frequency, fpeak for the spectral intensity given by equation (5) is proportional to the temperature
(8)
where
b = 2.89810-3 m.K is aWien constant
bf = (2.82 kB/h) K-1.s-1 is a Wien constant.
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 densities R 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 radial idea that was completely alien to classical notions. The energy of electromagnetic energy is quantized. Planck was able to derive the equations (6) and (7) 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. Energycan only exist in integer multiples of the lowest amount or quantum, h f.
This step marked the very beginning of quantum theory.
Cosmic Background Radiation and the Big-Bang
The early universe expanded from an initial catastrophic event – the big bang. As the universe expanded, it cooled, the greater the expansion the lower the temperature. The Planck formula is applicable to relating the radiation and its temperature. Ralph Alpher, George Gamov and Robert Herman in the late 1940’s predicted that as a consequence of the big-bang model, the existence of a this radiation. In 1963 Arno Penzias and Robert Wislon whilst using a large radio antenna to study the radio waves emitted in our galaxy, accidentally discovered that there was a background of electromagnetic radiation that was coming from no particular source and from all parts of the sky.
Satellite measurements (COBE: The Cosmic Orbiting Background Experiment) have been made of this background radiation with extraordinary precision over a range of frequencies from 61010 Hz to 61011 Hz. The measurement can be fitted to the Planck formula for blackbody radiation perfectly. The background temperature of the Universe can be determined to six significant figures. These results provide conclusive evidence to support the big-bang model of the Universe. This cosmic background radiation, the radiation that is the oldest directly observable relic of the explosion that possibly triggered the birth of the Universe.
Investigation 1: How efficient is a hot tungsten filament?
Blackbodies do not exist in nature. However, simple models are often used that assume an object such as the Sun or an incandescent lamp behave as a blackbody.
Often a car headlight uses a hot tungsten filament to emit electromagnetic radiation. We can estimate the percentage of the radiation in the visible part of the electromagnetic spectrum. Assume that the hot tungsten filament has a surface temperature of 2400 Kand the electrical power consumed by the filament is 55 W. Therefore, the thermal power radiated is also 55 W.
Step 1
Calculate the wavelength for the peak peakin the power density curve from Wien’s displacement law and state what part of the electromagnetic spectrum it is in.
Step 2
Plot the function for the thermal power radiated from the spectral intensity, Rgiven by equation (4) in the form
(11)
where N is a normalizing constant and includes a factor for the surface area. Its value is initially set to N =1.
How does the wavelength value at the peak of this function relate to that given by Wien’s displacement law?
Step 3
Numerically integrate P given by equation (11) with the limits 1 and2 so that P1 0 and P2 0.
Determine the value of N so that the thermal power radiated is, P = 55 W, i.e., the area under the curve for P is 55 W.Recalculate the normalized value of P and plot it.
Step 4
Shade the area under the P curve to show the visible part of the spectrum.
Numerically integrate the function P for the limits corresponding to only the visible part of the electromagnetic spectrum.
1 = 700 nm (red) and 2 = 400 nm (blue).
This gives only the total power radiated in the visible part of the electromagnetic spectrum, Pvisible.
Step 5
Calculate the filament efficiency, given as percentage of visible radiation emitted by the hot tungsten filament to the power consumed by the filament
.
Are you surprised by your answer?
Investigation 2: The Sun and the Earth as Blackbodies
The Sun can be assumed to be a blackbody and from its spectrum the maximum electromagnetic radiation is emitted at a wavelength, peak = 502.25 nm (yellow). The radius of the Sun is approximately, RS = 6.96108 m. From this data, we can estimate the Sun’s surface temperature, TS and its total solar power output, PS.
The mean distance between the Sun and the Earth is, RSE = 1.4961011 m and the radius of the Earth is, RE = 6.3741011 m. From this data, we can estimate the solar power per unit area at the top of the Earth’s atmosphereand the average temperature of the Earth’s surface assuming that there is no atmopshere.
