Project Help Notebook by Keith Stroyan

Help NoteBook Copyright 1998 by Academic Press, Inc. - All rights reserved.
Modify this Notebook to prepare your project report.

Project 20:

by A. Student and B. Pupil

with technical help from Da LabMonitor and Help Notebook

colors

Needs["Graphics`Colors`"];

Hot bodies Glow

Write this last.  Begin with the basic question and state your main finding clearly, but with few technical details.  You might even modify the title of this section to make it more interesting.

Planck's Radiation Law 
in Wavelength λ Form

Notebook Overview

Planck's Law is a formula for the intensity of each frequency radiating from a black body.  This notebook takes you through the computations using Planck's Law in wavelength form that are described in Project 20.

Main Computations

The general intensity differential

Clear[i, λ, a, b, T]; i[λ_] := 
      (a/((λ^5) 
          ((E^((b/(λ T)))) - 1)))i[λ] dλ

The derivative of i with respect to λ

D[i[λ],λ]

Together[%]

The c-g-s values of the physical constants

ℏ = 6.6255 10^(-27); (*Planck's constant in  erg sec*)
c = 2.9979 10^(10) ;  (*Speed of light in cm/sec*)
k = 1.3805 10^(-16);  (*Boltzman's constant in erg/deg*)
a = 8 N[Pi] ℏ c^2;
b = ℏ c/k;
i[λ]

The special case of T = 1000

T = 10^3 ;
i[λ]

Plot[i[λ],{λ,0,10},AxesLabel->{"λ","i"},PlotStyle->Red];

Plot[i[λ],{λ,0,6.5 10^(-5)},
Ticks ->{{2. 10^(-5),4. 10^(-5), 6. 10^(-5)},Automatic},AxesLabel->{"λ","i"},PlotStyle->Red];

Plot[i[λ],{λ,0,.002},
Ticks ->{{.0005,.001,.0015},Automatic},AxesLabel->{"λ","i"},PlotStyle->Red];

The surface plot of I vs. λ and T

Clear[T]
l := L/1000
Plot3D[i[l],{L,0,.5},{T,500,1500},BoxRatios -> {1,1,1}
		,ViewPoint->{-1.949, -4.924, 2.316},
		PlotPoints -> 30];

Find the Root of di/dλ = 0

Plot[{5 , (5 - x) Exp[x]},{x,0,6},PlotStyle->{Blue,DarkGreen}];

FindRoot[5 == (5 - x) Exp[x] , {x,5}]

Planck's Radiation Law 
in Frequency ω Form

Notebook Overview

Planck's Law is a formula for the intensity of each frequency radiating from a black body.  This  NoteBook helps with some computations using Planck's Law in frequency form.

Hint 26.5.

Enter your formula for i[ω] here (use i, because Math'ca uses I = Sqrt[-1]):

Planck's Law written in terms of frequency does not lead to Wien's Law.  We want you to show this by finding the maximum of i[ω].  Use this notebook to help with the computations and sketch curves like the ones we have done for wavelength.  The intensity formula with frequency is

i = ℏ ω3 / ( π2 c2 ((e^ℏ) ω / (k T) - 1) )
		   = α ω3 / (eβ ω / T - 1)

i[ω_] := ????? ; (*General formula for i[ω]*)
i[ω] dω

Compute the derivative

(di/(dω)) = a ω2  ((3 - b ω / T) eβ ω / T - 3) / (eβ ω / T - 1)2

D[i[ω],ω]       (*The derivative of your i[ω] formula*)

The c-g-s values of the physical constants

ℏ = 6.6255 10^(-27); (*Planck's constant in  erg sec*)
c = 2.9979 10^(10) ;  (*Speed of light in cm/sec*)
k = 1.3805 10^(-16);  (*Boltzman's constant in erg/deg*)
a = 8 N[Pi] ℏ c^2;
b = ℏ c/k; 
i[ω]

Your plot of  I vs. w at temperature 1000 degrees K

Run this cell as is, first.  You can not clearly see what the graph looks like using the given plot range.  Now, change the plot range by powers of 10 until you get a good graph.

T = 10^3 ;      (*A particular temperature*)
Plot[i[ω],{ω,.1,10}]

Continue solving Hint 26.5 with help from Mathematica.

Finally

(Optional) Ask your neighborhood quantum mechanic why maximum frequency gives a different Wien's law.