Question

Using a programming language of your choice: 1) generate a range of wavelengths, 2) compute corresponding monochromatic blackbody intensity using a) Planck function, b) Rayleigh-Jeans simplification, and c) Wien simplification, and 3) plot the three resulting spectra (i.e., a diagram that shows how B(λ) changes with λ). Using this figure explain the phenomenon of “ultraviolet catastrophe”. Please include the code (not only figure) in your answer.

Answer 1

Here is a Matlab code for the part 1-3 of the above question

% Program to demonstrate laws of blackbody radiation, ultraviolet

% catastrophe

% MKS units are used

clear all;

h=6.6*10^(-34);c=3*10^8;k=1.38*10^(-23);

lambda=linspace(0.00001*10^(-6),4*10^(-6),1000); % Generating wave length grid

nu=c./lambda;

%Blackbody intensity calculation

T=5000;

Bp=2*h*nu.^3/c^2./(exp(h.*nu/k/T)-1); %Planck's

Brj=2*nu.^2*k*T/c^2; %Rayleigh-Jeans'

Bw=2*h*nu.^3/c^2.*exp(-h.*nu/k/T); %Wien's

% Plot

figure;

plot(lambda,Bp,lambda,Brj,lambda,Bw)

xlabel('wave length','FontSize',20)

ylabel('black body intensity','FontSize',20)

set(gca,'FontSize',15)

ylim([0 3*10^-8])

grid on;

print('bbs', '-dpng', '-r600')

The plot of the spectra is shown below

From the above figure one can see that the blackbody radiation intensity as predicted by the Rayleigh and Jeans is increasing as the wavelength of the emission decreases (~i.e. towards ultraviolet from a long wavelength, e.g. infrared). For this case the value of the intensity is ~1400 as lambdaà0.00001 micrometer. This is contrary to the observation and termed as ultraviolet catastrophe.

Appendix

nu=c/lambda;

This along with the above expressions are implemented in the code.