Fishbane Solar energy output
total power radiated = 4x1026 W
power per unit area at the top of the earth’s atm = 1.4x103 W.m-2
Re = 6.374x106 m
sun-earth radius = 1.496x1011 m
Step 1
Estimate the temperature of the Sun’s photosphere from the Wien displacement law
(8)
and the frequency at the peak from the equation
(8)
Step 2
Estimate the total power output of the Sun from the Stefan-Boltzmann law
(2)
Step 3
Plot the spectral intensities given by the equations
(4)
(5)
How do the peaks in the plots correspond to the predictions of the Wien displacement law?
Shade in the colors of a rainbow the area under the curves to show the visible part of the electromagnetic spectrum.
Step 4
Calculate the total power radiated by the Sun by numerically integrating the spectral intensities given by equations (4) and (5). Compare these two results with the Stefan-Boltzmann prediction of equation (2) Also, by integrating only the portions of the spectral intensity, find the power output and percentages for visible, infrared and ultraviolet radiation.
MATLAB SCRIPTING
Investigation 1 How efficient is a hot tungsten filament?
Matlab file: filament.m
Matlab highlights:
- [1D] integration with Simpson’s 1/3 rule
- Shading the area under a curve
- Normalizing a function
Matlab screen outputs:
wavelength at peak (m) = 1.2075e-006
wavelength at peak (um) = 1.2075
P_total (W) = 5.500e+001
P_visible (W) = 1.925+000
efficiency (percentage) = 3.50
Matlab script:
close all
clear
nfile = 'filament.m';
% CONSTANTS declare all constants ------
c = 2.99792458e8; % speed of light
h = 6.62608e-34; % Planck constant
kB = 1.38066e-23; % Boltzmann constant
sigma = 5.6696e-8; % Stefan constant
b1 = 2.898e-3; % Wien constant - wavelength
b2 = 2.82*kB/h; % Wien constant - frequency
mu = 1e6; % convert m into micrometers
%INPUTS ------
T = 2400; % Temperature of filament
P_total = 55; % Total power radiated by filament
N = 1; % Normalizing constant, initial value N = 1
N = P_total/5.00185673e21; % adjusted normalizing constant
num = 201; % Number of data points fro wavelength range
% SETUP ------
wL_peak = b1/T; % wavelength at peak
wL1 = wL_peak/10; % min for wavelength range lambda1
wL2 = 10*wL_peak; % max for wavelength range lambda2
wL = linspace(wL1,wL2,num); % wavelengths
P_wL = zeros(num,1); % array for thermal power per unti wavelength interval
% CALCULATIONS ------
K = (h*c)/(kB*T); % constant to simply calcuation
P_wL = N ./ (wL.^5 .* (exp(K ./ wL)-1));
% Area under curve
P_total = simpson1d(P_wL,wL1,wL2);
%Find visible range
for cn = 1 : num
if wL(cn) > 400e-9
num1 = cn;
break
end; end
for cn = 1 : num
if wL(cn) > 800e-9;
num2 = cn;
break
end; end
% power radiated in visible part of spectrum
P_visible = simpson1d(P_wL,wL(num1),wL(num2));
% efficiency of filament
eff = 100*P_visible/P_total;
% GRAPHICS ______
figure(1)
h_area1 = area(mu*wL,P_wL);
hold on
h_area2 = area(mu*wL(num1:num2),P_wL(num1:num2));
set(h_area1,'FaceColor',[0 0 0]);
set(h_area2,'FaceColor',[1 1 0]);
tm1 = 'Tungsten filament: \eta = ';
tm2 = num2str(eff,3);
tm3 = ' %';
tm = [tm1 tm2 tm3]
title(tm);
xlabel('wavelength \lambda (\mum)');
ylabel('Power / d\lambda (W.m^{-1})');
% OUTPUTS ------
clc
disp(' ');
disp(nfile);
disp(' ');
fprintf(' wavelength at peak (m) = %0.4e \n',wL_peak);
fprintf(' wavelength at peak (um) = %0.4f \n',mu*wL_peak);
disp(' ');
fprintf(' P_total (W) = %0.8e \n',P_total);
disp(' ');
fprintf(' P_visible (W) = %0.8e \n',P_visible);
disp(' ')
fprintf(' efficiency (percentage) = %0.2f \n',eff);
disp(' ');
Comments:
The call to the function simpson1d.m integrates a function f between limits a and b by Simpson’s 1/3 rule.
