.MCAD 304020000 1 74 333 0
.CMD PLOTFORMAT
0 0 1 1 1 0 0 1 1 
0 0 1 1 1 0 0 1 1 
0 1 0 0 1 1 NO-TRACE-STRING
0 2 1 0 1 1 NO-TRACE-STRING
0 3 2 0 1 1 NO-TRACE-STRING
0 4 3 0 1 1 NO-TRACE-STRING
0 1 4 0 1 1 NO-TRACE-STRING
0 2 5 0 1 1 NO-TRACE-STRING
0 3 6 0 1 1 NO-TRACE-STRING
0 4 0 0 1 1 NO-TRACE-STRING
0 1 1 0 1 1 NO-TRACE-STRING
0 2 2 0 1 1 NO-TRACE-STRING
0 3 3 0 1 1 NO-TRACE-STRING
0 4 4 0 1 1 NO-TRACE-STRING
0 1 5 0 1 1 NO-TRACE-STRING
0 2 6 0 1 1 NO-TRACE-STRING
0 3 0 0 1 1 NO-TRACE-STRING
0 4 1 0 1 1 NO-TRACE-STRING
0 1 1 21 15 0 0 3 
.CMD FORMAT  rd=d ct=10 im=i et=3 zt=40 pr=3 mass length time charge temperature tr=0 vm=0
.CMD SET ORIGIN 0
.CMD SET TOL 0.001000000000000
.CMD SET PRNCOLWIDTH 8
.CMD SET PRNPRECISION 4
.CMD PRINT_SETUP 1.200000 1.218750 1.200000 1.200000 0
.CMD HEADER_FOOTER 1 1 *empty* *empty* *empty* 0 1 *empty* *empty* *empty*
.CMD HEADER_FOOTER_FONT fontID=14 family=Arial points=10 bold=0 italic=0 underline=0 colrid=1733264598
.CMD HEADER_FOOTER_FONT fontID=15 family=Arial points=10 bold=0 italic=0 underline=0 colrid=1733264598
.CMD DEFAULT_TEXT_PARPROPS 0 0 0
.CMD DEFINE_FONTSTYLE_NAME fontID=0 name=Variables
.CMD DEFINE_FONTSTYLE_NAME fontID=1 name=Constants
.CMD DEFINE_FONTSTYLE_NAME fontID=2 name=Text
.CMD DEFINE_FONTSTYLE_NAME fontID=4 name=User^1
.CMD DEFINE_FONTSTYLE_NAME fontID=5 name=User^2
.CMD DEFINE_FONTSTYLE_NAME fontID=6 name=User^3
.CMD DEFINE_FONTSTYLE_NAME fontID=7 name=User^4
.CMD DEFINE_FONTSTYLE_NAME fontID=8 name=User^5
.CMD DEFINE_FONTSTYLE_NAME fontID=9 name=User^6
.CMD DEFINE_FONTSTYLE_NAME fontID=10 name=User^7
.CMD DEFINE_FONTSTYLE fontID=0 family=Times^New^Roman points=10 bold=0 italic=0 underline=0 colrid=-1
.CMD DEFINE_FONTSTYLE fontID=1 family=Times^New^Roman points=10 bold=0 italic=0 underline=0 colrid=-1
.CMD DEFINE_FONTSTYLE fontID=2 family=Arial points=12 bold=0 italic=0 underline=0 colrid=5
.CMD DEFINE_FONTSTYLE fontID=4 family=Arial points=10 bold=0 italic=0 underline=0 colrid=-1
.CMD DEFINE_FONTSTYLE fontID=5 family=Courier^New points=10 bold=0 italic=0 underline=0 colrid=-1
.CMD DEFINE_FONTSTYLE fontID=6 family=System points=10 bold=0 italic=0 underline=0 colrid=-1
.CMD DEFINE_FONTSTYLE fontID=7 family=Script points=10 bold=0 italic=0 underline=0 colrid=-1
.CMD DEFINE_FONTSTYLE fontID=8 family=Roman points=10 bold=0 italic=0 underline=0 colrid=-1
.CMD DEFINE_FONTSTYLE fontID=9 family=Modern points=10 bold=0 italic=0 underline=0 colrid=-1
.CMD DEFINE_FONTSTYLE fontID=10 family=Times^New^Roman points=10 bold=0 italic=0 underline=0 colrid=-1
.CMD UNITS U=1
.CMD DIMENSIONS_ANALYSIS 0 0
.CMD COLORTAB_ENTRY 0 0 0
.CMD COLORTAB_ENTRY 128 0 0
.CMD COLORTAB_ENTRY 0 128 0
.CMD COLORTAB_ENTRY 128 128 0
.CMD COLORTAB_ENTRY 0 0 128
.CMD COLORTAB_ENTRY 128 0 128
.CMD COLORTAB_ENTRY 0 128 128
.CMD COLORTAB_ENTRY 128 128 128
.CMD COLORTAB_ENTRY 192 192 192
.CMD COLORTAB_ENTRY 255 0 0
.CMD COLORTAB_ENTRY 0 255 0
.CMD COLORTAB_ENTRY 255 255 0
.CMD COLORTAB_ENTRY 0 0 255
.CMD COLORTAB_ENTRY 255 0 255
.CMD COLORTAB_ENTRY 0 255 255
.CMD COLORTAB_ENTRY 255 255 255
.CMD COLORTAB_ENTRY 0 64 128
.TXT 2 1 7 0 0
Cg b73.000000,73.000000,514
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;\red0\green64\blue128;}{
\fonttbl{\f0\fcharset0\fnil Arial;}{\f1\fcharset0\fnil Century Gothic;}{
\f2\fcharset2\fnil Symbol;}}\plain\cf1\fs24 \pard COPYRIGHT MIRON 
KAUFMAN, 1997\par {\b \par               }{\f1\fs36\ul STATISTICAL 
PHYSICS COMPUTER LAB }{\f1\fs36\ul #3}{\f1\fs32 \par }{\fs28 \par }We 
study the black-body (electromagnetic) radiation in thermal 
equilibrium at a temperature T inside a container of volume V.  We 
will start with the free energy.  We will then get the energy U(V,T). 
Then we will study the spectral energy density \tab u{\f2\dn w}{\f2  }{
\cf2 = }dU/d{\f2 w } and the spectral energy density u{\f2\dn l} {\cf2 
= }dU/d{\f2 l}.  Next we will derive the Wien displacement law.  
Finally we will compute the number of photons in a volume V at a 
temperature T.}
.EQN 25 0 119 0 0
{0:c}NAME:3*(10)^(8)
.EQN 0 8 120 0 0
{0:k.B}NAME:1.381*(10)^(-23)
.EQN 0 13 121 0 0
{0:hbar}NAME:1.05457*(10)^(-34)
.EQN 0 18 193 0 0
{0:h}NAME:{0:hbar}NAME*2*{0:\p}NAME
.EQN 5 -18 330 0 0
{0:F}NAME({0:V}NAME,{0:T}NAME):(({0:k.B}NAME)^(4))/(({0:\p}NAME)^(2)*({0:c}NAME)^(3)*({0:hbar}NAME)^(3))*{0:V}NAME*({0:T}NAME)^(4)*(0&{0:\¥}NAME`({0:x}NAME)^(2)*{0:ln}NAME(1-({0:e}NAME)^(-{0:x}NAME))&{0:x}NAME)
.TXT 1 -21 264 0 0
Cg a72.000000,72.000000,19
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard The free energy is:}
.TXT 7 0 265 0 0
Cg a68.750000,68.750000,76
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;\red0\green64\blue128;}{
\fonttbl{\f0\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard The integral 
is done by using {\cf2\ul Floating Point Evaluation f}rom under {\cf2
\ul Symbolic}.}
.EQN 7 0 132 0 0
(0&{0:\¥}NAME`({0:x}NAME)^(2)*{0:ln}NAME(1-({0:e}NAME)^(-{0:x}NAME))&{0:x}NAME)
.EQN 6 0 133 0 0
-2.16464646742228
.TXT 4 0 134 0 0
Cg a73.000000,73.000000,27
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard The integral is equal 
to: {\object{\*\objclass \eqn}\rsltpict{\*\objdata 
.EQN 57 16 135 0 0
-((({0:\p}NAME)^(4))/(45))={19007}?_n_u_l_l_
}}}
.TXT 6 0 137 0 0
Cg a73.000000,73.000000,36
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard The Stefan constant is 
defined as: {\object{\*\objclass \eqn}\rsltpict{\*\objdata 
.EQN 63 23 138 0 0
{0:a}NAME:3*(({0:\p}NAME)^(4))/(45)*(({0:k.B}NAME)^(4))/(({0:\p}NAME)^(2)*({0:c}NAME)^(3)*({0:hbar}NAME)^(3))
}}}
.EQN 3 47 317 0 0
{0:a}NAME={19187}?_n_u_l_l_
.TXT 3 -47 144 0 0
Cg a73.000000,73.000000,7
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard Then: {\object{
\*\objclass \eqn}\rsltpict{\*\objdata 
.EQN 69 5 145 0 0
{0:F}NAME({0:V}NAME,{0:T}NAME):-(({0:a}NAME)/(3))*{0:V}NAME*({0:T}NAME)^(4)
}}}
.TXT 4 0 266 0 0
Cg a72.000000,72.000000,110
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard This is the fundamental 
equation. We find the pressure from: p = -dF/dV and the entropy from:      
S = -dF/dT.}
.EQN 7 0 147 0 0
{0:p}NAME({0:T}NAME):({0:a}NAME)/(3)*({0:T}NAME)^(4)
.EQN 0 19 148 0 0
{0:S}NAME({0:V}NAME,{0:T}NAME):(4)/(3)*{0:a}NAME*{0:V}NAME*({0:T}NAME)^(3)
.TXT 19 -19 298 0 0
Cg b73.000000,73.000000,30
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard {\fs16 COPYRIGHT MIRON 
KAUFMAN, 1997\par }}
.EQN 7 36 296 0 0
{0:V}NAME:1
.EQN 0 15 297 0 0
{0:T}NAME:0;500
.EQN 3 -51 295 0 0
&&(_n_u_l_l_&_n_u_l_l_)&{0:S}NAME({0:V}NAME,{0:T}NAME)@&&(_n_u_l_l_&_n_u_l_l_)&{0:T}NAME
0 1 1 1 1 0 0 1 1 
0 1 1 1 1 0 0 1 1 
0 1 0 0 1 1 NO-TRACE-STRING
0 2 1 0 1 1 NO-TRACE-STRING
0 3 2 0 1 1 NO-TRACE-STRING
0 4 3 0 1 1 NO-TRACE-STRING
0 1 4 0 1 1 NO-TRACE-STRING
0 2 5 0 1 1 NO-TRACE-STRING
0 3 6 0 1 1 NO-TRACE-STRING
0 4 0 0 1 1 NO-TRACE-STRING
0 1 1 0 1 1 NO-TRACE-STRING
0 2 2 0 1 1 NO-TRACE-STRING
0 3 3 0 1 1 NO-TRACE-STRING
0 4 4 0 1 1 NO-TRACE-STRING
0 1 5 0 1 1 NO-TRACE-STRING
0 2 6 0 1 1 NO-TRACE-STRING
0 3 0 0 1 1 NO-TRACE-STRING
0 4 1 0 1 1 NO-TRACE-STRING
0 1 1 57 23 10 0 3 
.TXT 34 1 273 0 0
Cg a72.000000,72.000000,64
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;\red0\green0\blue0;}{
\fonttbl{\f0\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard The energy 
is U = F +TS = aVT{\up 4}{\cf2\up  }and the heat capacity C{\dn V} = 4aVT{
\fs20\up 3}{\cf2\fs20\up  }}
.EQN 4 0 150 0 0
