{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 1 1 } 1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Norma l" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 16 "Laplace equation" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 873 "We solve the Laplace-Poisson problem in cylindrical coordinates to find the va riation of electrostatic force and potential with radial distance betw een two concentric cylinders. We need to solve Laplace's equation with the boundary condition that the potential is held constant (V(l)=0 at the inner cylinder, i.e, the filament of a diode, and V(d)=V0 at the \+ anode cylinder). No charges are present between the cylinders, i.e., P oisson eq. reduces to a Laplace eq. The situation is different, in pr inciple, if the electron plasma in a diode is taken into account, i.e. , once the filament is heated and a current flows. The electronic char ge cloud develops its own potential which can impede the flow of curre nt (space-charge-limited region of the current-voltage characteristic) . We ignore this effect here, i.e., we consider only the potential due to the external voltage." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 64 "We need to solve the equation given in Cartesian c oordinates as " }}{PARA 0 "" 0 "" {TEXT 19 39 "diff(V,x$2)+diff(V,y$2) +diff(V,z$2) = 0" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "We make the approximation that the cylinder is very long , and therefore the potential does not depend on the height " }{TEXT 276 1 "z" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 64 "The problem r educes to one dimension in cylindrical coordinates." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart; with(linalg):" }}{PARA 7 "" 1 " " {TEXT -1 80 "Warning, the protected names norm and trace have been r edefined and unprotected\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 153 "Ma ple can remind us of the Laplacian in various orthogonal coordinate sy stems (without explicit computations). We are interested in the radial part only:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "v1 := [r, th eta, z]:\nlaplacian(f(r), v1, coords=cylindrical);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#*&,&-%%diffG6$-%\"fG6#%\"rGF+\"\"\"*&F+F,-F&6$F(-%\"$ G6$F+\"\"#F,F,F,F+!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 " DE:=diff(V(r),r$2)+diff(V(r),r)/r=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#DEG/,&-%%diffG6$-%\"VG6#%\"rG-%\"$G6$F-\"\"#\"\"\"*&-F(6$F*F-F2F- !\"\"F2\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "This differential equation has a very simple solution:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dsolve(DE,V(r));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%\"VG6#%\"rG,&%$_C1G\"\"\"*&%$_C2GF*-%#lnGF&F*F*" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 199 "The boundary conditions of zero potential at the \+ filament and potential V0 at the cylinder can be imposed, but the thic kness of the filament has to be taken into account due to the log-dive rgence at " }{TEXT 257 1 "r" }{TEXT -1 3 "=0." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "BC:=V(l)=0,V(d)=V0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#BCG6$/-%\"VG6#%\"lG\"\"!/-F(6#%\"dG%#V0G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "sol:=dsolve(\{DE,BC\},V(r));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG/-%\"VG6#%\"rG,&*&*&%#V0G\"\"\" -%#lnG6#%\"lGF.F.,&F/F.