h_area1 = area(mu*wL,P_wL);
set(h_area1,'FaceColor',[0 0 0]);
For vectors X and Y, AREA(X,Y) is the same as PLOT(X,Y) except that the area between 0 and Y is filled. The fill color is determined by the vector for the FaceColor.
h_area1 = area(mu*wL,P_wL);
set(h_area1,'FaceColor',[0 0 0]);
Investigation 2: The Sun and the Earth as Blackbodies
Matlab file: sun.m
Matlab highlights:
- [1D] integration with Simpson’s 1/3 rule
- Shading the area under a curve – associating a color of the rainbow with its wavelength
Matlab screen outputs:
sun.m
Sun: temperature of photosphere (K) = 5770
Frequency at peak in spectral intenisty(f) (Hz) = 3.39e+014
P(Stefan Boltzmann) (W) = 3.79e+026
P(wL)_total (W) = 3.77e+026
P(f)_total (W) = 3.79e+026
------
P_visible (W) = 1.39e+026
Percentage visible radiation = 36.8
P_IR (W) = 1.92e+026
Percentage IR radiation = 51.0
P_UV (W) = 4.61e+025
Percentage UV radiation = 12.2
Matlab script:
close all
clear
nfile = 'sun.m';
% CONSTANTS declare all constants ------
c = 2.99792458e8; % speed of light
h = 6.62608e-34; % Planck constant
kB = 1.38066e-23; % Boltzmann constant
sigma = 5.6696e-8; % Stefan constant
b_wL = 2.898e-3; % Wien constant - wavelength
b_f = 2.82*kB/h; % Wien constant - frequency
R_sun = 6.93e8; % Radius - Sun
%INPUTS ------
wL_peak = 5.0225e-7 % Wavelength at peak of spectral intensity
num = 201; % Number of data points fro wavelength range
% SETUP ------
T = b_wL/wL_peak; % Sun's temperature
R_wL = zeros(num,1); % array for thermal power per unti wavelength interval
R_f = zeros(num,1); % array for thermal power per unti frequency interval
T = b_wL / wL_peak; % Sun's temperature
f_peak = b_f * T; % Frequency at peak of spectral intensity(f)
A_sun = 4*pi*R_sun^2; % Surface area of Sun
P_sb = A_sun * sigma * T^4; % Total power output of Sun
% CALCULATIONS: SPECTRAL INTENSITY (WAVELENGTH) ------
wL1 = wL_peak/10; % min for wavelength range lambda1
wL2 = 10*wL_peak; % max for wavelength range lambda2
wL = linspace(wL1,wL2,num); % wavelengths
K1 = (h*c)/(kB*T); % constants to simply calculation
K2 = (2*pi*h*c^2);
R_wL = K2 ./ (wL.^5 .* (exp(K1 ./ wL)-1)); % Spectral intensity
% Spectral intensity - wL: visible
wL1_vis = 400e-9; wL2_vis = 700e-9; num_wL = 111;
wL_vis = linspace(wL1_vis,wL2_vis,num_wL);
R_wL_vis = K2 ./ (wL_vis.^5 .* (exp(K1 ./ wL_vis)-1));
% Spectral intensity - wL: IR
wL1_IR = 700e-9; wL2_IR = wL2;
wL_IR = linspace(wL1_IR,wL2_IR,num);
R_wL_IR = K2 ./ (wL_IR.^5 .* (exp(K1 ./ wL_IR)-1));
% Spectral intensity - wL: UV
wL1_UV = wL1; wL2_UV = 400e-9;
