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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 277 15 "Magnetic Bottle" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 586 "In this worksh
eet we investigate charged-particle trajectories in magnetic fields. T
he magnetic field of a current loop was calculated in the worksheet Cu
rrentloop.mws. The magnetic field lines were calculated using a solver
 for systems of first-order differential equations. Then the current l
oop results were extended by combining the fields to form a Helmholtz \+
pair, and a solenoid. Here we investigate charged-particle trajectorie
s by integrating Newton's equations with the Lorentz force. We begin w
ith the study of trajectories in a magnetic field caused by a single c
urrent loop." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 240 "Subsequently we assemble a magnetic field made up of four curr
ent loops. Two of the loops are larger and form a Helmholtz coil. Two \+
loops that are smaller are added to generate a bottleneck. This will b
e a simple model of a magnetic bottle." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 89 "In principle, we work in MKSA (SI) un
its. We do, however use a charge-over-mass ratio of " }{TEXT 282 1 "e
" }{TEXT -1 1 "/" }{TEXT 281 1 "m" }{TEXT -1 231 "=1, i.e., we do not \+
specify the mass as 1 kg, and the charge unit as 1.6 * 10^(-19) C. For
 real-life applications this would have to be corrected. For the quali
tative analysis of charged-particle trajectories this is not a problem
." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "
" 0 "" {TEXT 272 60 "Charged-particle trajectories in the field of a c
urrent loop" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 391 "We carry out the s
olution of Newton's equation for a charged particle with the Lorentz f
orce. The usual textbook discussions deal only with homogeneous fields
, where the constant magnetic field imposes circular motion in a plane
 perpendicular to the magnetic field direction. In general this leads \+
to spiralling motion for particles which possess a velocity component \+
aligned with the field." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 697 "The motion of charged particles in inhomogeneous ma
gnetic fields is much more complicated. We are particularly interested
 in a demonstration that charged particles which move in the direction
 where the magnetic field becomes stronger are slowed down and turn ar
ound, i.e., that they prefer to travel in the direction where the magn
etic field becomes weaker. This has implications for cosmic charged pa
rticles (picking up energy in galactic magnetic fields), and charged p
articles travelling from the Sun to the Earth, which spiral towards th
e poles (and turn around where the field becomes too strong), and coll
ide with each other to produce optical airglow emissions (aurora, or n
orthern lights)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 19 "When we calculated " }{TEXT 261 1 "B" }{TEXT -1 1 "_" }
{TEXT 260 1 "x" }{TEXT -1 101 " we really obtained the transverse comp
onent of the magnetic field, as we worked specifically in the " }
{TEXT 263 1 "x" }{TEXT -1 1 "-" }{TEXT 262 1 "z" }{TEXT -1 135 " plane
. For a solution to Newton's equations we should generate three Cartes
ian components to be calculated anywhere in space (for any " }{TEXT 
266 1 "x" }{TEXT -1 1 "," }{TEXT 265 1 "y" }{TEXT -1 1 "," }{TEXT 264 
1 "z" }{TEXT -1 192 "). Given how the Lorentz force acts (at right ang
les to the velocity and to the B field) we cannot expect the motion to
 be confined to a plane for the case of a non-homogeneous magnetic fie
ld." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 208 "W
e define a procedure to calculate the Cartesian field components for t
he particular case of a current loop defined in Bxa,Bza. We start over
 and define the functions needed for the magnetic field components." }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "restart; with(linalg): wit
h(plots):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for \+
norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Qp:=(Q+R)^2+a^2: Qm:=(Q
-R)^2+a^2:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Qr:=2*sqrt(Q*R/Qp);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#QrG,$*$-%%sqrtG6#*&*&%\"QG\"\"\"%
\"RGF-\"\"\",&*$),&F,F-F.F-\"\"#F/F-*$)%\"aGF4F/F-!\"\"F/F4" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "Bxex:=i_L*a/(5*10^(6))/(sqrt
(Qp)*Q)*((R^2+Q^2+a^2)*EllipticE(Qr)/Qm-EllipticK(Qr));" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 46 "Bxa:=unapply(subs(i_L=1000,R=0.005,Bxex),Q,a):
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%BxexG,$*&*(%$i_LG\"\"\"%\"aGF),
&*&*&,(*$)%\"RG\"\"#\"\"\"F)*$)%\"QGF2F3F)*$)F*F2F3F)F)-%*EllipticEG6#
,$*$-%%sqrtG6#*&*&F6F3F1F3F3,&*$),&F6F)F1F)F2F3F)F7F)!\"\"F3F2F)F3,&*$
),&F6F)F1!\"\"F2F3F)F7F)FGF)-%*EllipticKGF;FLF)F3*&-F?6#,*F4F)*&F6F)F1
F)F2F/F)F7F)F3F6\"\"\"FG#F)\"(+++&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 77 "Bzex:=i_L/(5*10^(6))/sqrt(Qp)*(EllipticE(Qr)*(R^2-Q^2
-a^2)/Qm+EllipticK(Qr));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "Bza:=un
apply(subs(i_L=1000,R=0.005,Bzex),Q,a):" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%%BzexG,$*&*&%$i_LG\"\"\",&*&*&-%*EllipticEG6#,$*$-%%sqrtG6#*&*
&%\"QG\"\"\"%\"RGF8F8,&*$),&F7F)F9F)\"\"#F8F)*$)%\"aGF>F8F)!\"\"F8F>F)
,(*$)F9F>F8F)*$)F7F>F8!\"\"F?FHF)F8,&*$),&F7F)F9FHF>F8F)F?F)FBF)-%*Ell
ipticKGF/F)F)F8*$-F36#,*FFF)*&F7F)F9F)F>FDF)F?F)F8FB#F)\"(+++&" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 120 "To avoid the problems associated \+
with taking the limit at x=0,z=0, we add a tiny constant to the cylind
rical coordinate " }{TEXT 276 1 "r" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 18 "eta:=10^(-Digits):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 71 "Bfield:=proc(x,y,z) local r,Btr,Bx,By,Bz,cphi,sp
hi; global Bxa,Bza,eta;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "r:=sqrt(
x^2+y^2)+eta;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Bz:=evalf(Bza(z,r)
); Btr:=evalf(Bxa(z,r));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "cphi:=e
valf(x/r); sphi:=evalf(y/r);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "By:
=Btr*sphi; Bx:=Btr*cphi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "[Bx,By,
Bz]; end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Bfield(0.002,0
.003,0.004);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%$\"+A)Q#y?!#6$\"+M#e
t6$F&$\"+1r]hUF&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Bfield(
0.0,0.00,0.004);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"!F$$\"+/KLOG
!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 307 "NE:=diff(x(t),t)=p_
x(t),diff(p_x(t),t)=p_y(t)*Bfield(x(t),y(t),z(t))[3]-p_z(t)*Bfield(x(t
),y(t),z(t))[2],diff(y(t),t)=p_y(t),diff(p_y(t),t)=p_z(t)*Bfield(x(t),
y(t),z(t))[1]-p_x(t)*Bfield(x(t),y(t),z(t))[3],diff(z(t),t)=p_z(t),dif
f(p_z(t),t)=p_x(t)*Bfield(x(t),y(t),z(t))[2]-p_y(t)*Bfield(x(t),y(t),z
(t))[1]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "We start a trajectory
 in the " }{TEXT 268 1 "x" }{TEXT -1 1 "-" }{TEXT 267 1 "z" }{TEXT -1 
31 " plane with no momentum in the " }{TEXT 271 1 "x" }{TEXT -1 48 "-d
irection, and non-vanishing components in the " }{TEXT 270 1 "y" }
{TEXT -1 6 "- and " }{TEXT 269 1 "z" }{TEXT -1 12 "-directions." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "IC:=x(0)=0.02,p_x(0)=0,y(0)=
0,p_y(0)=0.001,z(0)=0,p_z(0)=0.001;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%#ICG6(/-%\"xG6#\"\"!$\"\"#!\"#/-%$p_xGF)F*/-%\"yGF)F*/-%$p_yGF)$\"
\"\"!\"$/-%\"zGF)F*/-%$p_zGF)F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 88 "sol:=dsolve(\{NE,IC\},\{x(t),p_x(t),y(t),p_y(t),z(t),p_z(t)\},
numeric,output=listprocedure):" }}{PARA 8 "" 1 "" {TEXT -1 82 "Error, \+
(in DEtools/convertsys) unable to convert to an explicit first-order s
ystem" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "This problem is solved i
n Maple6." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "sol(1);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$solG6#\"\"\"" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 131 "For faster calculations and to be able to work in M
aple5 we introduce our own Runge-Kutta procedure used before for the f
ieldlines." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "RKn:=proc(f,i
c::list,N,x1,h) local xn,yn,ynew,m1,m2,m3,m4,ii,nn;" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 13 "xn:=op(1,ic):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 
"yn:=[]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "for nn from 1 to N do:
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "yn:=[op(yn),op(1+nn,ic)]: od:" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "for ii from 1 to round((x1-xn)/h)
 do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "m1:=[]:" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 22 "for nn from 1 to N do:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "m1:=[op(m1),evalf(f(xn,yn)[nn])]; od:" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 7 "m2:=[]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f
or nn from 1 to N do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "m2:=[op(m2
),evalf(f(xn+h/2,evalm(yn+h/2*m1))[nn])]; od:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "m3:=[]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "for nn f
rom 1 to N do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "m3:=[op(m3),evalf
(f(xn+h/2,evalm(yn+h/2*m2))[nn])]; od:" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 9 "xn:=xn+h;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "m4:=[]:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "for nn from 1 to N do:" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 49 "m4:=[op(m4),evalf(f(xn,evalm(yn+h*m3))[nn])]
; od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "ynew:=[]:" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 22 "for nn from 1 to N do:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 83 "ynew:=[op(ynew),evalf(yn[nn]+h/6*(m1[nn]+2*(m2[nn]+m3
[nn])+m4[nn]))]; od: yn:=ynew;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od
:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "[xn,op(ynew)];" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "We
 define a RHS procedure for the problem." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 57 "fn:=proc(t,yn) local x,y,z,p_x,p_y,p_z,BF; global Bfi
eld;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "x:=yn[1]: p_x:=yn[2]: y:=yn
[3]: p_y:=yn[4]: z:=yn[5]: p_z:=yn[6]: BF:=Bfield(x,y,z);" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 74 "[p_x,p_y*BF[3]-p_z*BF[2],p_y,p_z*BF[1]-p_x*B
F[3],p_z,p_x*BF[2]-p_y*BF[1]];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "en
d:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 279 "The argument list contains
 the external RHS, a list that starts with the initial time (independe
nt variable), and the initial values, the number of dependent variable
s, the final value of the time (independent variable), and the time-st
ep (step-size of the independent variable)." }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "It seems t
hat we can use a large step. (at least intially)." }}{PARA 0 "" 0 "" 
{TEXT -1 94 "We prepare for a graph of the current loop which causes t
he magnetic field and the trajectory:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 68 "PL4:=spacecurve([0.005*cos(f),0.005*sin(f),0],f=0..2*
Pi,color=blue):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 113 "We pick initi
al conditions for a charged particle located above the current loop wi
thout initial momentum in the " }{TEXT 273 1 "z" }{TEXT -1 48 "-direct
ion, and a small amount of motion in the " }{TEXT 275 1 "x" }{TEXT -1 
1 "-" }{TEXT 274 1 "y" }{TEXT -1 115 " plane. The reader is encouraged
 to play with the initial conditions patiently in order to understand \+
the outcomes." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "Rt:=[]: d
t:=0.5: x0:=0.002: px0:=-0.0: y0:=eta: py0:=-0.00005: z0:=0.005: pz0:=
0.0: for i from 1 to 1000 do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "t
0:=(i-1)*dt; sol:=RKn(fn,[t0,x0,px0,y0,py0,z0,pz0],6,t0+dt,dt); x0:=so
l[2]: px0:=sol[3]: y0:=sol[4]: py0:=sol[5]: z0:=sol[6]: pz0:=sol[7]: R
t:=[op(Rt),[x0,y0,z0]]: od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
60 "PL5:=pointplot3d(Rt,axes=boxed,color=green,labels=[x,y,z]): " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "display(PL4,PL5);" }}{PARA 
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F*7%$\"+XlQg>F*$!+$[M#[JF*$\"+)*owf()F*7%$\"+2SgS>F*$!+,%\\B;$F*$\"+lN
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F*$\"+VBZ(y)F*7%$\"+fo'3%=F*$!+w_sKKF*$\"+If_#z)F*7%$\"+P4w?=F*$!+EYvY
KF*$\"+KuT(z)F*7%$\"+`Vg+=F*$!+%ym2E$F*$\"+Jq9-))F*7%$\"+_!)R!y\"F*$!+
w4wuKF*$\"+:\\r1))F*7%$\"+&)H9g<F*$!+\\kt)G$F*$\"+t776))F*7%$\"+6,%)R<
F*$!+wCp-LF*$\"+'Hm`\"))F*7%$\"+'R!\\><F*$!+^$GmJ$F*$\"+!=]%>))F*7%$\"
+8[4*p\"F*$!+\"RV0L$F*$\"+BJPB))F*7%$\"+UVly;F*$!+OpVWLF*$\"+C`8F))F*7
%$\"+p*p\"e;F*$!+]$3$eLF*$\"+')ptI))F*7%$\"+(oUwj\"F*$!+=q:sLF*$\"+9$y
T$))F*-F\\[l6&F^[lF+F_[lF+-%*AXESSTYLEG6#%$BOXG-%+AXESLABELSG6%%\"xG%
\"yG%\"zG" 1 2 0 1 0 2 1 1 2 2 1.000000 47.000000 -27.000000 0 }}}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Rt[nops(Rt)];" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#7%$\"+(oUwj\"!#7$!+=q:sLF&$\"+9$yT$))F&" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "The charged particle spiralled out
 again, but in a different direction." }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 271 "We observe qualitatively some spiralling of the charged-
particle orbits, and preferred motion in the z direction towards weake
r fields. This can be used for trapping of charged particles (magnetic
 bottle, confinement in fusion reactors), which is discussed further b
elow." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "A number of open questio
ns remains:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 232 "1) we have simply followed the mathematical convention for the
 traversal of the current loop. What happens to the magnetic field com
ponents if the current direction (direction in which the curve integra
l is carried out) is reversed?" }}{PARA 0 "" 0 "" {TEXT -1 109 "2) we \+
have considered positively charged particles; what happens if the sign
 of the Lorentz force is changed?" }}{PARA 0 "" 0 "" {TEXT -1 268 "3) \+
most importantly: we have been sloppy as far as the units are concerne
d, i.e., we are just looking qualitatively at the trajectories. For re
al-life applications we would need to understand better what our units
 (m=1, e=1) imply, i.e., how do we convert to SI units?" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" 
