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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 45 "The potential of a mass (
charge) distribution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 56 "First, we use a Monte-Carlo simulation to calculate the
 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 46 "restart: Digits:=14: with(stats): with(plots):" }}
{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been r
edefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "#?stats[rando
m]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "randomize();" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"+U&[F8\"" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 8 "N:=1500;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"
%+:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "RN:=stats[random, un
iform](3*N):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "seq(RN[j],j
=1..10);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6,$\"-o*y4&*y*!#7$\"-*pEG?E'
F%$\"-n#[q`<#F%$\"-^wFS9;F%$\"-\"Gf-@%pF%$\"-GIIRTyF%$\"-bGXj/KF%$\"-e
*))3WX$F%$\",:DWd*)*F%$\"-y$4.kG$F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 73 "Suppose we would like to calculate the potential of a sphere of
 radius a:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "a:=2;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"\"#" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 151 "We will need to scale the uniform random numbers such \+
that position vectors lie inside a volume V' that bounds the sphere fr
om above and is economical." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "tr:=x->(2*x-1)*a;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#trGj+6#%\"xG6\"6$%)operatorG%&arrowGF(*&,
&*&\"\"#\"\"\"9$F0F0F0!\"\"F0%\"aGF0F(F(F(6#\"\"!" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 58 "The boundaries of the interval [0,1] are transform
ed into:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "tr(1);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 6 "tr(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"#" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 18 "The sample volume:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "V_S:=(2*a)^
3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$V_SG\"#k" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 101 "VNx:=Vector(N,datatype=float[8]): VNy:=Vect
or(N,datatype=float[8]): VNz:=Vector(N,datatype=float[8]):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "for i from 1 to N do: VNx[i]:=tr(RN
[i]): VNy[i]:=tr(RN[N+i]): VNz[i]:=tr(RN[2*N+i]): od:" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 36 "Our first test mass position vector:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "VNx[1],VNy[1],VNz[1];" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6%$\"3%***>(e\"R!e\">!#<$\"3G++crNyta!#=
$\"3h***zWQLmi'F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Pick a probe
 position along the x-axis outside of the volume:" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 8 "x_p:=10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%$x_pG\"#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 107 "Work in units wher
e G=1 and the probe mass=1. We distribute a mass of M=1 over the volum
e, i.e., rho=1/Vol." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Measure th
e volume of the sphere first as a check:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 60 "fV:=i->if VNx[i]^2+VNy[i]^2+VNz[i]^2 < a^2 then 1 els
e 0 fi;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#fVGj+6#%\"iG6\"6$%)opera
