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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 13 "Lorentz force" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 836 "We simulate the \+
motion of a charged particle exposed to an electric and magnetic field
. The situation corresponds to that encountered in a vacuum tube with \+
a long filament surrounded by a cylindrical anode with a solenoid plac
ed on top. This is more complicated than the usual discussion of a cha
rged particle with constant velocity (acquired, e.g., upon acceleratio
n in an electric field) entering a homogeneous magnetic field (circula
r trajectory) or an electric field aligned with the magnetic field (he
lical motion as a result of independent influence of the fields on the
 position variables). In this example the motion is confined to a plan
e, the particles are accelerated by an electric field acting in the ra
dial direction. Once they acquire a non-zero velocity the Lorentz forc
e deflects the particle away from the radial path." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 481 "The Verlet algorithm use
d for the Coulomb problem requires a modification to account for the v
elocity dependence of the Lorentz force. The equations of motion are w
ritten in Cartesian coordinates (E and B are the magnitudes of the ele
ctric field acting in the radial direction and the magnetic field acti
ng in the z direction respectively). The radial dependence of the elec
tric field strength was determined from a solution to the Laplace equa
tion in cylindrical coordinates (see " }{TEXT 19 11 "Laplace.mws" }
{TEXT -1 23 "), or from Gauss' law (" }{TEXT 19 9 "Gauss.mws" }{TEXT 
-1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 
1 "m" }{TEXT -1 18 " diff(x(t),t$2) = " }{TEXT 260 1 "q" }{TEXT -1 35 
" [E*x(t)/(x(t)^2+y(t)^2) + vy(t)*B]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT 259 1 "m" }{TEXT -1 18 " diff(y(t),t$2) = " }
{TEXT 261 1 "q" }{TEXT -1 35 " [E*y(t)/(x(t)^2+y(t)^2) - vx(t)*B]" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 
1 "" {TEXT -1 80 "Warning, the protected names norm and trace have bee
n redefined and unprotected\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "
The Cartesian components for the Lorentz force are calculated as follo
ws:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "v:=vector([vx,vy,vz]
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'vectorG6#7%%#vxG%#vyG%#
vzG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Bf:=vector([Bx,By,Bz
]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#BfG-%'vectorG6#7%%#BxG%#ByG%
#BzG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "crossprod(v,Bf);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%,&*&%#vyG\"\"\"%#BzGF*F
**&%#vzGF*%#ByGF*!\"\",&*&F-F*%#BxGF*F**&%#vxGF*F+F*F/,&*&F4F*F.F*F**&
F)F*F2F*F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Now we restrict the
 homogeneous magnetic field to be aligned with the z axis:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "subs(Bx=0,By=0,%);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#-%'vectorG6#7%*&%#vyG\"\"\"%#BzGF),$*&%#vxGF)F*F)!
\"\"\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "We choose the mass (
units) and charge sign:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "
m:=1; q:=-1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG\"\"\"" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"qG!\"\"" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 131 "We worry about the step-size for small distances r as th
e force is inversely proportional to r. We should try various small va
lues." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "dt:=0.05; dt2:=dt^
2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dtG$\"\"&!\"#" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%$dt2G$\"#D!\"%" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "vx0:=0; vy0:=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
$vx0G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$vy0G\"\"!" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 181 "We have to start the trajectories at a f
inite distance (filament thickness or larger). We can also choose fini
te initial velocities (they depend on the temperature of the filament)
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "Let'
s measure distances in multiples of the filament radius." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "rx[0]:=1; rx[1]:=rx[0]+dt*vx0;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#rxG6#\"\"!\"\"\"" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>&%#rxG6#\"\"\"$F'\"\"!" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 30 "ry[0]:=1; ry[1]:=ry[0]+dt*vy0;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%#ryG6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>&%#ryG6#\"\"\"$F'\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 
"E:='E'; B:='B';" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"EGF$" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"BGF$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 178 "The electric field varies with distance according to the follo
wing functions. To obtain the electric force one needs to multiply wit
h the charge of the probe particle (electron)." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 17 "Ex:=-x/(x^2+y^2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#ExG,$*&%\"xG\"\"\",&*$)F'\"\"#F(F(*$)%\"yGF,F(F(!\"
\"F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Ey:=-y/(x^2+y^2);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#EyG,$*&%\"yG\"\"\",&*$)%\"xG\"\"#
F(F(*$)F'F-F(F(!\"\"F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "V
erlet:=proc(n1,E0,B0) local Fx,Fy,vx,vy; global rx,ry,dt,dt2,m,q,Ex,Ey
;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "vx:=(rx[n1]-rx[n1-1])/dt; vy:=
(ry[n1]-ry[n1-1])/dt;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "Fx:=evalf(
q*subs(x=rx[n1],y=ry[n1],E0*Ex)+q*B0*vy);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 51 "Fy:=evalf(q*subs(x=rx[n1],y=ry[n1],E0*Ey)-q*B0*vx);" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "rx[n1+1]:=2*rx[n1]-rx[n1-1]+dt2/m
*Fx;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "ry[n1+1]:=2*ry[n1]-ry[n1-1]
+dt2/m*Fy;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 7 "N:=200;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%\"NG\"$+#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "for i from \+
1 to N do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Verlet(i,1,2); od:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "nops(convert(rx,list));" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$-#" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "rx[40],ry[40];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"
+%zs)fa!#5$\"+7$>4u\"!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
66 "plot([seq([rx[i],ry[i]],i=0..N)],style=point,scaling=constrained);
" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6$
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e turn to Maple's dsolve[numeric] engine. It can handle the solution m
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in the time-step only when the charged particle returns to the radial \+
distance " }{TEXT 257 1 "r" }{TEXT -1 77 "=1, where we expect intuitiv
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> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 
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{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$de1G/-%%diffG6$-%\"xG6#%\"tG-%\"$G6
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{TEXT -1 104 "Maple tries for a long time to obtain an analytic soluti
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