-F06#%\"dG!\"\"F7F.*&*&F-F.-F0F(F.F.F3F7F7" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Vp1:=simplify(rhs(sol));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$Vp1G,$*&*&%#V0G\"\"\",&-%#lnG6#%\"l G!\"\"-F,6#%\"rGF)F)F),&F+F)-F,6#%\"dGF/F/F/" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 43 "We enter the equivalent expression by hand:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Vp:=V0*ln(r/l)/ln(d/l);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#VpG*&*&%#V0G\"\"\"-%#lnG6#*&%\"rGF(%\"lG! \"\"F(F(-F*6#*&%\"dGF(F.F/F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 181 " This expression is more natural, as it makes it obvious that the logs \+ are taken of dimensionless numbers. One way to read the equation is to say that we are measuring the distances " }{TEXT 271 1 "r" }{TEXT -1 5 " and " }{TEXT 270 1 "d" }{TEXT -1 37 " as multiples of the filament radius " }{TEXT 269 1 "l" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "Maple will not recognize immediate ly that the two expressions are identical:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 17 "simplify(Vp-Vp1);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#*&*&%#V0G\"\"\",**&-%#lnG6#*&%\"rGF&%\"lG!\"\"F&-F*6#F.F&F&*&F)F&-F* 6#%\"dGF&F/*&-F*6#*&F5F&F.F/F&F0F&F/*&F7F&-F*6#F-F&F&F&F&*&F7F&,&F0F&F 3F/F&F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "#assume(l>0,d>0, r>0); # with this command activated the line below simplifies to 0." } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "simplify(Vp-Vp1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&*&%#V0G\"\"\",**&-%#lnG6#*&%\"rGF&% \"lG!\"\"F&-F*6#F.F&F&*&F)F&-F*6#%\"dGF&F/*&-F*6#*&F5F&F.F/F&F0F&F/*&F 7F&-F*6#F-F&F&F&F&*&F7F&,&F0F&F3F/F&F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 171 "The electric field decreases inversely proportional with the radial distance (as expected from the divergence of the field lin es). Its radial component is calculated from:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "E:=-diff(Vp,r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"EG,$*&%#V0G\"\"\"*&%\"rGF(-%#lnG6#*&%\"dGF(%\"lG!\"\"F(F1F1" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "This result is in accord with the calculation based on Gauss' law carried out in " }{TEXT 19 9 "Gauss.m ws" }{TEXT -1 149 " , where a charge was distributed linearly on a lon g wire, and the electric field was calculated using cylindrical probe \+ surfaces at different radii " }{TEXT 277 1 "r" }{TEXT -1 215 ". The pr esent result was obtained directly for the electrostatic potential in \+ a slightly different setting: we assumed that a given outer cylinder s its at a fixed potential difference against the wire in the center." } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "We graph the electric field usin g as a distance scale the filament radius " }{TEXT 267 1 "l" }{TEXT -1 50 ", and assume that the anode cylinder has a radius " }{TEXT 268 1 "d" }{TEXT -1 34 " of 100 times the filament radius." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "plot(subs(l=1,d=100,V0=100,E),r=1.. 100);" }}{PARA 13 "" 1 "" {GLPLOT2D 427 147 147 {PLOTDATA 2 "6%-%'CURV ESG6$7jn7$$\"\"\"\"\"!$!3wei^4CZr@!#;7$$\"39DJqX\\Vn5!#<$!3\"\\pSG;!HM ?F-7$$\"30]iS\"*)p[8\"F1$!323\"*\\!