{TEXT 283 22 "Simple magnetic bottle" }}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 173 "We now proceed with the model of a magnetic bottle based on fo
ur current loops. First we restart and define the analytic expressions
 for the magnetic field of a single loop." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 35 "restart; with(linalg): with(plots):" }}{PARA 7 "" 
1 "" {TEXT -1 32 "Warning, new definition for norm" }}{PARA 7 "" 1 "" 
{TEXT -1 33 "Warning, new definition for trace" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 38 "We pick for the Helmholtz coil radius:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "R_Hh:=0.005;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%R_HhG$\"\"&!\"$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "Qp:=(Q+R)^2+a^2: Qm:=(Q-R)^2+a^2:" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 19 "Qr:=2*sqrt(Q*R/Qp);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#QrG,$*$-%%sqrtG6#*&*&%\"QG\"\"\"%\"RGF-\"\"\",&*$),&F,F-F.F-
\"\"#F/F-*$)%\"aGF4F/F-!\"\"F/F4" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 83 "Bxex:=i_L*a/(5*10^(6))/(sqrt(Qp)*Q)*((R^2+Q^2+a^2)*El
lipticE(Qr)/Qm-EllipticK(Qr));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "B
xa:=unapply(subs(i_L=1000,R=R_Hh,Bxex),Q,a):" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%BxexG,$*&*(%$i_LG\"\"\"%\"aGF),&*&*&,(*$)%\"RG\"\"#
\"\"\"F)*$)%\"QGF2F3F)*$)F*F2F3F)F)-%*EllipticEG6#,$*$-%%sqrtG6#*&*&F6
F3F1F3F3,&*$),&F6F)F1F)F2F3F)F7F)!\"\"F3F2F)F3,&*$),&F6F)F1!\"\"F2F3F)
F7F)FGF)-%*EllipticKGF;FLF)F3*&-F?6#,*F4F)*&F6F)F1F)F2F/F)F7F)F3F6\"\"
\"FG#F)\"(+++&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "Bzex:=i_L
/(5*10^(6))/sqrt(Qp)*(EllipticE(Qr)*(R^2-Q^2-a^2)/Qm+EllipticK(Qr));" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Bza:=unapply(subs(i_L=1000,R=R_Hh
,Bzex),Q,a):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%BzexG,$*&*&%$i_LG\"
\"\",&*&*&-%*EllipticEG6#,$*$-%%sqrtG6#*&*&%\"QG\"\"\"%\"RGF8F8,&*$),&
F7F)F9F)\"\"#F8F)*$)%\"aGF>F8F)!\"\"F8F>F),(*$)F9F>F8F)*$)F7F>F8!\"\"F
?FHF)F8,&*$),&F7F)F9FHF>F8F)F?F)FBF)-%*EllipticKGF/F)F)F8*$-F36#,*FFF)
*&F7F)F9F)F>FDF)F?F)F8FB#F)\"(+++&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 250 "We define a second function for the loops with smaller radius,
 which are introduced to confine the charged particles. The idea is to
 create a higher density of magnetic field lines inside the Helmholtz \+
coils. For the confining field radius we choose:" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 12 "R_cf:=0.002;" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%%R_cfG$\"\"#!\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 119 "We choo
se twice the current in the confining coils to obtain a large increase
 in the magnetic field at the bottlenecks." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 45 "Bxb:=unapply(subs(i_L=2000,R=R_cf,Bxex),Q,a):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Bzb:=unapply(subs(i_L=2000,R=R_cf,B
zex),Q,a):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "To avoid the proble
ms associated with taking the limit at " }{TEXT 279 1 "x" }{TEXT -1 3 
"=0," }{TEXT 280 2 " z" }{TEXT -1 57 "=0, we add a tiny constant to th
e cylindrical coordinate " }{TEXT 259 1 "r" }{TEXT -1 1 "." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "eta:=10^(-Digits):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "Bfield:=proc(x,y,z) local r,Btr,Bx,
By,Bz,cphi,sphi; global Bxa,Bza,eta;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 21 "r:=sqrt(x^2+y^2)+eta;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "Bz:
= evalf(Bza(z+0.5*R_Hh,r)+Bza(z-0.5*R_Hh,r)+Bzb(z+0.5*R_Hh,r)+Bzb(z-0.