torG%&arrowGF(@%2,(*$)&%$VNxG6#9$\"\"#\"\"\"F6*$)&%$VNyGF3F5F6F6*$)&%$
VNzGF3F5F6F6*$)%\"aGF5F6F6\"\"!F(F(F(6#FB" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 6 "fV(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "VNx[1]^2+VNy[1]^2+VNz[1]^2;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"/k25/04W!#8" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 63 "Vol:=evalf(add(fV(i),i=1..N)/N*V_S); Vol_ex
:=evalf(4/3*Pi*a^3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$VolG$\"/+++
+g<M!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'Vol_exG$\"/#HQ;K5N$!#7" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Now program the actual integral
 for the potential energy:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
98 "f:=i->if VNx[i]^2+VNy[i]^2+VNz[i]^2 < a^2 then 1/sqrt((VNx[i]-x_p)
^2+VNy[i]^2+VNz[i]^2) else 0 fi;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%
\"fGj+6#%\"iG6\"6$%)operatorG%&arrowGF(@%2,(*$)&%$VNxG6#9$\"\"#\"\"\"F
6*$)&%$VNyGF3F5F6F6*$)&%$VNzGF3F5F6F6*$)%\"aGF5F6*&F6F6-%%sqrtG6#,(*$)
,&F1F6%$x_pG!\"\"F5F6F6F7F6F;F6FK\"\"!F(F(F(6#FL" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 47 "To include the mass density we multiply by rho." }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Vp:=add(f(i),i=1..N)/N*V_S
*(1/Vol);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#VpG$\"/s&z\\06+\"!#9" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "The result is a number. It is \+
a measure of the energy. It became dimensionless by our choice of unit
s." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 263 "Ho
w accurate is this result? Let us test it by calculating the deviation
 without breaking the sample into subsamples, but by estimating the de
viation directly with the help of the average of the square of the fun
ction to be measured (see page 2-4 from the notes)." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "For the volume calcula
tion this wouldn't work because the function we are averaging in that \+
case is 1." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "f_avg:=add(f(
i),i=1..N)/N;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&f_avgG$\"/:4fL!fM&
!#:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "f_squared_avg:=add(f
(i)^2,i=1..N)/N;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%.f_squared_avgG$
\"/8H+](pR&!#;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "dev:=sqrt
((f_squared_avg-f_avg^2)/N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$dev
G$\"/o+\"e_5I\"!#;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 195 "This is th
e deviation for the average of f that was used to estimate the potenti
al energy. To obtain the standard deviation for the final result we ha
ve to multiply it by the appropriate factors:" }}{PARA 0 "" 0 "" 
{TEXT -1 59 "1/Vol represents the mass of each element to be summed ov
er" }}{PARA 0 "" 0 "" {TEXT -1 88 "V_S is the reference volume with wh
ich the measurement (and deviation) has to be scaled." }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "It is appropriate then
 to quote the measurement result as follows:" }}{PARA 0 "" 0 "" {TEXT 
-1 165 "[V-dev,V,V+dev] where V is our best estimate, and the brackets
 provide the 1-sigma interval. We have 68% confidence that the correct
 answer falls into this interval." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 68 "[V_S*(1/Vol)*(f_avg-dev),V_S*(1/Vol)*f_avg,V_S*(1/Vol
)*(f_avg+dev)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%$\"/.9u@Tn(*!#:$
\"/s&z\\06+\"!#9$\"//]y(pa-\"F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
72 "There appears to be a 2.5 % uncertainty associated with our measur
ement." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 
"Let us repeat such a measurement near the surface of the sphere:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 9 "x_p:=2.5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$x_pG$\"#D!\"\"" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "f_avg:=add(f(i),i=1..N)/N;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&f_avgG$\"/9&**Qj1:#!#9" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "f_squared_avg:=add(f(i)^2,i=
1..N)/N;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%.f_squared_avgG$\"/\"yiV
,%G5!#9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "dev:=sqrt((f_squ
ared_avg-f_avg^2)/N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$devG$\"/!)
QU*>?9'!#;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "[V_S*(1/Vol)*
(f_avg-dev),V_S*(1/Vol)*f_avg,V_S*(1/Vol)*(f_avg+dev)];" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#7%$\"/&*pJ/W7R!#9$\"/VuJ&fu-%F&$\"/!*yJ'yC9%F&" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "uncertainty:=(%[3]-%[2])/
%[2]*100;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,uncertaintyG$\"/zGsA(e
&G!#8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Now go inside the sphere
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "x_p:=1.0;" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%$x_pG$\"#5!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "f_avg:=add(f(i),i=1..N)/N;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&f_avgG$\"/D)\\B)[&o$!#9" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 36 "f_squared_avg:=add(f(i)^2,i=1..N)/N;" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%.f_squared_avgG$\"/I;\\R[eL!#9" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "dev:=sqrt((f_squared_avg-f_avg^2)/N
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$devG$\"/nqK(eZ:\"!#:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "[V_S*(1/Vol)*(f_avg-dev),V_S
*(1/Vol)*f_avg,V_S*(1/Vol)*(f_avg+dev)];" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#7%$\"/,q2kT&o'!#9$\"/1RhLm,pF&$\"/53:.\"z6(F&" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 34 "uncertainty:=(%[3]-%[2])/%[2]*100;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,uncertaintyG$\"/py'\\eK8$!#8" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "Now repeat this in a loop to show \+
a graph of the potential as a function of distance:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 25 "Np:=101: dx:=0.1: PO:=[]:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "for j from 1 to Np do: x_p:=(j-1)*
dx; Vp:=add(f(i),i=1..N)/N*V_S*(1/Vol); PO:=[op(PO),[x_p,Vp]]: od:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "PL1:=plot(PO,style=point,co
lor=red,symbolsize=20): display(PL1);" }}{PARA 13 "" 1 "" {GLPLOT2D 
590 304 304 {PLOTDATA 2 "6%-%'CURVESG6&7aq7$$\"\"!F)$\"3i*4$[S6+ut!#=7
$$\"3/+++++++5F,$\"3w*H;W&H-,uF,7$$\"35+++++++?F,$\"3\\**fR'=\\&RuF,7$
$\"3))**************HF,$\"3++r4U$fQY(F,7$$\"3A+++++++SF,$\"3++/\"[9jSX
(F,7$$\"3++++++++]F,$\"3f*\\\"pLDz2uF,7$$\"3w**************fF,$\"3u*\\
t9,.@L(F,7$$\"3a**************pF,$\"3%)*>/I`-[B(F,7$$\"3U+++++++!)F,$
\"3&**ptuyo$HrF,7$$\"3A+++++++!*F,$\"3?+\"o#=[H?qF,7$$\"\"\"F)$\"3$**R
!RhLm,pF,7$$\"33+++++++6!#<$\"3/+E(4EVww'F,7$$\"3%**************>\"F\\
o$\"3=+B++^(=i'F,7$$\"3/+++++++8F\\o$\"3c*\\Bs,=zY'F,7$$\"3!**********
****R\"F\\o$\"3!)*f#GUE(GI'F,7$$\"3++++++++:F\\o$\"3^*H_([!>_8'F,7$$\"
33+++++++;F\\o$\"3Q+ypsIagfF,7$$\"3%**************p\"F\\o$\"3_+[dJ)GRw
&F,7$$\"3/+++++++=F\\o$\"35+$)y=<O_bF,7$$\"3!***************=F\\o$\"3Y
+$)>z77:`F,7$$\"\"#F)$\"3))*fnU,GZ0&F,7$$\"33+++++++@F\\o$\"3u*R\"3=b5
3[F,7$$\"3;+++++++AF\\o$\"3;+zHeSv%e%F,7$$\"3#)*************H#F\\o$\"3
1+J8<a8#Q%F,7$$\"3!**************R#F\\o$\"39+V'*4H<(>%F,7$$\"3++++++++
DF\\o$\"3C+UuJ&fu-%F,7$$\"33+++++++EF\\o$\"3++5(G>(3rQF,7$$\"3;+++++++
FF\\o$\"3D+I@!f\"\\EPF,7$$\"3#)*************z#F\\o$\"3!)*fU.;iBf$F,7$$
\"3!***************GF\\o$\"37+KX7benMF,7$$\"\"$F)$\"3'***)e4C17N$F,7$$
\"33+++++++JF\\o$\"3@+z6D!)RUKF,7$$\"3;+++++++KF\\o$\"3:+Kc#\\T/9$F,7$
$\"3#)*************H$F\\o$\"3y**)yd$oqWIF,7$$\"3!**************R$F\\o$
\"3!**HhF\"*RY&HF,7$$\"3++++++++NF\\o$\"3u**4hx4vpGF,7$$\"33+++++++OF
\\o$\"3?+k0Sag*y#F,7$$\"3;+++++++PF\\o$\"3%)*>&G<i\"Qr#F,7$$\"3#)*****
********z$F\\o$\"3C+)HHNP?k#F,7$$\"3!***************QF\\o$\"3++L8c'eRd
#F,7$$\"\"%F)$\"3D+X))p6I4DF,7$$\"3k*************4%F\\o$\"3'***G>>L\"y
W#F,7$$\"3;+++++++UF\\o$\"3!**zN2in#*Q#F,7$$\"3#)*************H%F\\o$
\"33+EDfyXLBF,7$$\"3M+++++++WF\\o$\"33+DCam>!G#F,7$$\"3++++++++XF\\o$
\"3$**ftZN8$HAF,7$$\"3k*************f%F\\o$\"35+p[6Al!=#F,7$$\"3;+++++
++ZF\\o$\"3$**pAQyqS8#F,7$$\"3#)*************z%F\\o$\"3#**\\/xaQ%*3#F,
7$$\"3M+++++++\\F\\o$\"3'**>6ZnNm/#F,7$$\"\"&F)$\"37+T]e>b0?F,7$$\"3k*
************4&F\\o$\"3'**RH\"oe3m>F,7$$\"3;+++++++_F\\o$\"3++8+6P9G>F,
7$$\"3#)*************H&F\\o$\"37+^*4*)Q;*=F,7$$\"3M+++++++aF\\o$\"3/+a
D\\7\\c=F,7$$\"3++++++++bF\\o$\"3-+i9xkiA=F,7$$\"3k*************f&F\\o
$\"3)**\\*G#fv**y\"F,7$$\"3;+++++++dF\\o$\"3)**f<*pWZe<F,7$$\"3#)*****
********z&F\\o$\"3'**HSGUj!G<F,7$$\"3M+++++++fF\\o$\"3#**>WU#oo)p\"F,7
$$\"\"'F)$\"3#**R5$oFHq;F,7$$\"3k*************4'F\\o$\"3#**R.VwKGk\"F,
7$$\"3;+++++++iF\\o$\"3#****Q.Yhih\"F,7$$\"3#)*************H'F\\o$\"37
+$GNRO0f\"F,7$$\"3M+++++++kF\\o$\"3++hgjxhl:F,7$$\"3++++++++lF\\o$\"3
\"***p]F#o9a\"F,7$$\"3k*************f'F\\o$\"37+`m7F0=:F,7$$\"3;++++++
+nF\\o$\"3))**f2Z#Q`\\\"F,7$$\"3#)*************z'F\\o$\"3))*f*Q/QHt9F,
7$$\"3M+++++++pF\\o$\"3-++^i,*=X\"F,7$$\"\"(F)$\"3****[Mu(*4J9F,7$$\"3
k*************4(F\\o$\"3****e]\\m*3T\"F,7$$\"3;+++++++sF\\o$\"3-+B*eCc
7R\"F,7$$\"3#)*************H(F\\o$\"3%**44-Pb@P\"F,7$$\"3M+++++++uF\\o
$\"3'**Hrl3sNN\"F,7$$\"3++++++++vF\\o$\"37+\"RCj&[N8F,7$$\"3k*********
****f(F\\o$\"3,+i?Tj(yJ\"F,7$$\"3;+++++++xF\\o$\"3$**pO3dD2I\"F,7$$\"3
#)*************z(F\\o$\"3++Y+Qc,%G\"F,7$$\"3M+++++++zF\\o$\"3++)[svHxE
\"F,7$$\"\")F)$\"3++Aq%)>&=D\"F,7$$\"3k*************4)F\\o$\"3$**\\,g;
njB\"F,7$$\"3G*************>)F\\o$\"3***p:!))3E@7F,7$$\"3q+++++++$)F\\
o$\"3$**HvSV>l?\"F,7$$\"3M+++++++%)F\\o$\"3/+fSU(H@>\"F,7$$\"3++++++++
&)F\\o$\"3/+P]n$z!y6F,7$$\"3k*************f)F\\o$\"3++I@WkNk6F,7$$\"3G
*************p)F\\o$\"3'**p'Ga'\\4:\"F,7$$\"3q+++++++))F\\o$\"3%**\\K_
>[y8\"F,7$$\"3M+++++++*)F\\o$\"3,+XE^</D6F,7$$\"\"*F)$\"3&**z'>m/_76F,
7$$\"3k*************4*F\\o$\"3***HF\"=\\F+6F,7$$\"3G*************>*F\\
o$\"3-+\"yd4'H)3\"F,7$$\"3q+++++++$*F\\o$\"3/+[Ow`dw5F,7$$\"3M+++++++%
*F\\o$\"3-+D)[]/^1\"F,7$$\"3++++++++&*F\\o$\"3)**\\-ZdvQ0\"F,7$$\"3k**
***********f*F\\o$\"3-+cQ45)G/\"F,7$$\"3G*************p*F\\o$\"3/+;iXN
6K5F,7$$\"3q+++++++)*F\\o$\"3-+p@<ic@5F,7$$\"3M+++++++**F\\o$\"3'**Ha+
MK7,\"F,7$$\"#5F)$\"3(**>dz\\06+\"F,-%'COLOURG6&%$RGBG$\"*++++\"!\")F(
F(-%&STYLEG6#%&POINTG-%'SYMBOLG6$%(DEFAULTG\"#?-%+AXESLABELSG6$Q!6\"F_
\\m-%%VIEWG6$Fj[mFj[m" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 138 "What \+
should we think of this result? What is the meaning of the structures?
 How do we demonstrate that these are statistical fluctuations?" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "First of \+
all, we can switch the roles of x, y, z:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 99 "f1:=i->if VNx[i]^2+VNy[i]^2+VNz[i]^2 < a^2 then 1/sqr
t((VNy[i]-x_p)^2+VNx[i]^2+VNz[i]^2) else 0 fi:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 99 "f2:=i->if VNx[i]^2+VNy[i]^2+VNz[i]^2 < a^2 the
n 1/sqrt((VNz[i]-x_p)^2+VNy[i]^2+VNx[i]^2) else 0 fi:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "PO:=[]:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 101 "for j from 1 to Np do: x_p:=(j-1)*dx; Vp:=add(f1(i),
i=1..N)/N*V_S*(1/Vol); PO:=[op(PO),[x_p,Vp]]: od:" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 51 "PL2:=plot(PO,style=point,color=blue,symbolsize=20):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "PO:=[]:" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 101 "for j from 1 to Np do: x_p:=(j-1)*dx; Vp:=add(f
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "PL3:=plot(PO,style=point,color=gree
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)4\"F,7$F`hl$\"3++\"=jGRm3\"F,7$Fehl$\"3++AYe>'\\2\"F,7$Fjhl$\"3/+'\\D
%G`j5F,7$F_il$\"3'**4g96WB0\"F,7$Fdil$\"30+)pNE)QT5F,7$Fiil$\"3-+e,1\"
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6#Fj[m-%%VIEWG6$Fj[mFj[m" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 130 "We recognize a great deal of consistency between the t
hree calculations for probe mass locations outside of the mass distrib
ution." }}{PARA 0 "" 0 "" {TEXT -1 369 "Inside the volume we have a po
tential problem from data points where x_p happens to nearly coincide \+
with a test mass location (whose y and z coordinates would have to be \+
very small for the probe and test masses to be on top of each other) a
nd contributes a big amount. In order to smooth out the data after suc
h a statistical fluke one would need to use a very large N." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 112 "We would like \+
to compare this result to the analytic answer. This can be obtained in
 one of three standard ways:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 44 "1) use of Gauss' law for the field strength;" }
}{PARA 0 "" 0 "" {TEXT -1 34 "2) solution of Poisson's equation;" }}
{PARA 0 "" 0 "" {TEXT -1 69 "3) direct calculation of the integral in \+
spherical polar coordinates." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 171 "All three methods are usually explored in cour
ses on electrostatics, where the problem of determining the potential \+
associated with a charge distribution is quite central." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 285 "Let us use method (
1), as it makes us appreciate some fundamental physics concepts. The f
ield strength is given by the force experienced by the probe mass divi
ded by the probe mass. The field strength is related to the force exac
tly as the potential is related to the potential energy." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 277 "Gauss' law states
 that the surface integral of the field strength is related to the vol
ume integral of the charge distribution enclosed by the surface. For o
bjects of high symmetry it can be applied by using probe surfaces with
 symmetry, such that two properties are satisfied:" }}{PARA 0 "" 0 "" 
{TEXT -1 106 "a) the field strength vector is perpendicular to the sur
face (no need to worry about orientation effects);" }}{PARA 0 "" 0 "" 
{TEXT -1 126 "b) the magnitude of the field strength is constant on th
e surface (allows to calculate the surface integral in a trivial way).