*3T8>F-7$$\"3(\\P4r$[I-7F1$!3Y#G*)e Q\"41=F-7$$\"35+D\"GyR(p7F1$!3xVn9,7<5#4=F1$!3$ p&)Q,uST[=vF17$$\"3'*****\\i#=z:$F1$!3kAC'*p!yi(oF17$$\"3')*\\PM$[ r'4%F1$!3^5'[fE@0I&F17$$\"3y**\\P/9^N]F1$!35`8sGvJ7VF17$$\"3_+D1MjG\"4 'F1$!3K2RIjJ)[c$F17$$\"3S++vj71ZrF1$!3L5\\(e6t#QIF17$$\"3c++DYygs#*F1$ !3\"y`M?89=M#F17$$\"3*)*\\(o2_!)Q6F-$!3Mk[K0))z1>F17$$\"3\"**\\Pu'R$\\ L\"F-$!3'))Gv;b^mi\"F17$$\"3))*\\Pz$R,Q:F-$!3kWX\\Ny'=T\"F17$$\"33+v$R `R![F-$!3,IS')o/P46F17$$\"37++D#)e ts@F-$!39%Qpl([=%***!#=7$$\"3=+]7l7TiBF-$!3+h))fYGw\">*F\\s7$$\"3E++v@ 3%fd#F-$!3Y#o&fwD#)H%)F\\s7$$\"3y***\\7:Z.z#F-$!3+3QtHp3#y(F\\s7$$\"3G ++Dyt'p*HF-$!3qq/X=dcXsF\\s7$$\"3-+v$*Q$)f%=$F-$!3l_%fpmp'=oF\\s7$$\"3 J++D*p4xS$F-$!37L(\\I9MAP'F\\s7$$\"3O+++(G9nf$F-$!3)HAF\\s7$$\"$+\"F*$!3!) ei^4CZr@F\\s-%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*Fb_l-%+AXESLABELSG6$Q\"r6 \"Q!6\"-%%VIEWG6$;F(Fg^l%(DEFAULTG" 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 42 "We verify that this is just a graph of -1/" }{TEXT 274 1 "r" } {TEXT -1 65 " (up to some factor) by graphing a log-log plot (the nega tive of " }{TEXT 275 1 "E" }{TEXT -1 49 " has to be graphed so that th e log can be taken):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "wit h(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoor ds has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 " loglogplot(subs(l=1,d=100,V0=100,-E),r=1..100,scaling=constrained);" } }{PARA 13 "" 1 "" {GLPLOT2D 442 142 142 {PLOTDATA 2 "6'-%'CURVESG6$7jn 7$$\"\"!F)$\"+;VvO8!\"*7$$\"+w;9MG!#6$\"+**GT38F,7$$\"+*R2Y\\&F0$\"+U# 3=G\"F,7$$\"+K%f9+)F0$\"+@(RnD\"F,7$$\"+ps9P5!#5$\"+*eRIB\"F,7$$\"+_lb v9F@$\"+g')>*=\"F,7$$\"+)GRP(=F@$\"+(Q!Q\\6F,7$$\"+NA\"\\d#F@$\"+#4j#z 5F,7$$\"+>&f%yJF@$\"+k$3*=5F,7$$\"+qMF3PF@$\"+(op#f'*F@7$$\"+1TU!=%F@$ \"+\\!>r=*F@7$$\"+y-C1YF@$\"+\")GIh()F@7$$\"+\\)3S*\\F@$\"+2V`t$)F@7$$ \"+UtNChF@$\"+8e=VsF@7$$\"+ReV?qF@$\"+;t5ZjF@7$$\"+\\,4ZyF@$\"+2IX?bF@ 7$$\"+[]FT&)F@$\"+3\"oi#[F@7$$\"+J*=?n*F@$\"+DU_&p$F@7$$\"+X%\\k0\"F,$ \"+6([I!GF@7$$\"+%yfa7\"F,$\"+>`%H6#F@7$$\"+r-'p=\"F,$\"+V/%z\\\"F@7$$ \"+^7bU7F,$\"+FlI?%*F07$$\"+:xn\"H\"F,$\"+&4gw]%F07$$\"+Pp+P8F,$!+8/@E D!#87$$\"+0bNt8F,$!+C*=,m$F07$$\"+#)e$4T\"F,$!+*ec\"=uF07$$\"+Q#ecW\"F ,$!+A#R!*3\"F@7$$\"+:?ow9F,$!+,qF*R\"F@7$$\"+kY0.:F,$!+'[.Im\"F@7$$\"+ !fiC`\"F,$!+ZF3d>F@7$$\"+Sf!fb\"F,$!+Wi^\">#F@7$$\"+4Mn\"e\"F,$!+N4>\\ CF@7$$\"+@CF.;F,$!+^5=lEF@7$$\"+KgzD;F,$!+jrT!*GF@7$$\"+&y4ik\"F,$!+)o aX4$F@7$$\"+1N`m;F,$!+6>z(H$F@7$$\"+(o%R%o\"F,$!+;PSwMF@7$$\"+Q4(Gq\"F ,$!+Ai;hOF@7$$\"+v\\E@VF@7$$\"+'zgPy\"F,$!+2[1qWF@7$$\" +Iz'yz\"F,$!+Th86YF@7$$\"+pE,8=F,$!+MNeiZF@7$$\"+pX=E=F,$!+KDI%*[F@7$$ \"+44#)R=F,$!+JfmI]F@7$$\"+M#==&=F,$!+'=R1:&F@7$$\"+Lgck=F,$!+xr6y_F@7 $$\"+!RGi(=F,$!+X2u%R&F@7$$\"+LJ4))=F,$!+y\")Q8bF@7$$\"+:zQ**=F,$!+')f LEcF@7$$\"+pf!4\">F,$!+Pl^TdF@7$$\"+5xr@>F,$!+\\Rj\\eF@7$$\"+cF]K>F,$! 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oT**f)*F77$$\"3)**\\PWNFZe*F7$\"3Q0ag<))*y!**F77$$\"3!3]i&3Q*[y*F7$\"3 UG*)*Re!y_**F77$$\"$+\"F*F]\\l-%'COLOURG6&%$RGBG$\"#5!