5*R_Hh,r)); Btr:=evalf(Bxa(z+0.5*R_Hh,r)+Bxa(z-0.5*R_Hh,r)+Bxb(z+0.5*R
_Hh,r)+Bxb(z-0.5*R_Hh,r));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "cphi:
=evalf(x/r); sphi:=evalf(y/r);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "B
y:=Btr*sphi; Bx:=Btr*cphi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "[Bx,B
y,Bz]; end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Bfield(0.002
,0.003,0.004);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%$\"+a&4y\\%!#6$\"+
KVrYnF&$\"+G[fI7!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Bfie
ld(0.0,0.00,0.004);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"!F$$\"+Vi
?`7!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 168 "For fast calculations
 and to be able to work in Maple5 we introduce our own Runge-Kutta pro
cedure used before for the fieldlines rather than relying on dsolve[nu
meric]." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "RKn:=proc(f,ic::
list,N,x1,h) local xn,yn,ynew,m1,m2,m3,m4,ii,nn;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "xn:=op(1,ic):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "yn
:=[]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "for nn from 1 to N do:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "yn:=[op(yn),op(1+nn,ic)]: od:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "for ii from 1 to round((x1-xn)/h) d
o:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "m1:=[]:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "for nn from 1 to N do:" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 37 "m1:=[op(m1),evalf(f(xn,yn)[nn])]; od:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "m2:=[]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "for nn f
rom 1 to N do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "m2:=[op(m2),evalf
(f(xn+h/2,evalm(yn+h/2*m1))[nn])]; od:" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 7 "m3:=[]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "for nn from 1 to \+
N do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "m3:=[op(m3),evalf(f(xn+h/2
,evalm(yn+h/2*m2))[nn])]; od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "xn:
=xn+h;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "m4:=[]:" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 22 "for nn from 1 to N do:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 49 "m4:=[op(m4),evalf(f(xn,evalm(yn+h*m3))[nn])]; od:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "ynew:=[]:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "for nn from 1 to N do:" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 83 "ynew:=[op(ynew),evalf(yn[nn]+h/6*(m1[nn]+2*(m2[nn]+m3[nn])+m4[
nn]))]; od: yn:=ynew;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "[xn,op(ynew)];" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "We defi
ne a RHS procedure for the problem." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 57 "fn:=proc(t,yn) local x,y,z,p_x,p_y,p_z,BF; global Bfi
eld;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "x:=yn[1]: p_x:=yn[2]: y:=yn
[3]: p_y:=yn[4]: z:=yn[5]: p_z:=yn[6]: BF:=Bfield(x,y,z);" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 74 "[p_x,p_y*BF[3]-p_z*BF[2],p_y,p_z*BF[1]-p_x*B
F[3],p_z,p_x*BF[2]-p_y*BF[1]];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "en
d:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 279 "The argument list contains
 the external RHS, a list that starts with the initial time (independe
nt variable), and the initial values, the number of dependent variable
s, the final value of the time (independent variable), and the time-st
ep (step-size of the independent variable)." }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 220 "It seems that we can use a large step dt=0.5. The read
er should check occasionally whether a reduction in the time step resu
lts in reproduced trajectories. This is the price to be paid for using
 a non-adaptive algorithm." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 94 "We prepare for a graph of the current loop which \+
causes the magnetic field and the trajectory:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 73 "PL1:=spacecurve([R_Hh*cos(f),R_Hh*sin(f),0.5*R_H
h],f=0..2*Pi,color=blue):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "PL2:=s
pacecurve([R_Hh*cos(f),R_Hh*sin(f),-0.5*R_Hh],f=0..2*Pi,color=blue):" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "PL3:=spacecurve([R_cf*cos(f),R_cf
*sin(f),0.5*R_Hh],f=0..2*Pi,color=red):" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 73 "PL4:=spacecurve([R_cf*cos(f),R_cf*sin(f),-0.5*R_Hh],f=0..2*Pi,
color=red):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 121 "We pick initial c
onditions for a charged particle located in between the current loops \+
with some initial momentum in the " }{TEXT 256 1 "z" }{TEXT -1 48 "-di
rection, and a small amount of motion in the " }{TEXT 258 1 "x" }
{TEXT -1 1 "-" }{TEXT 257 1 "y" }{TEXT -1 154 " plane. We start the tr
ajectory in the middle of the bottle (z=0) so that we are in a relativ
ely weaker magnetic field region with our initial conditions." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "The read
er is encouraged to play with the initial conditions patiently in orde
r to understand the outcomes." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 35 "p_long:=-0.00004; p_trans:=0.00002;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'p_longG$!\"%!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%(p_transG$\"\"#!\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "R
t:=[]: dt:=0.50: x0:=0.0005: px0:=-0.0: y0:=eta: py0:=p_trans: z0:=0.0
: pz0:=p_long: for i from 1 to 400 do:" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 168 "t0:=(i-1)*dt; sol:=RKn(fn,[t0,x0,px0,y0,py0,z0,pz0],6,t0+dt,d
t); x0:=sol[2]: px0:=sol[3]: y0:=sol[4]: py0:=sol[5]: z0:=sol[6]: pz0:
=sol[7]: Rt:=[op(Rt),[x0,y0,z0]]: od:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 60 "PL5:=pointplot3d(Rt,axes=boxed,color=green,labels=[x,
y,z]): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "display(PL1,PL2,
PL3,PL4,PL5);" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6
)-%'CURVESG6$7T7%$\"+++++]!#7\"\"!$\"+++++DF*7%$\"+p+&*e\\F*$\"+&3eQR'
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{PARA 0 "" 0 "" {TEXT -1 196 "p_long=-0.00004; p_trans=0.00002 work. T
oo much longitudinal momentum: violates the condition below. Too much \+
transverse momentum gets the charged particle out of the strong-field \+
region sideways." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 16 "J.D. Jackson in " }{TEXT 278 25 "Classical Electrodynamic
s" }{TEXT -1 270 " derives an expression for the trapping condition. T
he ratio of the longitudinal to transverse initial velocity of the par
ticle has to be bounded from above by an expression that involves the \+
ratio of magnetic field strengths at maximum as compared to the middle
 region." }}{PARA 0 "" 0 "" {TEXT -1 31 "v_L(0)/v_T(0) < sqrt(B_max/B-
1)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 250 "Th
e above example shows that the trajectories can be complicated. We can
 check whether the confinement result is consistent with the above for
mula. The reader should change the initial conditions to see that not \+
all of them lead to particle trapping." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "abs(p_long/p_trans);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#$\"+++++?!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "B0:=nor
m(Bfield(0.,0.,0.));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#B0G$\"+V=9!
z'!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "B_m:=norm(Bfield(e
ta,eta,0.0024));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$B_mG$\"+jgR]G!
\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "sqrt(B_m/B0-1);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+V=D)y\"!\"*" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 70 "We can make the particle escape by exceeding this ra
tio substantially:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "p_lon
g:=-0.00008; p_trans:=0.00002;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'p
_longG$!\")!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(p_transG$\"\"#!
\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "Rt:=[]: dt:=0.50: x
0:=0.0000: px0:=-0.0: y0:=eta: py0:=p_trans: z0:=0.0: pz0:=p_long: for
 i from 1 to 400 do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "t0:=(i-1)*
dt; sol:=RKn(fn,[t0,x0,px0,y0,py0,z0,pz0],6,t0+dt,dt); x0:=sol[2]: px0
:=sol[3]: y0:=sol[4]: py0:=sol[5]: z0:=sol[6]: pz0:=sol[7]: Rt:=[op(Rt
),[x0,y0,z0]]: od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "PL5:=
pointplot3d(Rt,axes=boxed,color=green,labels=[x,y,z]): " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "display(PL1,PL2,PL3,PL4,PL5);" }}
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