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 "For a
 sphere of mass M the best probe surfaces are spheres (inside or outsi
de the massive sphere)." }}{PARA 0 "" 0 "" {TEXT -1 168 "Point (a): th
e field strength is perpendicular to the probe sphere; point (b): the \+
field strength is of equal magnitude everywhere on a probe sphere of c
onstant radius." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 319 "For probe spheres inside the massive sphere the amount o
f enclosed mass changes with the radius; for probe spheres outside the
 massive sphere it remains constant. This will turn out to be crucial \+
for the determination of the variation of the field strength magnitude
 with distance from the centre of the massive sphere." }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 415 "The mass density has \+
to have the following properties: 1) roportional to the constant: Mass
/Volume; The total volume for a sphere of radius R (=a) is known from \+
geometry; 2) it should scale like r^2, such that upon integration over
 r the result will depend on r^3 - this is really the volume element r
^2 in SPC; 3) a factor of 4 Pi, which stems from the fact that we have
 integrated over polar and azimuthal angles." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 36 "rho_inside:=4*Pi*r^2*M/(4*Pi/3*R^3);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%+rho_insideG,$**\"\"$\"\"\"%\"rG\"\"#%\"MG
F(%\"RG!\"$F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "It is interestin
g to verify how much mass is included in partial spheres:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "int(rho_inside,r=0..R);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%\"MG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 25 "int(rho_inside,r=0..R/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,
$*&\"\")!\"\"%\"MG\"\"\"F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 158 "Ga
uss law (up to constants) relates directly the product of field streng
th magnitude times surface area at probe sphere radius r to the volume
 integral above:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "E:='E':
 Gauss:=4*Pi*r^2*E=int(rho_inside,r=0..r);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&GaussG/,$**\"\"%\"\"\"%#PiGF))%\"rG\"\"#F)%\"EGF)F)*
(%\"MGF)%\"RG!\"$F,\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 
"E_inside:=solve(Gauss,E);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)E_ins
ideG,$*,\"\"%!\"\"%\"MG\"\"\"%\"rGF*%#PiGF(%\"RG!\"$F*" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 91 "For probe spheres with radius r larger th
an the massive sphere the right-hand side changes:" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 49 "E:='E': Gauss:=4*Pi*r^2*E=int(rho_inside,r=
0..R);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&GaussG/,$**\"\"%\"\"\"%#P
iGF))%\"rG\"\"#F)%\"EGF)F)%\"MG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 26 "E_outside:=solve(Gauss,E);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%*E_outsideG,$**\"\"%!\"\"%\"MG\"\"\"%#PiGF(%\"rG!\"#F*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "R:=a;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"RG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "M:=1;" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG\"\"\"" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 79 "PLE1:=plot(E_inside,r=0..R): PLE2:=plot(E_outs
ide,r=R..10): display(PLE1,PLE2);" }}{PARA 13 "" 1 "" {GLPLOT2D 634 
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y******z]rf\"*Fcs$\"3K[TI[pv%[*F07$$\"3rlmmc%GpL*Fcs$\"3OJla2a8G\"*F07
$$\"39KLL8-V&\\*Fcs$\"3JID&HTPf#))F07$$\"3=+++XhUk'*Fcs$\"3@$pTPdo*>&)
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 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 456 "The field strenght magnit
ude is zero at the origin, rises linearly, has a cusp at r=R, and then
 falls of with the inverse distance squared. The latter is known from \+
Newton's law of gravity, and from Coulomb's law of electrostatics. Now
 construct the potential from the field strength. Due to spherical sym
metry we just need to integrate along r, and keep track of the sign. W
e begin the integration at infinity, such that the potential is zero a
t infinity:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "U_outside:=-
4*Pi*int(E_outside,r=infinity..s) assuming s>0;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%*U_outsideG*&\"\"\"F&%\"sG!\"\"" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 106 "In order to obtain a continuous potential we appl
y the following trick to calculate the potential for s<R:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "U_inside:=eval(U_outside,s=R)-4*Pi*
int(E_inside,r=R..s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)U_insideG,
&#\"\"$\"\"%\"\"\"*&\"#;!\"\"%\"sG\"\"#F," }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 112 "We find a parabolic form inside the sphere. This is not \+
surprising given that the field strength rises linearly." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "PLU1:=plot(U_inside,s=0..R,color=b
lack): PLU2:=plot(U_outside,s=R..10,color=black): display(PLU1,PLU2);
" }}{PARA 13 "" 1 "" {GLPLOT2D 538 234 234 {PLOTDATA 2 "6&-%'CURVESG6$
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*****\\c9W;)Fcs$\"3.3ghjv#[A\"F,7$$\"3Lmmmmd'*G$)Fcs$\"3g&))*Rf#H1?\"F
,7$$\"3j*****\\iN7])Fcs$\"3w?8L#f*Hw6F,7$$\"3kKLLt>:n')Fcs$\"3q&pHca\"
y`6F,7$$\"3AKLL.a#o$))Fcs$\"3?8Ll1\"G;8\"F,7$$\"3ammm^Q40!*Fcs$\"3*R-k
if#[56F,7$$\"3y******z]rf\"*Fcs$\"39yQ.:qt\"4\"F,7$$\"3rlmmc%GpL*Fcs$
\"3kHM&[-;52\"F,7$$\"39KLL8-V&\\*Fcs$\"3?Mh=q\"QJ0\"F,7$$\"3=+++XhUk'*
Fcs$\"3&p&*eYeAZ.\"F,7$$\"3=+++:o<E)*Fcs$\"3YENQ3)*o<5F,7$$\"#5F)$\"3/
+++++++5F,Fiz-%+AXESLABELSG6%Q\"s6\"Q!F^[m-%%FONTG6#%(DEFAULTG-%%VIEWG
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0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Finally,
 we can put our comparison in place:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "display(PLU1,PLU2,PL1,PL2,PL3);" }}{PARA 13 "" 1 "" 
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H$*4huF,7$$\"3p****\\i'y]!HF,$\"3W&HzsB`sW(F,7$$\"3,LL$ezs$HLF,$\"3!3y
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$\"3++$pl*Hd)4\"F,7$Fagn$\"3++\"=jGRm3\"F,7$Ffgn$\"3++AYe>'\\2\"F,7$F[
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n$\"3-+e,1\"e1.\"F,7$F_in$\"3&**>\")*Qn9?5F,7$Fdin$\"3/+4$4aZ)45F,7$Ff
jl$\"3K*p$HZ:a(***F0-Fjz6&F\\[lF(F]jnF(F`jnFdjn-%+AXESLABELSG6%Q\"s6\"
Q!F_aq-%%FONTG6#Fgjn-%%VIEWG6$;F(FfjlFgjn" 1 2 0 1 10 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+
4" "Curve 5" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 181 "Why is the maxim
um in the potential at zero displacement below the analytical result? \+
Is the symbolic result wrong? Is the statistical simulation converged?
 What is its uncertainty?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 65 "Concerning the symbolic answer we may have the fol
lowing concern:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 101 "how can the field strength have a kink at r=R? aren't ph
ysical variables supposed to be well-behaved?" }}{PARA 0 "" 0 "" 
{TEXT -1 74 "the potential looks smooth, but its second derivative is \+
undefined at r=R." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 83 "Hint: think about the assumption made about the mass dist
ribution. Is it realistic?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Aside:" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 120 "how can we use t
he above code with the home-made uniform random number generator? Here
 are the required first few lines:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "randx:=()->rand()/999999999999.;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&randxGj+6\"F&6$%)operatorG%&arrowGF&*&-%%randGF&\"\"
\"$\"-************\"\"!!\"\"F&F&F&6\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "rand();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"->/wIfz" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "randx();" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#$\"/U-&*)R$)>%!#9" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "N:=1500;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"%+
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "RN:=[seq(randx(),i=1..
3*N)]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "From here on its is the
 same as at the beginning of the worksheet." }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 
2 33 1 1 }