\"\"F+F+-%+AXESL ABELSG6$Q\"r6\"Q!6\"-%%VIEWG6$;F(F]\\l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "This result generalizes a simple result known from t he parallel-plate condenser: when a voltage " }{TEXT 272 1 "V" }{TEXT -1 596 "0 is applied across two parallel plates, the electric field be tween the plates is constant (the field lines are parallel, i.e., the \+ field is homogeneous) and the electric potential changes linearly as o ne goes from one plate to the other. In the case of the cylindrical ge ometry the situation is different: the field lines follow radial beams that connect the two circular cross sections, i.e., the distance betw een them increases as one goes from the inner cylinder to the outer on e. Correspondingly, the electric field weakens with radial distance. W e note that the electric field falls like 1/" }{TEXT 273 1 "r" }{TEXT -1 133 " (as opposed to being constant), and that the potential grows \+ only logarithmically (rather than linearly) in this different geometry ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "The following graph illustrates the cylinders (filament and anode cylinde r) via their cross sections." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with(plottools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "c 1 := disk([0,0], 1, color=red):\nc2 := circle([0,0], 10, color=blue): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "for i from 1 to 20 do: phi:=2*Pi*i/20: l1[i]:=line([1*cos(phi),1*sin(phi)],[10*cos(phi),10*s in(phi)],color=green): od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "display(c1,c2,seq(l1[i],i=1..20),scaling=constrained,axes=boxed); " }}{PARA 13 "" 1 "" {GLPLOT2D 431 151 151 {PLOTDATA 2 "6:-%)POLYGONSG 6$7U7$$\"\"\"\"\"!$F*F*7$$\"+8q9@**!#5$\"+OBL`7F/7$$\"+6;$eo*F/$\"+t)) *o[#F/7$$\"+f[w(H*F/$\"+FbC\"o$F/7$$\"++o1j()F/$\"+Tn`<[F/7$$\"+V*p,4) F/$\"+CD&y(eF/7$$\"+tio*G(F/$\"+g5ZXoF/7$$\"+'*)RUP'F/$\"+HC80xF/7$$\" +^zEe`F/$\"+a#zKW)F/7$$\"+=HzdUF/$\"+B0F[!*F/7$$\"+Q*p,4$F/$\"+l^c5&*F /7$$\"+UJ\"Q(=F/$\"+3D(G#)*F/7$$\"+D>0zi!#6$\"+%Gn-)**F/7$$!+m>0ziFaoF bo7$$!+YJ\"Q(=F/$\"+2D(G#)*F/7$$!+U*p,4$F/$\"+j^c5&*F/7$$!+8HzdUF/$\"+ E0F[!*F/7$$!+YzEe`F/$\"+d#zKW)F/7$$!+$*)RUP'F/$\"+JC80xF/7$$!+$G'o*G(F /$\"+]5ZXoF/7$$!+]*p,4)F/$\"+9D&y(eF/7$$!+0o1j()F/$\"+Kn`<[F/7$$!+i[w( H*F/$\"+>bC\"o$F/7$$!+8;$eo*F/$\"+l))*o[#F/7$$!+9q9@**F/$\"+IBL`7F/7$$ !\"\"F*$!+:w1-T!#>7$$!+8q9@**F/$!+QBL`7F/7$$!+6;$eo*F/$!+t))*o[#F/7$$! 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The force on a charg ed particle of charge " }{TEXT 266 1 "q" }{TEXT -1 13 " is given as " }{TEXT 264 1 "F" }{TEXT -1 3 " = " }{TEXT 265 1 "q" }{TEXT -1 1 " " } {TEXT 263 1 "E" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "An interesting problem to be solved concerns the trajectory of a char ged particle." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 127 "We develop a simple numerical algorithm which leads to a recursion formula based on an approximation for the second derivative ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "y\"( t_i) = [y(t_\{i+1\}) + y(t_\{i-1\}) - 2 y(t_\{i\})]/dt^2 + O(dt^2), wh ere dt = t_\{i+1\}-t_i ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 180 "This formula can be turned into a recursion by so lving it for y(t_\{i+1\}), and using Newton's law to replace the seco nd derivative of position (the acceleration) by the force/mass." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 124 "We wish to solve Newton's equatio n for the radial motion in cylindrical coordinates. The electric field is given as follows:" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 178 "The electric field varies with distance \+ according to the following functions. To obtain the electric force one needs to multiply with 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 "" {TEXT -1 100 "To understand this assignment differentiate the expressi on for the field in cylindrical coordinates " }{TEXT 19 14 "V(r) = A*l n(r)" }{TEXT -1 17 " with respect to " }{TEXT 278 1 "x" }{TEXT -1 5 " \+ and " }{TEXT 279 1 "y" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "We pick units in which the charge and the mass of the par ticle equal unity." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "q:=-1 : m:=1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "dt:=0.01; dt2:=d t^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dtG$\"\"\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$dt2G$\"\"\"!\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 161 "We are interested in an initial condition which correspo nds to particles at rest. We translate this into two subsequent postio ns to start the two-step recursion." }}}{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 "" {TEXT -1 329 "We c onstruct a procedure that implements a single step of the recursion. T he first argument is the index at which the recursion is carried out, \+ the second is a parameter that allows one to change the overall streng th of the electric field which is proportional to the potential differ ence applied between filament and outer ring." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "Verlet:=proc(n1,E0) local Fx,Fy,vx,vy; global rx ,ry,dt,dt2,m,q,Ex,Ey;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Fx:=evalf( q*subs(x=rx[n1],y=ry[n1],E0*Ex));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Fy:=evalf(q*subs(x=rx[n1],y=ry[n1],E0*Ey));" }}{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:=100;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"$+# " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "for i from 1 to N do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Verlet(i,10); od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "nops(convert(rx,list));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$-#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "rx[40],ry[40];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+c^Qo8!\" *F#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "plot([seq([rx[i],ry[ i]],i=0..N)],style=point,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 357 126 126 {PLOTDATA 2 "6'-%'CURVESG6$7ew7$$\"\"\"\"\"!F(F' 7$$\"3%***********\\+5!#L=C-\"F .FE7$$\"3\"*******zxPF5F.FH7$$\"3,+++/R#G.\"F.FK7$$\"30+++QTvQ5F.FN7$$ \"3-+++=d;X5F.FQ7$$\"33+++\"pb?0\"F.FT7$$\"35+++C4Uf5F.FW7$$\"31+++8\" es1\"F.FZ7$$\"33+++#zjb2\"F.Fgn7$$\"33+++VVL%3\"F.Fjn7$$\"33+++1gc$4\" F.F]o7$$\"3-+++*)[D.6F.F`o7$$\"3)******f(pR86F.Fco7$$\"3%*******Q\"))R 7\"F.Ffo7$$\"3)******p9C]8\"F.Fio7$$\"3))*****Rn+l9\"F.F\\p7$$\"3!**** **4J8%e6F.F_p7$$\"3%******Hdd2<\"F.Fbp7$$\"31+++4*GN=\"F.Fep7$$\"3'*** ****4Fs'>\"F.Fhp7$$\"3********=VL57F.F[q7$$\"3$******p.fVA\"F.F^q7$$\" 3%******>8#zQ7F.Faq7$$\"3-+++Y)GOD\"F.Fdq7$$\"3%******>Sk)o7F.Fgq7$$\" 3/+++6S\\%G\"F.Fjq7$$\"3$******z(G^+8F.F]r7$$\"3&*******3i\"pJ\"F.F`r7 $$\"3*******\\@*pL8F.Fcr7$$\"3'******z6d3N\"F.Ffr7$$\"3)******f:&Qo8F. 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For later times the spa cing between the points becomes more equidistant, which means that the velocity is approaching a constant due to the decrease in the acceler ation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "56 1 \+ 0" 2 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }