Question

Using the programming language Python, 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')

% screenshot : The plot of the spectra :-

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.