{0:U}NAME({0:V}NAME,{0:T}NAME):{0:a}NAME*{0:V}NAME*({0:T}NAME)^(4)
.EQN 0 16 321 0 0
{0:C.V}NAME({0:V}NAME,{0:T}NAME):4*{0:a}NAME*{0:V}NAME*({0:T}NAME)^(3)
.EQN 4 -17 69 0 0
&&(_n_u_l_l_&_n_u_l_l_)&{0:U}NAME({0:V}NAME,{0:T}NAME)@&&(_n_u_l_l_&_n_u_l_l_)&{0:T}NAME
0 1 1 1 1 0 0 1 1 
0 1 1 1 1 0 0 1 1 
0 1 0 0 1 1 NO-TRACE-STRING
0 2 1 0 1 1 NO-TRACE-STRING
0 3 2 0 1 1 NO-TRACE-STRING
0 4 3 0 1 1 NO-TRACE-STRING
0 1 4 0 1 1 NO-TRACE-STRING
0 2 5 0 1 1 NO-TRACE-STRING
0 3 6 0 1 1 NO-TRACE-STRING
0 4 0 0 1 1 NO-TRACE-STRING
0 1 1 0 1 1 NO-TRACE-STRING
0 2 2 0 1 1 NO-TRACE-STRING
0 3 3 0 1 1 NO-TRACE-STRING
0 4 4 0 1 1 NO-TRACE-STRING
0 1 5 0 1 1 NO-TRACE-STRING
0 2 6 0 1 1 NO-TRACE-STRING
0 3 0 0 1 1 NO-TRACE-STRING
0 4 1 0 1 1 NO-TRACE-STRING
0 1 1 24 33 10 0 3 
.EQN 2 35 324 0 0
&&(_n_u_l_l_&_n_u_l_l_)&{0:C.V}NAME({0:V}NAME,{0:T}NAME)@&&(_n_u_l_l_&_n_u_l_l_)&{0:T}NAME
0 1 1 1 1 0 0 1 1 
0 1 1 1 1 0 0 1 1 
0 1 0 0 1 1 NO-TRACE-STRING
0 2 1 0 1 1 NO-TRACE-STRING
0 3 2 0 1 1 NO-TRACE-STRING
0 4 3 0 1 1 NO-TRACE-STRING
0 1 4 0 1 1 NO-TRACE-STRING
0 2 5 0 1 1 NO-TRACE-STRING
0 3 6 0 1 1 NO-TRACE-STRING
0 4 0 0 1 1 NO-TRACE-STRING
0 1 1 0 1 1 NO-TRACE-STRING
0 2 2 0 1 1 NO-TRACE-STRING
0 3 3 0 1 1 NO-TRACE-STRING
0 4 4 0 1 1 NO-TRACE-STRING
0 1 5 0 1 1 NO-TRACE-STRING
0 2 6 0 1 1 NO-TRACE-STRING
0 3 0 0 1 1 NO-TRACE-STRING
0 4 1 0 1 1 NO-TRACE-STRING
0 1 1 24 32 10 0 3 
.TXT 52 -35 300 0 0
Cg b73.000000,73.000000,30
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard {\fs16 COPYRIGHT MIRON 
KAUFMAN, 1997}{\fs20 \par }}
.TXT 7 1 151 0 0
Cg a59.375000,59.375000,51
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}{\f1\fcharset2\fnil Symbol;}}\plain\cf1\fs24 
\pard u{\f1\fs20\dn w} = dU/d{\f1\fs20 w} is the frequency spectral 
energy density}
.EQN 6 -1 73 0 0
{0:u.\w}NAME({0:\w}NAME,{0:V}NAME,{0:T}NAME):({0:V}NAME*{0:hbar}NAME)/(({0:\p}NAME)^(2)*({0:c}NAME)^(3))*(({0:\w}NAME)^(3))/(({0:e}NAME)^(({0:hbar}NAME*{0:\w}NAME)/({0:k.B}NAME*{0:T}NAME))-1)
.EQN 0 27 75 0 0
{0:\w}NAME:5*(10)^(11),6*(10)^(11);5*(10)^(14)
.EQN 6 -27 74 0 0
&&(_n_u_l_l_&_n_u_l_l_)&{0:u.\w}NAME({0:\w}NAME,1,300)@&&(_n_u_l_l_&_n_u_l_l_)&{0:\w}NAME
0 1 1 1 1 0 0 1 1 
0 1 1 1 1 0 0 1 1 
0 1 0 0 1 1 NO-TRACE-STRING
0 2 1 0 1 1 NO-TRACE-STRING
0 3 2 0 1 1 NO-TRACE-STRING
0 4 3 0 1 1 NO-TRACE-STRING
0 1 4 0 1 1 NO-TRACE-STRING
0 2 5 0 1 1 NO-TRACE-STRING
0 3 6 0 1 1 NO-TRACE-STRING
0 4 0 0 1 1 NO-TRACE-STRING
0 1 1 0 1 1 NO-TRACE-STRING
0 2 2 0 1 1 NO-TRACE-STRING
0 3 3 0 1 1 NO-TRACE-STRING
0 4 4 0 1 1 NO-TRACE-STRING
0 1 5 0 1 1 NO-TRACE-STRING
0 2 6 0 1 1 NO-TRACE-STRING
0 3 0 0 1 1 NO-TRACE-STRING
0 4 1 0 1 1 NO-TRACE-STRING
0 1 1 59 47 10 0 3 
.TXT 83 0 301 0 0
Cg b73.000000,73.000000,29
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard {\fs16 COPYRIGHT MIRON 
KAUFMAN, 1997}}
.TXT 9 1 157 0 0
Cg a72.000000,72.000000,191
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;\red0\green0\blue0;
\red0\green64\blue128;}{\fonttbl{\f0\fcharset0\fnil Arial;}{\f1
\fcharset2\fnil Symbol;}}\plain\cf1\fs24 \pard The spectral energy 
density has a maximum.  We determine it by setting the derivative of u{
\cf2\f1\fs20\dn w}{\cf3\fs20  }{\cf3 equal to zero.}{\cf3\fs20 \par }{
\cf3 Derivative of: }{\object{\*\objclass \eqn}\rsltpict{\*\objdata 
.EQN 322 20 159 0 0
((({0:x}NAME)^(3))/(({0:e}NAME)^({0:x}NAME)-1))
}}{\cf3  is obtained by using }{\cf3\ul Differentiate on Variable}{\cf3  
from under }{\cf3\ul Symbolic}{\cf3 .}}
.EQN 15 0 104 0 0
3*(({0:x}NAME)^(2))/(({0:exp}NAME({0:x}NAME)-1))-(({0:x}NAME)^(3))/((({0:exp}NAME({0:x}NAME)-1))^(2))*{0:exp}NAME({0:x}NAME)
.TXT 6 0 162 0 0
Cg a72.000000,72.000000,38
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard Then use {\ul Simplify} 
from under {\ul Symbolic}:}
.EQN 4 0 168 0 0
-({0:x}NAME)^(2)*((-3*{0:exp}NAME({0:x}NAME)+3+{0:x}NAME*{0:exp}NAME({0:x}NAME)))/((({0:exp}NAME({0:x}NAME)-1))^(2))
.TXT 7 0 169 0 0
Cg a72.000000,72.000000,99
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard Next set the numerator 
equal to zero. We use the root function to solve the transcedental equation.}
.EQN 5 2 166 0 0
{0:x}NAME:3
.EQN 3 -3 167 0 0
{0:x.max}NAME:{0:root}NAME(-3*{0:exp}NAME({0:x}NAME)+3+{0:x}NAME*{0:exp}NAME({0:x}NAME),{0:x}NAME)
.EQN 3 4 173 0 0
{0:x.max}NAME={0}?_n_u_l_l_
.TXT 6 -4 260 0 0
Cg a71.000000,71.000000,23
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;\red0\green64\blue128;
\red0\green0\blue0;}{\fonttbl{\f0\fcharset0\fnil Arial;}{\f1\fcharset2
\fnil Symbol;}}\plain\cf1\fs24 \pard The frequency {\cf2\f1\fs20 w}{
\cf3\fs20\dn max}{\cf3\fs20  is:}{\object{\*\objclass \eqn}\rsltpict{
\*\objdata 
.EQN 365 16 177 0 0
{0:\w.max}NAME({0:T}NAME):({0:x.max}NAME*{0:k.B}NAME)/({0:hbar}NAME)*{0:T}NAME
}}}
.TXT 7 1 180 0 0
Cg a72.000000,72.000000,6
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard Now: {\object{
\*\objclass \eqn}\rsltpict{\*\objdata 
.EQN 372 6 181 0 0
({0:x.max}NAME*{0:k.B}NAME)/({0:hbar}NAME)={84723}?_n_u_l_l_
}}}
.TXT 6 -1 278 0 0
Cg a43.000000,43.000000,8
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard So at: {\object{
\*\objclass \eqn}\rsltpict{\*\objdata 
.EQN 378 5 184 0 0
{0:T}NAME:300
}}}
.TXT 2 14 290 0 0
Cg a47.375000,47.375000,23
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}{\f1\fcharset2\fnil Symbol;}}\plain\cf1\fs24 
\pard the frequency {\f1 w}{\dn max} is: }
.EQN 0 21 291 0 0
{0:\w.max}NAME({0:T}NAME)={0}?_n_u_l_l_
.TXT 32 -35 292 0 0
Cg b73.000000,73.000000,30
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard {\fs16 COPYRIGHT MIRON 
KAUFMAN, 1997}{\fs20 \par }}
.TXT 7 2 190 0 0
Cg a71.000000,71.000000,58
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;\red0\green0\blue0;
\red0\green64\blue128;}{\fonttbl{\f0\fcharset0\fnil Arial;}{\f1
\fcharset2\fnil Symbol;}}\plain\cf1\fs24 \pard The wavelength spectral 
energy frequency is u{\cf2\f1\fs20\dn l}{\cf3\fs20  = u}{\cf2\f1\fs20
\dn w}{\cf3\fs20 |d}{\cf3\f1\fs20 w}{\cf3\fs20 /d}{\cf3\f1\fs20 l}{\cf3
\fs20 |}}
.EQN 4 1 191 0 0
{0:u.\l}NAME({0:\l}NAME,{0:V}NAME,{0:T}NAME):(8*{0:\p}NAME*{0:h}NAME*{0:c}NAME*{0:V}NAME)/(({0:\l}NAME)^(5)*(({0:e}NAME)^(({0:h}NAME*{0:c}NAME)/({0:k.B}NAME*{0:T}NAME*{0:\l}NAME))-1))
.EQN 9 1 194 0 0
{0:\l}NAME:(10)^(-6),1.1*(10)^(-6);(10)^(-4)
.EQN 1 -4 196 0 0
&&(_n_u_l_l_&_n_u_l_l_)&{0:u.\l}NAME({0:\l}NAME,1,200),{0:u.\l}NAME({0:\l}NAME,1,300),{0:u.\l}NAME({0:\l}NAME,1,400)@1*(10)^(-4)&&(_n_u_l_l_&_n_u_l_l_)&{0:\l}NAME
0 1 1 1 1 0 0 1 1 
0 1 1 1 1 0 0 1 1 
0 1 0 0 1 1 NO-TRACE-STRING
0 3 1 0 1 1 NO-TRACE-STRING
0 4 3 0 1 1 NO-TRACE-STRING
0 4 3 0 1 1 NO-TRACE-STRING
0 1 4 0 1 1 NO-TRACE-STRING
0 2 5 0 1 1 NO-TRACE-STRING
0 3 6 0 1 1 NO-TRACE-STRING
0 4 0 0 1 1 NO-TRACE-STRING
0 1 1 0 1 1 NO-TRACE-STRING
0 2 2 0 1 1 NO-TRACE-STRING
0 3 3 0 1 1 NO-TRACE-STRING
0 4 4 0 1 1 NO-TRACE-STRING
0 1 5 0 1 1 NO-TRACE-STRING
0 2 6 0 1 1 NO-TRACE-STRING
0 3 0 0 1 1 NO-TRACE-STRING
0 4 1 0 1 1 NO-TRACE-STRING
0 1 1 57 67 10 0 3 
.TXT 82 0 83 0 0
Cg b73.000000,73.000000,29
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard {\fs16 COPYRIGHT MIRON 
KAUFMAN, 1997}}
.TXT 4 0 203 0 0
Cg a68.000000,68.000000,138
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;\red0\green0\blue0;
\red0\green64\blue128;}{\fonttbl{\f0\fcharset0\fnil Arial;}{\f1
\fcharset2\fnil Symbol;}}\plain\cf1\fs24 \pard The maximum in u{\cf2\f1
\dn l}{\cf3  is obtained by setting the derivative du}{\cf2\f1\dn l}{
\cf3 /d}{\cf3\f1 l}{\cf3  equal to zero. Equivalently we differentiate }{
\object{\*\objclass \eqn}\rsltpict{\*\objdata 
.EQN 522 16 212 0 0
(({0:e}NAME)^({0:x}NAME)-1)/(({0:x}NAME)^(5))
}}{\cf3  with respect to x and get:  }}
.EQN 9 0 213 0 0
({0:exp}NAME({0:x}NAME))/(({0:x}NAME)^(5))-5*(({0:exp}NAME({0:x}NAME)-1))/(({0:x}NAME)^(6))
.TXT 8 1 214 0 0
Cg a72.000000,72.000000,9
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard Simplify:}
.EQN 5 2 206 0 0
(({0:x}NAME*{0:exp}NAME({0:x}NAME)-5*{0:exp}NAME({0:x}NAME)+5))/(({0:x}NAME)^(6))
.TXT 5 -3 215 0 0
Cg a73.000000,73.000000,12
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard Solve for x:}
.EQN 6 0 216 0 0
{0:x}NAME:5
.EQN 3 1 223 0 0
{0:x.max}NAME:{0:root}NAME(({0:x}NAME*{0:exp}NAME({0:x}NAME)-5*{0:exp}NAME({0:x}NAME)+5),{0:x}NAME)
.EQN 3 2 225 0 0
{0:x.max}NAME={0}?_n_u_l_l_
.TXT 6 -2 219 0 0
Cg a72.000000,72.000000,129
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;\red0\green0\blue0;
\red0\green64\blue128;}{\fonttbl{\f0\fcharset0\fnil Arial;}{\f1
\fcharset2\fnil Symbol;}}\plain\cf1\fs24 \pard Since: x = hc/k{\cf2\dn B}T{
\cf3\f1\fs20 l }it follows: {\cf3\f1\fs20 l}{\cf2\dn max}*T = hc/k{\cf2
\dn B}x{\cf2\dn max                         } {\object{\*\objclass 
\eqn}\rsltpict{\*\objdata 
.EQN 564 46 229 0 0
({0:h}NAME*{0:c}NAME)/({0:k.B}NAME*{0:x.max}NAME)={0}?_n_u_l_l_
}}{\cf3\f1\fs20  }This equation is known as the {\b Wien displacement law}.}
.EQN 12 3 228 0 0
{0:\l.max}NAME({0:T}NAME):(2.899*(10)^(-3))/({0:T}NAME)
.TXT 8 -2 302 0 0
Cg a71.000000,71.000000,103
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}{\f1\fcharset2\fnil Symbol;}}\plain\cf1\fs24 
\pard The Sun' s spectrum has a maximum at {\f1 l} = 470nm.  Then by 
Wien's law the Sun's surface temperature is: {\object{\*\objclass \eqn}
\rsltpict{\*\objdata 
.EQN 587 18 303 0 0
(2.899*(10)^(-3))/(4.7*(10)^(-7))={0}?_n_u_l_l_
}}}
.TXT 10 1 305 0 0
Cg a70.000000,70.000000,45
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard The energy per volume 
from all spectrum is: {\object{\*\objclass \eqn}\rsltpict{\*\objdata 
.EQN 594 33 306 0 0
{0:a}NAME*(6168)^(4)={0}?_n_u_l_l_
}}}
.TXT 5 -3 326 0 0
Cg a70.000000,70.000000,80
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard The energy per volume 
from the visible part of the spectrum, 400nm to 700nm, is:}
.EQN 7 3 308 0 0
(4*(10)^(-7)&7*(10)^(-7)`{0:u.\l}NAME({0:\l}NAME,1,6168)&{0:\l}NAME)={19187}?_n_u_l_l_
.TXT 7 0 309 0 0
Cg a70.000000,70.000000,64
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard Hence the visible energy 
represents: {\object{\*\objclass \eqn}\rsltpict{\*\objdata 
.EQN 613 28 310 0 0
(0.417)/(1.094)={0}?_n_u_l_l_
}}of the total solar energy.}
.TXT 6 -3 281 0 0
Cg b73.000000,73.000000,30
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard {\fs16 COPYRIGHT MIRON 
KAUFMAN, 1997}{\fs20 \par }}
.TXT 6 0 117 0 0
Cg a67.625000,67.625000,82
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard Next we calculate the 
average number of photons in a volume V at temperature T\tab  {\object{
\*\objclass \eqn}\rsltpict{\*\objdata 
.EQN 630 15 232 0 0
{0:N}NAME({0:V}NAME,{0:T}NAME):(({0:k.B}NAME)^(3))/(({0:\p}NAME)^(2)*({0:c}NAME)^(3)*({0:hbar}NAME)^(3))*{0:V}NAME*({0:T}NAME)^(3)*(0&{0:\¥}NAME`(({0:x}NAME)^(2))/(({0:e}NAME)^({0:x}NAME)-1)&{0:x}NAME)
}}.}
.TXT 16 2 251 0 0
Cg a71.000000,71.000000,74
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;\red0\green64\blue128;}{
\fonttbl{\f0\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard We do the 
integral by using {\cf2\ul Floating Point Evaluation} from under {\cf2
\ul Symbolic}.}
.EQN 8 1 236 0 0
(0&{0:\¥}NAME`(({0:x}NAME)^(2))/(({0:e}NAME)^({0:x}NAME)-1)&{0:x}NAME)
.EQN 8 0 237 0 0
2.40411380631918
.TXT 5 -1 254 0 0
Cg a71.000000,71.000000,12
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard Then:{\object{
\*\objclass \eqn}\rsltpict{\*\objdata 
.EQN 662 6 256 0 0
2.40411380631918*(({0:k.B}NAME)^(3))/(({0:\p}NAME)^(2)*({0:c}NAME)^(3)*({0:hbar}NAME)^(3))={0}?_n_u_l_l_
}} or: {\object{\*\objclass \eqn}\rsltpict{\*\objdata 
.EQN 662 40 258 0 0
{0:N}NAME({0:V}NAME,{0:T}NAME):2.026032049498775*(10)^(7)*{0:V}NAME*({0:T}NAME)^(3)
}}}
.TXT 9 -2 311 0 0
Cg a71.500000,71.500000,195
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard The average number of 
photons inside a one cubic meter volume at room temperature 300K is:
\tab {\object{\*\objclass \eqn}\rsltpict{\*\objdata 
.EQN 673 13 327 0 0
{0:N}NAME(1,300)={0}?_n_u_l_l_
}}\par The average number of photons inside a one cubic meter volume 
at Sun's surface temperature 6168K is:\tab {\object{\*\objclass \eqn}
\rsltpict{\*\objdata 
.EQN 678 22 328 0 0
{0:N}NAME(1,6168)={0}?_n_u_l_l_
}}}
.TXT 50 0 282 0 0
Cg b73.000000,73.000000,30
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard {\fs16 COPYRIGHT MIRON 
KAUFMAN, 1997}{\fs20 \par }}