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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "This Maple code attempts t
o solve Physics 442 problem 1.04, the magnetic diffusion problem." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart:with(plots):" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 71 "Define the constants that we will use for
 copper and a 1 mm radius wire" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "s
igma:=1/1.68e-8;mu:=4*Pi*1e-7;a:=.001;Eo:=1;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&sigmaG$\"+_4Q_f!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%#muG,$%#PiG$\"\"%!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG$
\"\"\"!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#EoG\"\"\"" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 97 "Now calculate the diffusion time so we wi
ll know over what time scale to plot the solutions later" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 39 "diffuse:=1/(mu*sigma);tau:=a^2/diffuse;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(diffuseG,$*&\"\"\"F'%#PiG!\"\"$\"++
+++U!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$tauG,$%#PiG$\"+\"Q_4Q#!#
9" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "We now need the zeroes of t
he Jo Bessel functions.  Fortunately, online help tells us that they a
re given by a" }}{PARA 0 "" 0 "" {TEXT -1 111 "Maple function,  Bessel
JZeros(order,n), where order is 0 for Jo.  It will make the code look \+
less ugly later if" }}{PARA 0 "" 0 "" {TEXT -1 64 "these values are de
fined in terms of a simple function named j0n" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "j0n:= n ->BesselJZeros(0,n);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$j0nGR6#%\"nG6\"6$%)operatorG%&arrowGF(-%-BesselJZero
sG6$\"\"!9$F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Make sure it
 works" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "evalf(j0n(1));  # OK" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+eb#[S#!\"*" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 76 "Now define the function gamma(n)  that gives values \+
that go with these zeros" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "gam:= n
 -> j0n(n)^2/(mu*sigma*a^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$gam
GR6#%\"nG6\"6$%)operatorG%&arrowGF(*&*$)-%$j0nG6#9$\"\"#\"\"\"F4*(%#mu
G\"\"\"%&sigmaG\"\"\")%\"aG\"\"#F4!\"\"F(F(F(" }}}{EXCHG {PARA 11 "" 
1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "Now define a
 function An(n) which gets the coefficients in the sum for the magneti
c field" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "An:= n -> -2*sigma*Eo/(j
0n(n)*BesselJ(1,j0n(n)))*mu*a/j0n(n);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%#AnGR6#%\"nG6\"6$%)operatorG%&arrowGF(,$*&**%&sigmaG\"\"\"%#EoGF0
%#muGF0%\"aGF0\"\"\"*&)-%$j0nG6#9$\"\"#F4-%(BesselJG6$F0F7\"\"\"!\"\"!
\"#F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "Now we define the fu
nction Jz(rho,t), where rho=r/a by doing the Bessel function sum." }}
{PARA 0 "" 0 "" {TEXT -1 95 "The number of terms needed varies with t,
 but 20 is enough for the plots I do below.  The later" }}{PARA 0 "" 
0 "" {TEXT -1 87 "in time you plot the fewer you need and the earlier \+
in time you plot the more you need." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
95 "Jz:=(rho,t)-> sigma*Eo + sum(j0n(n)/(mu*a)*An(n)*BesselJ(0,j0n(n)*
rho)*exp(-gam(n)*t),n=1..20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#Jz
GR6$%$rhoG%\"tG6\"6$%)operatorG%&arrowGF),&*&%&sigmaG\"\"\"%#EoGF0F0-%
$sumG6$*&**-%$j0nG6#%\"nG\"\"\"-%#AnGF9F0-%(BesselJG6$\"\"!*&F7F09$F0F
0-%$expG6#,$*&-%$gamGF9F09%F0!\"\"F0F;*&%#muG\"\"\"%\"aG\"\"\"!\"\"/F:
;F0\"#?F0F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "We also define
 the magnetic field function the same way" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 94 "Bphi:=(rho,t)-> sigma*Eo*mu/2*a*rho + sum(An(n)*Besse
lJ(1,j0n(n)*rho)*exp(-gam(n)*t),n=1..20);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%BphiGR6$%$rhoG%\"tG6\"6$%)operatorG%&arrowGF),&*,%&s
igmaG\"\"\"%#EoGF0%#muGF0%\"aGF09$F0#F0\"\"#-%$sumG6$*(-%#AnG6#%\"nGF0
-%(BesselJG6$F0*&-%$j0nGF=F0F4\"\"\"F0-%$expG6#,$*&-%$gamGF=F09%F0!\"
\"F0/F>;F0\"#?F0F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "Define \+
the times at which to look at the curves of Jz and Bphi" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 106 "times:=array(1..5):times[1]:=.01*tau;times[2]
:=.05*tau;times[3]:=.1*tau;times[4]:=.3*tau;times[5]:=.6*tau;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&timesG6#\"\"\",$%#PiG$\"+\"Q_4Q#!#
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&timesG6#\"\"#,$%#PiG$\"+\">w/
>\"!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&timesG6#\"\"$,$%#PiG$\"+
\"Q_4Q#!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&timesG6#\"\"%,$%#PiG
$\"+Vr&G9(!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&timesG6#\"\"&,$%#
PiG$\"+H9dG9!#9" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "Finally, it's \+
time to plot this function for various times of order tau" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 76 "for m from 1 to 5 do pn[m]:= plot(Jz(rho,tim
es[m]),rho=0..1,0..sigma*Eo) od:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 52 "display(\{pn[1],pn[2],pn[3],pn[4],pn[5]\},axes=boxed)
;" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 30 "And now plot Bphi the same way" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 85 "for m from 1 to 5 do bn[m]:= plot(Bphi(rho,times[m]),rho=0..1,0.
.sigma*Eo*mu*a/2) od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "di
splay(\{bn[1],bn[2],bn[3],bn[4],bn[5]\},axes=boxed);" }}{PARA 13 "" 1 
"" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "9 2 0" 87 }
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