wL_UV = linspace(wL1_UV,wL2_UV,num);
R_wL_UV = K2 ./ (wL_UV.^5 .* (exp(K1 ./ wL_UV)-1));
% Area under curves
P_total = A_sun * simpson1d(R_wL,wL1,wL2);
P_vis = A_sun * simpson1d(R_wL_vis,wL1_vis,wL2_vis);
P_IR = A_sun * simpson1d(R_wL_IR,wL1_IR,wL2_IR);
P_UV = A_sun * simpson1d(R_wL_UV,wL1_UV,wL2_UV);
% Percentage radiation in visible, IR and UV
E_vis = 100 * P_vis / P_total;
E_IR = 100 * P_IR / P_total;
E_UV = 100 * P_UV / P_total;
% CALCULATIONS: SPECTRAL INTENSITY (WAVELENGTH) ------
f1 = f_peak/20; % min for frequency range
f2 = 5*f_peak; % max for frequency range
f = linspace(f1,f2,num); % frequencies
K3 = h/(kB*T); % constants to simply calcuation
K4 = (2*pi*h/c^2);
R_f = (K4 .* f.^3) ./ (exp(K3.*f)-1); % Spectral intensity
% Spectral intensity - f: visible
f1_vis = c/700e-9; f2_vis = c/400e-9; num_f = 111;
f_vis = linspace(f1_vis,f2_vis,num_f);
R_f_vis = (K4 .* f_vis.^3) ./ (exp(K3.*f_vis)-1);
% Area under curve
P_f = A_sun * simpson1d(R_f,f1,f2);
% GRAPHICS ______
figure(1)
h_area1 = area(wL,R_wL);
set(h_area1,'FaceColor',[0 0 0]);
set(h_area1,'EdgeColor','none');
hold on
thisColorMap = hsv(128);
for cn = 1 : num_wL-1
thisColor = thisColorMap(num_wL-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);
end
tm = 'Spectral Intensity - Sun';
title(tm);
xlabel('wavelength \lambda (m)');
ylabel('Spectral Intensity (W.m^{-1})');
figure(2)
h_area1 = area(f,R_f);
set(h_area1,'FaceColor',[0 0 0]);
set(h_area1,'EdgeColor','none');
tm = 'Spectral Intensity - Sun';
title(tm);
xlabel('frequency f (Hz)');
ylabel('Spectral Intensity (W.m^{-2}.Hz^{-1})');
hold on
thisColorMap = hsv(128);
for cn = 1 : num_f-1
thisColor = thisColorMap(cn,:);
h_area = area(f_vis(cn:cn+1),R_f_vis(cn:cn+1));
set(h_area,'FaceColor',thisColor);
set(h_area,'EdgeColor',thisColor);
end
% OUTPUTS ------
clc
disp(' ');
disp(nfile);
disp(' ');
fprintf(' Sun: temperature of photosphere (K) = %0.0f \n',T);
disp(' ');
fprintf(' Frequency at peak in spectral intenisty(f) (Hz) = %0.2e \n',f_peak);
disp(' ');
fprintf(' P(Stefan Boltzmann) (W) = %0.2e \n',P_sb);
disp(' ');
fprintf(' P(wL)_total (W) = %0.2e \n',P_total);
fprintf(' P(f)_total (W) = %0.2e \n',P_f);
disp(' ');
disp('------');
disp(' ');
fprintf(' P_visible (W) = %0.2e \n',P_vis);
disp(' ');
fprintf(' Percentage visible radiation = %0.1f \n',E_vis);
disp(' ');
fprintf(' P_IR (W) = %0.2e \n',P_IR);
disp(' ');
fprintf(' Percentage IR radiation = %0.1f \n',E_IR);
disp(' ');
fprintf(' P_UV (W) = %0.2e \n',P_UV);
disp(' ');
fprintf(' Percentage UV radiation = %0.1f \n',E_UV);
Comments: