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The current loop serves to understan d solenoids and other applications. We will be able to study the magne tic field in detail based on the expressions generated by Maple and ex plain how field lines are calculated. Further applications involve the calculation of particle trajectories in magnetic fields, which is lef t to a subsequent worksheet (MagBottle.mws)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 176 "We integrate the Biot-Savart l aw to calculate the magnetic field created by a current loop. We pick \+ a plane parallel to the current loop of radius R, displaced by a dista nce a." }}{PARA 0 "" 0 "" {TEXT -1 31 "The Biot-Savart law states that " }}{PARA 0 "" 0 "" {TEXT -1 1 "d" }{TEXT 256 1 "B" }{TEXT -1 30 " = m u0/(4*Pi) *I * crossprod(d" }{TEXT 257 1 "s" }{TEXT -1 1 "," }{TEXT 258 1 "r" }{TEXT -1 5 ")/r^3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "where I is the current in the conductor, d" } {TEXT 259 1 "s" }{TEXT -1 40 " is a line segment along the conductor, \+ " }{TEXT 260 1 "r" }{TEXT -1 46 " is the position vector to the locati on where " }{TEXT 261 1 "B" }{TEXT -1 341 " is generated taking the se gment location as the origin. The geometry of a loop for the conductor will allow us to perform the line integral in a simple way (integrati ng the polar angle that traces out the loop). The set-up of the proble m requires us to define the line segment and the position vector as a \+ function of the polar angle (phi)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 264 "We start with the special case of findin g the magnetic field components on a circle of radius R above the loop . This result is needed to understand the attraction/repulsion between two magnets (idealized as current loops). We define the current loop \+ to lie in the " }{TEXT 307 1 "x" }{TEXT -1 1 "-" }{TEXT 308 1 "y" } {TEXT -1 21 " plane. Then we have:" }}}{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 "" {MPLTEXT 1 0 35 "ds:=vector([-sin(phi),cos(phi),0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dsG-%'vectorG6#7%,$-%$sinG6#%$phiG!\"\"-% $cosGF,\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "We fix the locati on where we calculate the transverse and longitudinal (B_" }{TEXT 311 1 "z" }{TEXT -1 39 ") components of the magnetic field as (" }{TEXT 309 1 "R" }{TEXT -1 3 ",0," }{TEXT 310 1 "a" }{TEXT -1 132 "). The tra nsverse field will point along the x-axis, and no magnetic field compo nent exists along y there (no tangential component)." }}{PARA 0 "" 0 " " {TEXT -1 56 "The initial and final points of the position vector are :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "P1:=vector([R*cos(phi) ,R*sin(phi),0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#P1G-%'vectorG6# 7%*&%\"RG\"\"\"-%$cosG6#%$phiGF+*&F*\"\"\"-%$sinGF.F+\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "P2:=vector([R,0,a]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#P2G-%'vectorG6#7%%\"RG\"\"!%\"aG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "assume(R>0); assume(a,real);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "rv:=evalm(P2-P1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#rvG-%'vectorG6#7%,&%\"RG\"\"\"*&%#R|irGF+ -%$cosG6#%$phiGF+!\"\",$*&F-\"\"\"-%$sinGF0F+F2%\"aG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "r2:=simplify(add(rv[i]^2,i=1..3));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r2G,(*$)%#R|irG\"\"#\"\"\"F)*&F'F*- %$cosG6#%$phiG\"\"\"!\"#*$)%#a|irGF)F*F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "r3:=r2*sqrt(r2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#r3G*$),(*$)%#R|irG\"\"#\"\"\"F+*&F)F,-%$cosG6#%$phiG\"\"\"!\"#*$)%#a |irGF+F,F2#\"\"$F+F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "#mu 0:=evalf(4*Pi*10^(-7));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 " dB0:=simplify(crossprod(ds,rv));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% $dB0G-%'vectorG6#7%*&-%$cosG6#%$phiG\"\"\"%#a|irGF.*&-%$sinGF,F.F/\"\" \",&%#R|irGF.*&F5F.F*F3!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Bx:=10^(-7)*i_L*R*int(dB0[1]/r3,phi=0..2*Pi);" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#>%#BxG,$*(%$i_LG\"\"\"%#R|irGF(,**&*&,(*&%#a|irG\"\" \"*$)F)\"\"#F0!\"\"!\"\"*&,&*$)F)\"\"#F0F:*$)F/F:F0F(F0*&F/\"\"\")F)\" \"#F0F4F(*&F7F0*&F/\"\"\",&F8\"\"%F;F(\"\"\"F4!\"%F(-%*EllipticKG6#,$* &F)F0*$-%%sqrtG6#FDF0F4F:F(F0*$-FO6#FDF0F4F(*&*(F7F(F/F(FHF0F0*&)FD#\" \"$F:F0)F)\"\"#F0F4!\"#*&*&F7F0-%*EllipticEGFJF(F0*(F/\"\"\"-FO6#FDF0) F)\"\"#F0F4F:*&*&,(F.F(F6F5FAFEF(FHF0F0*$-FO6#FDF0F4F5F(#F(\")+++5" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "By:=10^(-7)*i_L*R*int(dB0[2 ]/r3,phi=0..2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ByG\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Bz:=10^(-7)*i_L*R*int(dB0[3] /r3,phi=0..2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#BzG,$*(%$i_LG \"\"\"%#R|irG\"\"\",(*&*&F)F(-%*EllipticKG6#,$*&F)F**$-%%sqrtG6#,&*$)F )\"\"#F*\"\"%*$)%#a|irGF:F*F(F*!\"\"F:F(F**$)F7#\"\"$F:F*F?\"\")*&*&F= F*F.F*F**&F)\"\"\")F7#\"\"$F:F*F?F:*&-%*EllipticEGF0F**&F)\"\"\"-F56#F 7F*F?!\"#F(#F(\")+++5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 157 "We have demonstrated that the field components can be calculated in a straigh tforward way on a circle the size of the current loop which is centere d at [0,0," }{TEXT 272 1 "a" }{TEXT -1 2 "]." }}{PARA 0 "" 0 "" {TEXT -1 145 "Before we introduce the general result (which can be calculate d analogously in Maple 6.0, but not in 5.1) we explore some limits and some graphs:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "limit(Bx,a =0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%*undefinedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "limit(Bx,R=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "lim it(Bz,a=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%'signumG6#%$i_LG\" \"\"%)infinityGF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "series (Bx,a=0,1);" }}{PARA 8 "" 1 "" {TEXT -1 38 "Error, (in series/ln) divi sion by zero" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "series(Bx,R =infinity,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%$i_LG\"\"\"%#a|i rG!\"\"#\"\"\"\"(+++&-%\"OG6#*&F&F&*$)%#R|irG\"\"#F&F(F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "The following result shows that in leadin g order the B_z component is that of a dipole (" }{TEXT 274 1 "a" } {TEXT -1 30 " plays the role of separation " }{TEXT 273 1 "z" }{TEXT -1 122 "). The area of the current loop times the current is proportio nal to the dipole moment. Note that the displacement in the " }{TEXT 278 1 "x" }{TEXT -1 1 "-" }{TEXT 277 1 "y" }{TEXT -1 53 " plane is fix ed to be a point above the current loop!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "series(Bz,a=infinity,11);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&*(%$i_LG\"\"\")%#R|irG\"\"#\"\"\"%#PiGF'F+*$)%#a|ir G\"\"$F+!\"\"#F'\"(+++&*&*(F&F+)F)\"\"%F+F,F+F+*$)F/\"\"&F+F1#!\"*\")+ ++5*&*(F&F+)F)\"\"'F+F,F+F+*$)F/\"\"(F+F1#\"\"$\"'++!)-%\"OG6#*&F+F+*$ )F/\"\"*F+F1F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "We can evaluate the fields numerically once the parameters are specified. Some exampl es are:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "evalf(subs(R=0.0 05,a=0.04,Bz));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%$i_LG$\"+]uN#H#! #;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "evalf(subs(R=0.005,a= 0.04,Bx));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%$i_LG$\"++]ZlU!#<" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "evalf(subs(R=0.005,a=0.04,d iff(Bz,a)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%$i_LG$!+DdAV;!#9" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Bz1:=evalf(subs(R=0.005,a= 0.04,i_L=1000,Bz));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$Bz1G$\"*[dBH #!#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Bx1:=evalf(subs(R=0 .005,a=0.04,i_L=1000,Bx));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$Bx1G$ \"(xaE%!#6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "We can generate a m apping for " }{TEXT 276 1 "B" }{TEXT -1 1 "_" }{TEXT 275 1 "x" }{TEXT -1 38 " as a function of height and graph it:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Bx1a:=unapply(evalf(subs(R=0.005,i_L=1000,Bx)),a ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Bx1a(0.04);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"*lxaE%!#8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "logplot(Bx1a(z),z=0.001..0.2);" }}{PARA 13 "" 1 "" {GLPLOT2D 695 185 185 {PLOTDATA 2 "6&-%'CURVESG6$7_o7$$\"\"\"!\"$$!+h# 4)yr!#57$$\"+`5bN6!#7$!+>&=cx(F-7$$\"+0@5r7F1$!+a;]8$)F-7$$\"+eJl19F1$ !+&y**[!))F-7$$\"+5U?U:F1$!+hmxe#*F-7$$\"+j_vx;F1$!+mvz\"o*F-7$$\"+:jI 8=F1$!+pZ!z+\"!\"*7$$\"+ot&)[>F1$!+y!\\a/\"FM7$$\"+?%3W3#F1$!+Ys7\"3\" FM7$$\"+D0^bBF1$!+;G&y9\"FM7$$\"+IEhEEF1$!+,Ah47FM7$$\"+NZr(*GF1$!+4wY n7FM7$$\"+So\")oJF1$!+RA=A8FM7$$\"+g_A`UF1$!+%)*f!>:FM7$$\"+zOjP`F1$!+ ;WA$p\"FM7$$\"+:'4ZA(F1$!+))>FM7$$\"+]by6\"*F1$!+3z'o?#FM7$$\"+o#*G <5!#6$!+yROLBFM7$$\"+!)**RB6F]q$!+p;\\`CFM7$$\"+$p5&H7F]q$!+cMtnDFM7$$ \"+09iN8F]q$!+w'3ln#FM7$$\"+s&\\#\\:F]q$!+4BfF]q$!+hSJM]FM7$$\"+OUN+jF] q$!+-RzR^FM7$$\"+21$)[nF]q$!+HR`d_FM7$$\"+?*[(GrF]q$!+%zc9N&FM7$$\"+NU oqvF]q$!+I>vaaFM7$$\"+s=(='zF]q$!+H/QTbFM7$$\"+5n1\"R)F]q$!+rKvJcFM7$$ \"+\"ph(*z)F]q$!+KKn8dFM7$$\"+H?>E#*F]q$!+*z]_z&FM7$$\"+(>)y<'*F]q$!+S C&p'eFM7$$\"+dx,/5F-$!+339TfFM7$$\"+T@*y/\"F-$!+&QU],'FM7$$\"+A\\3'3\" F-$!+&p=o2'FM7$$\"+rTLF6F-$!+u!o79'FM7$$\"+#o[*p6F-$!+D4W0iFM7$$\"+n'Q ;@\"F-$!+C/-miFM7$$\"+re(>D\"F-$!+)>)fAjFM7$$\"+fMw'H\"F-$!+uUT$Q'FM7$ $\"+*42qL\"F-$!+!)yDOkFM7$$\"+vd(*z8F-$!+;Z9\"\\'FM7$$\"+\"y6*=9F-$!+0 &p$RlFM7$$\"+?\"z9Y\"F-$!+%H21f'FM7$$\"+p;`,:F-$!+82PPmFM7$$\"+B\")RV: F-$!+Y+\"\\o'FM7$$\"+M-L%e\"F-$!+i.m5K0pFM7$$\" +E\"z4z\"F-$!+[i!G%pFM7$$\"+`41N=F-$!+*)fI&)pFM7$$\"+l#)[u=F-$!+A[!=-( FM7$$\"+.g_;>F-$!+`gZgqFM7$$\"+$[hn&>F-$!+^)*\\'4(FM7$$\"\"#!\"\"$!+%) GUMrFM-%'COLOURG6&%$RGBG$\"#5Fc`l\"\"!F\\al-%+AXESLABELSG6$Q\"z6\"%!G- %%VIEWG6$;F(Fa`l%(DEFAULTG-%*AXESTICKSG6$Fgal7]p/$!+)*******zFMQ&1e-08 Faal/$!+.+(*)p(FMQ!6\"/$!+Z(yG_(FMFcbl/$!+2+%zR(FMFcbl/$!+&**H5I(FMFcb l/$!+[([=A(FMFcbl/$!+e>!\\:(FMFcbl/$!+7+\"p4(FMFcbl/$!+#\\dd/(FMFcbl/$ !+++++qFMQ&1e-07Faal/$!+/+(*)p'FMFcbl/$!+W(yG_'FMFcbl/$!+4+%zR'FMFcbl/ $!+'**H5I'FMFcbl/$!+\\([=A'FMFcbl/$!+f>!\\:'FMFcbl/$!+8+\"p4'FMFcbl/$! 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That wo uld allow to draw the field lines." }}{PARA 0 "" 0 "" {TEXT -1 323 "We repeat the calculation of the field components with a generalized poi nt P2. The integrals are calculated by Maple only from version 6.0 on. To show what one can do in such a case we define functions that evalu ate the parameter integral numerically (the curve integral has been re duced by parametrizing the current loop)." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 20 "P2:=vector([Q,0,a]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#P2G-%'vectorG6#7%%\"QG\"\"!%#a|irG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "rv:=evalm(P2-P1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#rvG-%'vectorG6#7%,&%\"QG\"\"\"*&%#R|irGF+-%$cosG6#%$phiGF+!\"\", $*&F-\"\"\"-%$sinGF0F+F2%#a|irG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "r2:=simplify(add(rv[i]^2,i=1..3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r2G,**$)%\"QG\"\"#\"\"\"\"\"\"*(F(F+%#R|irGF+-%$cosG 6#%$phiGF+!\"#*$)F-F)F*F+*$)%#a|irGF)F*F+" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 16 "r3:=r2*sqrt(r2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#r3G*$),**$)%\"QG\"\"#\"\"\"\"\"\"*(F*F-%#R|irGF--%$cosG6#%$phiGF-! \"#*$)F/F+F,F-*$)%#a|irGF+F,F-#\"\"$F+F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "dB0:=simplify(crossprod(ds,rv));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$dB0G-%'vectorG6#7%*&-%$cosG6#%$phiG\"\"\"%#a|irGF.*& -%$sinGF,F.F/\"\"\",&%#R|irGF.*&%\"QGF.F*F3!\"\"" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 147 "We fix the loop radius as 0.005, the current at 1 000 A, and wish to vary a and Q. The remaining parameter integral will be carried out numerically." }}{PARA 0 "" 0 "" {TEXT -1 96 "We also s ubstitute numerical values for the current and radius so that we can g raph the results." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "Bx:=un apply(subs(R=0.005,i_L=1000,10^(-7)*i_L*R*Int(dB0[1]/r3,phi=0..2*Pi)), Q,a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#BxGR6$%\"QG%#a|irG6\"6$%)o peratorG%&arrowGF),$-%$IntG6$*&*&-%$cosG6#%$phiG\"\"\"9%\"\"\"F7*$),** $)9$\"\"#F7F9*&F?F9F3F9$!#5!\"$$\"#D!\"'F9*$)F8F@F7F9#\"\"$F@F7!\"\"/F 6;\"\"!,$%#PiGF@$\"+++++]!#;F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "evalf(Bx(0.,0.01));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#$\"+-+&)z9!#C" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "evalf(Bx (0.01,0.01));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+vO$R%R!#7" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "evalf(Bx(-0.01,0.01));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$!+vO$R%R!#7" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 46 "We define the z-component in an analogous way:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "Bz:=unapply(subs(R=0.005,i_L =1000,10^(-7)*i_L*R*Int(dB0[3]/r3,phi=0..2*Pi)),Q,a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#BzGR6$%\"QG%#a|irG6\"6$%)operatorG%&arrowGF),$- %$IntG6$*&,&$\"\"&!\"$\"\"\"*&9$F6-%$cosG6#%$phiGF6!\"\"\"\"\"*$),**$) F8\"\"#F>F6F7$!#5F5$\"#D!\"'F6*$)9%FDF>F6#\"\"$FDF>!\"\"/F<;\"\"!,$%#P iGFD$\"+++++]!#;F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 279 "On the surface it looks as if numerical integration is not much different fr om knowing the answer in terms of special functions. We show later tha t we pay a price in computational speed. We perform some simple explor ations of the field components using the numerical integration." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 281 34 "Exploring B_x and B_y by 2d graphs" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 65 "L1:=[seq([(i-10.5)/10*0.02,Bx((i-10.5)/10*0.02 ,0.005)],i=1..20)]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "plot(L1,titl e=\"B_x as a function of x (above the wire)\");" }}{PARA 13 "" 1 "" {GLPLOT2D 693 199 199 {PLOTDATA 2 "6&-%'CURVESG6$767$$!+++++>!#6$!+D(o R\\)!#87$$!+++++F*$\"+D(oR\\)F--%'COLOURG6&%$R GBG$\"#5!\"\"\"\"!F_r-%+AXESLABELSG6$%!GFcr-%&TITLEG6#QHB_x~as~a~funct ion~of~x~(above~the~wire)6\"-%%VIEWG6$%(DEFAULTGF\\s" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "L1:=[seq([(i-10.5)/10*0.02,Bx((i-10.5)/10*0.02,-0.005 )],i=1..20)]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "plot(L1,title=\"B_ x as a function of x (below the wire)\");" }}{PARA 13 "" 1 "" {GLPLOT2D 693 199 199 {PLOTDATA 2 "6&-%'CURVESG6$767$$!+++++>!#6$\"+D( oR\\)!#87$$!+++++F*$!+D(oR\\)F--%'COLOURG6&%$RG BG$\"#5!\"\"\"\"!F_r-%&TITLEG6#QHB_x~as~a~function~of~x~(below~the~wir e)6\"-%+AXESLABELSG6$%!GFhr-%%VIEWG6$%(DEFAULTGF\\s" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 379 "Note the change in sign: above the wire and below the wire the transverse component of the magnetic field has opposite sign! This is the first step towards understanding that the field lines are closed \+ curves around the wire. We can also show that the strength of the B_x \+ component falls off with height. Due to the divergence of the function at z=0 we do not connect the points." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "L1:=[seq([(i-10.5)/10*0.005,Bx(0.005,(i-10.5)/10*0.00 5)],i=1..20)]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "plot(L1,style=poi nt,symbol=cross,title=\"B_x as a function of height (crossing the wire )\");" }}{PARA 13 "" 1 "" {GLPLOT2D 691 221 221 {PLOTDATA 2 "6(-%'CURV ESG6$767$$!++++]Z!#7$!+dy11D!#67$$!++++]UF*$!+D@IIIF-7$$!++++]PF*$!+@3 %Rq$F-7$$!++++]KF*$!+$Q;:f%F-7$$!++++]FF*$!++$4E!eF-7$$!++++]AF*$!+]EW TvF-7$$!++++]ozFL7$$\"*+++]#F*$\"++t>ozFL7$$\"*+++](F*$\"+^g: 'f#FL7$$\"++++]7F*$\"+ZO2-:FL7$$\"++++] " 0 "" {MPLTEXT 1 0 65 "L1:=[seq([(i-20.5)/20*0.04,Bz((i-20.5)/20*0.04,0 .005)],i=1..40)]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "plot(L1,title= \"B_z as a function of x (above the loop)\");" }}{PARA 13 "" 1 "" {GLPLOT2D 653 299 299 {PLOTDATA 2 "6&-%'CURVESG6$7J7$$!+++++R!#6$!+/gS \\7!#87$$!+++++PF*$!+`TI`9F-7$$!+++++NF*$!+!G4Kq\"F-7$$!+++++LF*$!+U-] 7?F-7$$!+++++JF*$!+d(4$*R#F-7$$!+++++HF*$!+7TZ))GF-7$$!+++++FF*$!+,zJ9 NF-7$$!+++++DF*$!+T\"HXK%F-7$$!+++++BF*$!+v];&Q&F-7$$!+++++@F*$!+bNu&y 'F-7$$!+++++>F*$!+&y=#R')F-7$$!+++++*=wcF_o7$$!*++++&F*$\"+vZmH>F*7$$!*++++$F*$ \"+8dh&\\$F*7$$!*++++\"F*$\"+x#=?M%F*7$$\"*++++\"F*Ffq7$$\"*++++$F*Faq 7$$\"*++++&F*F\\q7$$\"*++++(F*Fgp7$$\"*++++*F*Fbp7$$\"+++++6F*F]p7$$\" +++++8F*Fho7$$\"+++++:F*Fco7$$\"+++++F*Fhn7$$\"+++++@ F*FY7$$\"+++++BF*FT7$$\"+++++DF*FO7$$\"+++++FF*FJ7$$\"+++++HF*FE7$$\"+ ++++JF*F@7$$\"+++++LF*F;7$$\"+++++NF*F67$$\"+++++PF*F17$$\"+++++RF*F+- %'COLOURG6&%$RGBG$\"#5!\"\"\"\"!F[v-%+AXESLABELSG6$%!GF_v-%&TITLEG6#QH B_z~as~a~function~of~x~(above~the~loop)6\"-%%VIEWG6$%(DEFAULTGFhv" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 142 "We obtain a strong z-component for the magnetic f ield at the center of the loop. Outside of the current loop the direct ion of B_z is reversed." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 " L1:=[seq([(i-20.5)/20*0.01,Bz((i-20.5)/20*0.01,0.0)],i=1..40)]:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "plot(L1,style=point,symbol=cross,ti tle=\"B_z as a function of x (crossing the loop)\");" }}{PARA 13 "" 1 "" {GLPLOT2D 653 299 299 {PLOTDATA 2 "6(-%'CURVESG6$7J7$$!++++](*!#7$! +?P(=>\"!#67$$!++++]#*F*$!+k$[)f9F-7$$!++++]()F*$!+;/TA=F-7$$!++++]#)F *$!+X'*pHBF-7$$!++++]xF*$!+[(z52$F-7$$!++++]sF*$!+:v`?UF-7$$!++++]nF*$ !+v9'=;'F-7$$!++++]iF*$!+!)ys?**F-7$$!++++]dF*$!+v;wN>!#57$$!++++]_F*$ !+!yKd,(FV7$$!++++]ZF*$\"++D(z/*FV7$$!++++]UF*$\"+OC\\SNFV7$$!++++]PF* $\"+E/#4T#FV7$$!++++]KF*$\"+f8sB>FV7$$!++++]FF*$\"+()*zkl\"FV7$$!++++] AF*$\"+\\Fj#\\\"FV7$$!++++]8FV7$$! *+++](F*$\"+F,Iy7FV7$$!)+++DF-$\"+#z)**e7FV7$$\")+++DF-Ffq7$$\")+++vF- Faq7$$\"*+++D\"F-F\\q7$$\"*+++v\"F-Fgp7$$\"*+++D#F-Fbp7$$\"*+++v#F-F]p 7$$\"*+++D$F-Fho7$$\"*+++v$F-Fco7$$\"*+++D%F-F^o7$$\"*+++v%F-Fin7$$\"* +++D&F-FZ7$$\"*+++v&F-FT7$$\"*+++D'F-FO7$$\"*+++v'F-FJ7$$\"*+++D(F-FE7 $$\"*+++v(F-F@7$$\"*+++D)F-F;7$$\"*+++v)F-F67$$\"*+++D*F-F17$$\"*+++v* F-F+-%'COLOURG6&%$RGBG$\"#5!\"\"\"\"!F[v-%'SYMBOLG6#%&CROSSG-%&TITLEG6 #QKB_z~as~a~function~of~x~(crossing~the~loop)6\"-%+AXESLABELSG6$%!GFhv -%&STYLEG6#%&POINTG-%%VIEWG6$%(DEFAULTGF`w" 1 5 1 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "In the immediate vicinity of the current wire " }{TEXT 266 1 "B" } {TEXT -1 1 "_" }{TEXT 265 1 "z" }{TEXT -1 55 " diverges and changes si gn, which is in analogy to the " }{TEXT 264 1 "B" }{TEXT -1 1 "_" } {TEXT 263 1 "x" }{TEXT -1 40 " component's behaviour as a function of \+ " }{TEXT 262 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 53 "At t he center the field the field is nearly constant." }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 66 "L1:=[seq([(i-20.5)/20*0.02,Bz((i-20.5)/20*0. 02,-0.001)],i=1..40)]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "plot(L1,t itle=\"B_z as a function of x (just below the loop)\");" }}{PARA 13 " " 1 "" {GLPLOT2D 653 299 299 {PLOTDATA 2 "6(-%'CURVESG6$7J7$$!++++]>!# 6$!+a)>z7\"!#77$$!++++]=F*$!+\\TCI8F-7$$!++++]F-7$$!++++]:F*$!+(pR4L#F-7$$!++++]9F*$!+g9x))GF-7$$!++++]8F *$!+Q!pRk$F-7$$!++++]7F*$!+zu:%p%F-7$$!++++]6F*$!+X\"QA?'F-7$$!++++]5F *$!+:E\"QX)F-7$$!*+++]*F*$!+O2$y>\"F*7$$!*+++])F*$!+!pW5y\"F*7$$!*+++] (F*$!+\"y)f&z#F*7$$!*+++]'F*$!+dK29WF*7$$!*+++]&F*$!+8b$)\\FF*7$$!*+++ ]%F*$\"+NsU#Q\"!#57$$!*+++]$F*$\"+V\"\\Ve\"Fcp7$$!*+++]#F*$\"+(RW3Q\"F cp7$$!*+++]\"F*$\"+SO1]7Fcp7$$!)+++]F*$\"+i(z<>\"Fcp7$$\")+++]F*Ffq7$$ \"*+++]\"F*Faq7$$\"*+++]#F*F\\q7$$\"*+++]$F*Fgp7$$\"*+++]%F*Fap7$$\"*+ ++]&F*F\\p7$$\"*+++]'F*Fgo7$$\"*+++](F*Fbo7$$\"*+++])F*F]o7$$\"*+++]*F *Fhn7$$\"++++]5F*FY7$$\"++++]6F*FT7$$\"++++]7F*FO7$$\"++++]8F*FJ7$$\"+ +++]9F*FE7$$\"++++]:F*F@7$$\"++++];F*F;7$$\"++++]F*F+-%'COLOURG6&%$RGBG$\"#5!\"\"\"\"!F[v-%'SYMBOLG6#%&CROSSG -%&TITLEG6#QMB_z~as~a~function~of~x~(just~below~the~loop)6\"-%+AXESLAB ELSG6$%!GFhv-%&STYLEG6#%&POINTG-%%VIEWG6$%(DEFAULTGF`w" 1 6 1 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "L1:=[seq([(i-20.5)/20*0.01,Bz(0.005,(i-20.5)/20*0.01) ],i=1..40)]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "plot(L1,title=\"B_z as a function of z (crossing the wire)\");" }}{PARA 13 "" 1 "" {GLPLOT2D 653 299 299 {PLOTDATA 2 "6&-%'CURVESG6$7J7$$!++++](*!#7$\"+] \"zkX(F*7$$!++++]#*F*$\"+le])=)F*7$$!++++]()F*$\"+l@V1!*F*7$$!++++]#)F *$\"+X,I@**F*7$$!++++]xF*$\"+,Tc%4\"!#67$$!++++]sF*$\"+P=P47FA7$$!++++ ]nF*$\"+%=&=Q8FA7$$!++++]iF*$\"+%)p)G[\"FA7$$!++++]dF*$\"+\"exck\"FA7$ $!++++]_F*$\"+vrF#FA7 $$!++++]PF*$\"+#\\#pSDFA7$$!++++]KF*$\"+()3F]GFA7$$!++++]FF*$\"+@$ FA7$$!++++]AF*$\"+#=(zVOFA7$$!++++] " 0 "" {MPLTEXT 1 0 65 "L1:=[seq([(i-20.5)/20*0.01,B z(0.001,(i-20.5)/20*0.01)],i=1..40)]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "plot(L1,title=\"B_z as a function of z (slightly off center)\"); " }}{PARA 13 "" 1 "" {GLPLOT2D 653 299 299 {PLOTDATA 2 "6&-%'CURVESG6$ 7J7$$!++++](*!#7$\"+F>7s6!#67$$!++++]#*F*$\"+7q2D8F-7$$!++++]()F*$\"+$ =_N]\"F-7$$!++++]#)F*$\"+.v\\7F-7$$!++++]sF*$ \"+mVvYAF-7$$!++++]nF*$\"+w[e(e#F-7$$!++++]iF*$\"+q1:!*HF-7$$!++++]dF* $\"+KGelMF-7$$!++++]_F*$\"+*fEg-%F-7$$!++++]ZF*$\"+)HTQo%F-7$$!++++]UF *$\"+b,+]aF-7$$!++++]PF*$\"+IvDJjF-7$$!++++]KF*$\"+]/hDtF-7$$!++++]FF* $\"+q&)z:%)F-7$$!++++]AF*$\"+I)*Rh&*F-7$$!++++] " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 267 36 "Exploring B_x and B_y by arrow plots" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 16 "with(plottools);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7F%$arcG%&arrowG%'circleG%%coneG%'cuboidG%&curveG%&cuti nG%'cutoutG%)cylinderG%%diskG%-dodecahedronG%(ellipseG%,ellipticArcG%+ hemisphereG%+hexahedronG%*homothetyG%*hyperbolaG%,icosahedronG%%lineG% +octahedronG%)piesliceG%&pointG%(polygonG%(projectG%*rectangleG%(refle ctG%'rotateG%&scaleG%*semitorusG%'sphereG%)stellateG%,tetrahedronG%&to rusG%*transformG%*translateG%%vrmlG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "l2 := arrow([10,10], vector([0,10]), .2, .4, .1, colo r=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "s:=0.05; 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We provide the results of the analytic integration so that Maple 5 \+ users can do the work (expressions transferred by hand!). We start by \+ defining some auxiliary variables." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Qp:=(Q+R)^2+a^2: Qm:=(Q-R)^2+a^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Qr:=2*sqrt(Q*R/Qp);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#QrG,$*$-%%sqrtG6#*&*&%\"QG\"\"\"%\"RGF-\"\"\",&*$),& F,F-F.F-\"\"#F/F-*$)%\"aGF4F/F-!\"\"F/F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "Bxex:=i_L*a/(5*10^(6))/(sqrt(Qp)*Q)*((R^2+Q^2+a^2)*El lipticE(Qr)/Qm-EllipticK(Qr));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%B xexG,$*&*(%$i_LG\"\"\"%\"aGF),&*&*&,(*$)%\"RG\"\"#\"\"\"F)*$)%\"QGF2F3 F)*$)F*F2F3F)F)-%*EllipticEG6#,$*$-%%sqrtG6#*&*&F6F3F1F3F3,&*$),&F6F)F 1F)F2F3F)F7F)!\"\"F3F2F)F3,&*$),&F6F)F1!\"\"F2F3F)F7F)FGF)-%*EllipticK GF;FLF)F3*&-F?6#,*F4F)*&F6F)F1F)F2F/F)F7F)F3F6\"\"\"FG#F)\"(+++&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "Bxa:=unapply(subs(i_L=1000,R =0.005,Bxex),Q,a):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "a is the he ight above the loop, and Q is the displacement." }}{PARA 0 "" 0 "" {TEXT -1 95 "We compare the previous definition based on numerical int egration with the new analytic result:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "evalf(Bx(0.002,0.003));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#BxG6$$\"\"#!\"$$\"\"$F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "evalf(Bxa(0.002,0.003));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+?%*\\gA!#6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 " Wonderful! Let's repeat for the " }{TEXT 282 2 "z-" }{TEXT -1 10 "comp onent:" }}}{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 11 "" 1 "" {XPPMATH 20 "6#>%%BzexG,$*&*&%$i_LG\"\"\",&*&*&-%*EllipticE G6#,$*$-%%sqrtG6#*&*&%\"QG\"\"\"%\"RGF8F8,&*$),&F7F)F9F)\"\"#F8F)*$)% \"aGF>F8F)!\"\"F8F>F),(*$)F9F>F8F)*$)F7F>F8!\"\"F?FHF)F8,&*$),&F7F)F9F HF>F8F)F?F)FBF)-%*EllipticKGF/F)F)F8*$-F36#,*FFF)*&F7F)F9F)F>FDF)F?F)F 8FB#F)\"(+++&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "Bza:=unapp ly(subs(i_L=1000,R=0.005,Bzex),Q,a):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "evalf(Bz(0.002,0.003));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+&\\Paf(!#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "eva lf(Bza(0.002,0.003));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+)\\Paf(!# 6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 235 "Marvellous! Does the analyt ic result work faster? (We could also accelerate the numerical integra tion by selecting options, such as 10 Digits and _Dexp, but would need to re-define the mapping so that evalf(Int,...) is called directly)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "t1:=time():" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "seq(evalf(Bz(0.002,0.001*i)),i=1..1 0);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6,$\"+^')308!#5$\"+PvyG5F%$\"+&\\ Paf(!#6$\"+!3K%=bF*$\"+(>r/.%F*$\"+$*fT#)HF*$\"+9HdTAF*$\"+IY+7 " 0 "" {MPLTEXT 1 0 10 "time( )-t1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$l$!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "t1:=time():" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "seq(evalf(Bza(0.002,0.001*i)),i=1..10);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6,$\"+^')308!#5$\"+PvyG5F%$\"+)\\Paf(!#6$\"+y?V=bF *$\"+%>r/.%F*$\"+!*fT#)HF*$\"+;HdTAF*$\"+IY+7 " 0 "" {MPLTEXT 1 0 10 "time()-t1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"#I!\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "We have 10 times faster results now! Let us attempt a plot of the \+ strength of the magnetic field:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 287 "PL1:=plot([[-0.005,0],[0.005,0]],style=point,color=blue,symbo l=circle): PL2:=contourplot(evalf(sqrt(Bxa(x,z)^2+Bza(x,z)^2)),x=-0.04 ..0.04,z=-0.04..0.04,grid=[50,50],scaling=constrained,contours=[0.0005 ,0.001,0.002,0.005,0.01,0.02,0.05,0.1],filled=true,coloring=[white,gre en],axes=boxed):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "display(PL1,PL2 );" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "68-%'CURVESG 6&7$7$$!1+++++++]!#=\"\"!7$$\"1+++++++]F*F+-%'COLOURG6&%$RGBGF+F+$\"*+ +++\"!\")-%&STYLEG6#%&POINTG-%'SYMBOLG6#%'CIRCLEG-F$6H7$7$$!1HdG9dG9dF *$!1AB\"*=]qL;F*7$$!1O&GZ`\\eN'F*$!1!fAhIlK;)!#>7$FF7$$!1[(3h`\\eN'F*$ !1-C*y3Fwu\"FK7$7$$!1$\\8od\\eN'F*$\"13B71`Ej\")FKFM7$FS7$$!1!\\B%fia0 jF*$\"1$**\\d2(e29F*7$7$FB$\"1%[)et]qL;F*FY7$FA7$$!1$\\xX>S[S%F*$!1&\\ ?.0sd7#F*7$7$$!1R71`Ej\"3%F*$!1HRYcvsBBF*F]o7$Fin7$$!1?;8!G))QI&F*$\"1 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F^h[nF^chmFdchm7&F^h[nFig[nF^bhmF^chm7&Fig[nFig[nFhahmF^bhm-F06&F2F_j^ lF_j^lF_j^lF`j^l-%*AXESSTYLEG6#%$BOXG-%(SCALINGG6#%,CONSTRAINEDG-%+AXE SLABELSG6$%!GF^[]o-%%VIEWG6$%(DEFAULTGFb[]o" 1 2 0 1 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 142 "The magnetic field strength is largest at the center. Will this b e true also if we increase the radius of the loop? Try it out as an ex ercise!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Let us incorporate an \+ algorithm that plots field lines:" }}{PARA 0 "" 0 "" {TEXT -1 1280 "In the case of the electric field we used a simple algorithm to use the \+ direction of the field to step from location (x_i,y_i) to (x_i+1,y_i+1 ) in such a way that the field line 'trajectory' would have the proper ty that the field vector form always a tangent to the field line. The \+ stepping was done by multiplying the field vector by a step-size dt an d using x_i+1 = x_i + dt*E_x(x_i,y_i) and likewise for the y-position. This numerical technique amounts to an explicit Euler (extrapolation) method for solving a pair of first-order differential equations for t he field line 'trajectory' [x(t),y(t)]. Note that we put the word 'tra jectory' in quotes, as the field lines are fictitious objects: in the \+ electric field case they indicate the direction of the acceleration, i .e., they are not real trajectories, as for any real (charged-particle ) trajectory the velocity vector forms a tangent. For magnetic field l ines the 'trajectory' does not even point in the direction of accelera tion of the charged particle, as the Lorentz force is orthogonal to th e magnetic field vector (and the velocity of the particle). Therefore, dt is not a time-step, but merely the step for the advancement along \+ the fictitious field line 'trajectory'. So we can think of it as a fic titious time-step." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 820 "For the magnetic field line case the method fails due to an accumulation of errors along the circular field lines (they become spirals in the numerical stepping technique). One remedy would be to \+ use a very small step dt. This would cost an exorbitant amount of comp uter time and memory, and would eventually lead to other numerical pro blems. Another remedy would be to code explictly a higher-order method (implicit, or explicit-implicit (i.e., predictor-corrector), or a mul ti-step method such as Runge-Kutta). Alternatively we can try to use M aple's dsolve[numeric] technique. It has the advantage of using an ada ptive time-step, and carries out error-checking (it reduces the time-s tep and repeats calculations if necessary, or increases dt when desira ble) in a way that is hidden from the user (black-box approach)." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "The diffe rential equations for the components of the field-line 'trajectory' ar e:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "FE:=diff(xf(t),t)=Bxa (xf(t),zf(t)),diff(zf(t),t)=Bza(xf(t),zf(t)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "sol:=dsolve(\{FE,xf(0)=0.004,zf(0)=0\},\{xf(t),z f(t)\},numeric,output=listprocedure):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "X:=subs(sol,xf(t)): Z:=subs(sol,zf(t)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "[X(0.01),Z(0.01)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+27WR_!#7$\"1+++e\"e.S\"!#=" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 174 "We need to figure out how much fictitious time is needed to advance the 'trajectory' for a closed loop. 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We leave these applications for a subs ection." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 268 40 "A pair of current loops: Helmholtz coils " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 253 "We consider an arrangement of two current loops that are in parallel. We place one loop at height z 0, and the other at height -z0. The fields point in the same direction . 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numerical displacement." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "BHH(0.00,0.002,0.0025); #we can't pick x=0 or z=0, we can fake it by adding a small nr." }}{PARA 8 "" 1 "" {TEXT -1 32 "Error, (in B xa) division by zero" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "FE: =diff(xf(t),t)=BHH(xf(t),zf(t),0.0025)[1],diff(zf(t),t)=BHH(xf(t),zf(t ),0.0025)[2]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "sol:=dsolv e(\{FE,xf(0)=0.002,zf(0)=-0.0025\},\{xf(t),zf(t)\},numeric,output=list procedure):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "X:=subs(sol, xf(t)): Z:=subs(sol,zf(t)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "Rt:=[]: dt:=0.002: f or it from 1 to 40 do: t_i:=it*dt: Rt:=[op(Rt),[X(t_i),Z(t_i)]]: od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "PL3:=plot(Rt,style=point,symbol= cross,scaling=constrained,axes=boxed,view=[-0.01..0.01,-0.01..0.01]): \+ display(PL1,PL2,PL3);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 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F*$\"1+++Uoy)[%F*7$$\"1+++I?u[CF*$\"1+++*R_#HZF*7$$\"1+++:&y'GDF*$\"1+ ++cTg^\\F*7$$\"1+++QAf3EF*$\"1+++[,dd^F*7$$\"1+++\\$Ryo#F*$\"1+++%yF)[ `F*7$$\"1+++!4&)fw#F*$\"1+++,;$p_&F*7$$\"1+++fwuUGF*$\"1+++$*QG$p&F*7$ $\"1+++zZ&z\"HF*$\"1+++\"zH\"\\eF*7$$\"1+++x8^\"*HF*$\"1+++[]c&*fF*7$$ \"1+++&3wL1$F*$\"1+++t&\\N8'F*7$$\"1+++%QWN8$F*$\"1+++FG#RE'F*7$$\"1++ +9m.-KF*$\"1+++R!>uQ'F*7$$\"1++++&*))oKF*$\"1+++s7o/lF*7$$\"1+++&=]TL$ F*$\"1+++`RF;mF*7$$\"1+++U?(yR$F*$\"1+++2QpAnF*7$$\"1+++g<6gMF*$\"1+++ s#zV#oF*7$$\"1+++Ft#4_$F*$\"1+++_&=<#pF*7$$\"1+++YmP!e$F*$\"1+++sk0:qF *7$$\"1+++6m^QOF*$\"1+++9.q/rF*7$$\"1+++$\\-ap$F*$\"1+++wZ#4>(F*7$$\"1 +++jv3^PF*$\"1+++#>wRF(F*-F56&F7$\"#5!\"\"F8F8F<-FA6#%&CROSSG-%*AXESST YLEG6#%$BOXG-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6$%!GF_fbn-%%VIEW G6$;$F`ebn!\"#$F\\^blFefbnFcfbn" 1 2 0 1 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 171 "Note ho w straight the field line is in the region between the coils. This is \+ a consequence of the vanishing transverse (x-) component of the magnet ic field in this region." }}{PARA 0 "" 0 "" {TEXT -1 417 "Also note th at the contours display strength of the magnetic field. They become co mplicated outside of the two coils as the transverse component of the \+ magnetic field becomes non-zero, while the longitudinal z-component be comes weaker. These are not equipotential lines (we have not introduce d the magnetic vector potential at all so far), and thus there is no n eed for the field lines to cross them at a right angle." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "We add a number of f ield lines:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "sol:=dsolve( \{FE,xf(0)=0.001,zf(0)=-0.0025\},\{xf(t),zf(t)\},numeric,output=listpr ocedure):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "X:=subs(sol,xf(t)): Z: =subs(sol,zf(t)):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "Rt:=[]: dt:=0. 002: for it from 1 to 40 do: t_i:=it*dt: Rt:=[op(Rt),[X(t_i),Z(t_i)]]: od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "PL4:=plot(Rt,style=point,s ymbol=cross,scaling=constrained,axes=boxed,view=[-0.01..0.01,-0.01..0. 01]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "sol:=dsolve(\{FE,xf(0)=0.0 0001,zf(0)=-0.0025\},\{xf(t),zf(t)\},numeric,output=listprocedure):" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "X:=subs(sol,xf(t)): Z:=subs(sol,zf (t)):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "Rt:=[]: dt:=0.002: for it \+ from 1 to 40 do: t_i:=it*dt: Rt:=[op(Rt),[X(t_i),Z(t_i)]]: od:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "PL5:=plot(Rt,style=point,symbol=cr oss,scaling=constrained,axes=boxed,view=[-0.01..0.01,-0.01..0.01]): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "sol:=dsolve(\{FE,xf(0)=0.003,zf(0 )=-0.0025\},\{xf(t),zf(t)\},numeric,output=listprocedure):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "X:=subs(sol,xf(t)): Z:=subs(sol,zf(t)):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "Rt:=[]: dt:=0.002: for it from 1 to 40 do: t_i:=it*dt: Rt:=[op(Rt),[X(t_i),Z(t_i)]]: od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "PL6:=plot(Rt,style=point,symbol=cross,scaling=c onstrained,axes=boxed,view=[-0.01..0.01,-0.01..0.01]):" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 87 "sol:=dsolve(\{FE,xf(0)=0.004,zf(0)=-0.0025\}, \{xf(t),zf(t)\},numeric,output=listprocedure):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "X:=subs(sol,xf(t)): Z:=subs(sol,zf(t)):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "Rt:=[]: 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$$\"1+++c+?#p\"F*$\"1+++>jD%>&F*7$$\"1+++N0aA`SSy\"F*$\"1+++0IRkbF*7$$\"1+++4y( \\\"=F*$\"1+++#*>\\ycF*7$$\"1+++*eCf%=F*$\"1+++6dM)y&F*7$$\"1+++mG\"o( =F*$\"1+++-D=%*eF*7$$\"1+++1ve2>F*$\"1+++K;A'*fF*7$$\"1+++\\Q?Q>F*$\"1 +++R?n%4'F*7$$\"1+++xfio>F*$\"1+++v;t*='F*7$$\"1+++\\_#))*>F*$\"1+++us e\"G'F*7$$\"1+++u!z(G?F*$\"1+++FWTqjF*7$$\"1+++&))p%e?F*$\"1+++gyPckF* 7$$\"1+++\"G%)y3#F*$\"1+++]=jRlF*7$$\"1+++6B,<@F*$\"1+++A2K?mF*-F56&F7 $\"#5!\"\"F8F8F<-FA6#%&CROSSG-%*AXESSTYLEG6#%$BOXG-%(SCALINGG6#%,CONST RAINEDG-%+AXESLABELSG6$%!GF]_hn-%%VIEWG6$;$F^^hn!\"#$F^aelFc_hnFa_hn" 1 2 0 1 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 }}}}{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 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 " " 0 "" {TEXT 269 41 "A sequence of current loops: the solenoid" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 210 "We can use the results for the cu rrent loop to understand the magnetic field of a solenoid. This is onl y true approximately, as the winding of the wire in a solenoid makes t he intersection of the wire with the " }{TEXT 271 1 "x" }{TEXT -1 1 "- " }{TEXT 270 1 "z" }{TEXT -1 458 " plane left-right asymmetric. The in tersections on the left should be half a spacing higher (or lower) tha n the ones on the right. Nevertheless, we should be able to obtain the main features. Note that we have a freedom to choose the spacing, as \+ our current loop is idealized as being generated by an arbitrarily thi n conductor. In practice, for the conductor has to have some cross sec tion to be able to carry a decent amount of current without overheatin g." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Nturn:=6;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&NturnG\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "d_s:=0.001;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$d_s G$\"\"\"!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "LL:=[seq((i -Nturn/2-0.5)*d_s,i=1..Nturn)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# LLG7($!#D!\"%$!#:F($!\"&F($\"\"&F($\"#:F($\"#DF(" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 117 "We define the two field components separately, an d displace the argument by a tiny number to avoid the singularities:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "eta:=evalf(10^(-Digits)): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "BSOx:=(x,z)->evalf(simp lify(add(Bxa(x+eta,z-LL[i]+eta),i=1..Nturn)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%BSOxGR6$%\"xG%\"zG6\"6$%)operatorG%&arrowGF)-%&evalf G6#-%)simplifyG6#-%$addG6$-%$BxaG6$,&9$\"\"\"%$etaGF;,(9%F;&%#LLG6#%\" iG!\"\"F " 0 "" {MPLTEXT 1 0 69 "BSOz:=(x,z)->evalf(simplify(add(Bza(x+eta,z-LL[i]+eta),i=1..Nt urn)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%BSOzGR6$%\"xG%\"zG6\"6$% )operatorG%&arrowGF)-%&evalfG6#-%)simplifyG6#-%$addG6$-%$BzaG6$,&9$\" \"\"%$etaGF;,(9%F;&%#LLG6#%\"iG!\"\"F " 0 "" {MPLTEXT 1 0 19 "BSOx(-0.005,0.002);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$!+(=D)pD!#5" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 313 "We show an alternative to the contourplot: we generate a dataset and create a surface plot of the data (this technique can b e used also for numerical data imported into Maple). It is faster than using contourplot, but it generates 3d graphs as opposed to contourpl ot. We need to adjust our pointplots accordingly." }}{PARA 0 "" 0 "" {TEXT -1 110 "Suppose we wish to superimpose field lines on a graph of B_z. 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The color coding is tr icked by the very large (positive and negative) values of the B_z comp onent near the wires (displaying the surface by draging the graph help s to visualize this). The different color coding of the field near the conductor crossings for x=R and x=-R is an artificial result caused b y the displacement of the mesh by eta. As one approaches a conductor c rossing from the left and right (either at x=R, or at x=-R) the B_z co mponent blows up with a branch going to positive infinity inside the c oil, and a branch going to negative infinity outside the coil. In real ity the field is finite, as the conductor has a finite radius. Note th at we cannot constrain " }{TEXT 286 1 "x" }{TEXT -1 5 " and " }{TEXT 287 1 "z" }{TEXT -1 117 " to be displayed using the same scale, as the scaling=constrained option works on all three coordinates in 3d graph s." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 336 "We leave it as an exercise to draw negative contour lines for B_z, and t o perform graphs for B_x (positive and negative). The latter graphs re quire a careful interpretation. It may help to choose a grid in such a way that one stays away from the conductor as much as possible, so th at the color-coding helps in understanding the height." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 403 "The calculation of field lines in analogy to the case of two c urrent loops is very slow. This must be caused at least in part by rep eated calculations of the field in Maple, and also by the attempt at v ery high accuracy in dsolve[numeric]. We can try to get faster results with a fixed-step algorithm, but have to be nervous about the accurac y. We could also try different algorithms in dsolve[numeric]." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 172 "We define our own Runge-Kutta procedure. For a simple on e-variable version consult: a) Numerical Methods texts for the theory \+ behind the algorithm, and b) Robert Israel's: " }{TEXT 283 22 "Calculu s the Maple Way" }{TEXT -1 398 " for an implementation in Maple. Our p rocedure works with an argument that contains a list [t,y1,...,yn] whe re x is the independent variable, and yn the dependent variables, n=1. .N. N has to be specified, as well as a starting value of the independ ent variable (called t1), and a fixed step-size h for stepping t. f is a procedure that calculates the right-hand sides of the differential \+ equations." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "RKn:=proc(f,i c:: list,N,t1,h) local tn,yn,ynew,m1,m2,m3,m4,ii,nn;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "tn:=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((t1-tn)/ 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(tn,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(tn+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(tn+h/2,evalm(yn+h/2*m2))[nn])]; od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "tn:=tn+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(tn,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 "[tn,op(ynew)];" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 306 "T he procedure defining the RHS is straightforward: A list is expected w ith the derivatives, and we have no direct dependence on the independe nt variable t (just the implicit dependence through the field line 'tr ajectory'). Note that we make a single call to [BSOx(x,z),BSOz(x,z)] p er evaluation of the RHS." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "fn:=proc(t,yn) global BSO;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "[ BSOx(yn[1],yn[2]),BSOz(yn[1],yn[2])];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 419 "The procedure RKn r eturns a list which contains the independent and dependent variables. \+ We can define a loop to store the intermediate results. In contrast to our previous calculations based on dsolve[numeric] we should check th e dependence of the results on the choice of dt! We could, in fact cho ose a smaller value of dt and store only every jth value (if we think \+ that dt is too large for an accurate integration)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 163 "Implementation details: \+ x_i, and z_i hold the initial values for each Runge-Kutta step; at the end of the RK step they are updated for the next cycle (as is t_i). \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "Rt:=[]: dt:=0.002: x_i :=0.004: z_i:=-0.002: for it from 1 to 120 do: t_i:=(it-1)*dt: sol:=RK n(fn,[t_i,x_i,z_i],2,t_i+dt,dt); x_i:=sol[2]: z_i:=sol[3]: Rt:=[op(Rt) ,[x_i,z_i,0.]]: od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "PL3: =pointplot3d(Rt,style=point,symbol=cross,axes=boxed,color=white):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "Rt:=[]: dt:=0.001: x_i:=0.0 02: z_i:=-0.002: for it from 1 to 120 do: t_i:=(it-1)*dt: sol:=RKn(fn, [t_i,x_i,z_i],2,t_i+dt,dt); x_i:=sol[2]: z_i:=sol[3]: Rt:=[op(Rt),[x_i ,z_i,0.]]: od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "PL4:=poin tplot3d(Rt,style=point,symbol=cross,axes=boxed,color=white):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 180 "Rt:=[]: dt:=0.001: x_i:=0.0 01: z_i:=-0.002: for it from 1 to 80 do: t_i:=(it-1)*dt: sol:=RKn(fn,[ t_i,x_i,z_i],2,t_i+dt,dt); x_i:=sol[2]: z_i:=sol[3]: Rt:=[op(Rt),[x_i, z_i,0.]]: od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "PL5:=point plot3d(Rt,style=point,symbol=cross,axes=boxed,color=white):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "Rt:=[]: dt:=0.001: x_i:=0.0 03: z_i:=-0.002: for it from 1 to 100 do: t_i:=(it-1)*dt: sol:=RKn(fn, [t_i,x_i,z_i],2,t_i+dt,dt); x_i:=sol[2]: z_i:=sol[3]: Rt:=[op(Rt),[x_i ,z_i,0.]]: od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "PL6:=poin tplot3d(Rt,style=point,symbol=cross,axes=boxed,color=white):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 180 "Rt:=[]: dt:=0.001: x_i:=0.0 0: z_i:=-0.002: for it from 1 to 100 do: t_i:=(it-1)*dt: sol:=RKn(fn,[ t_i,x_i,z_i],2,t_i+dt,dt); x_i:=sol[2]: z_i:=sol[3]: Rt:=[op(Rt),[x_i, z_i,0.]]: od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "PL7:=point plot3d(Rt,style=point,symbol=cross,axes=boxed,color=white):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 182 "Rt:=[]: dt:=0.001: x_i:=-0. 001: z_i:=-0.002: for it from 1 to 100 do: t_i:=(it-1)*dt: sol:=RKn(fn ,[t_i,x_i,z_i],2,t_i+dt,dt); x_i:=sol[2]: z_i:=sol[3]: Rt:=[op(Rt),[x_ i,z_i,0.]]: od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "PL8:=poi ntplot3d(Rt,style=point,symbol=cross,axes=boxed,color=white):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "display(PL2,PL3,PL4,PL5,PL6, PL7,PL8);" }}{PARA 13 "" 1 "" {GLPLOT3D 475 457 457 {PLOTDATA 3 "6--%% MESHG6%7K7K7%$!$+\"!\"%F)$\"+5>%\\u*!#77%$!#&*F+F)$\"+?q\\!>\"!#67%$!# !*F+F)$\"+Y!QwV\"F47%$!#&)F+F)$\"+E>$yr\"F47%$!#!)F+F)$\"+&f@A.#F47%$! #vF+F)$\"+mB'3Q#F47%$!#qF+F)$\"+VESiFF47%$!#lF+F)$\"+,i!Q<$F47%$!#gF+F )$\"+-.<5OF47%$!#bF+F)$\"+nGpkSF47%$!#]F+F)$\"+4`#)GXF47%$!#XF+F)$\"+7 jh#*\\F47%$!#SF+F)$\"+,O@XaF47%$!#NF+F)$\"+llXveF47%$!#IF+F)$\"+2hZsiF 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We started the field lines with equal spacing in \+ " }{TEXT 288 1 "x" }{TEXT -1 77 ". They remain equally spaced inside t he solenoid on account of the fact that " }{TEXT 290 1 "B" }{TEXT -1 1 "_" }{TEXT 289 1 "x" }{TEXT -1 57 " is very small there. We suspect \+ that the field strength " }{TEXT 291 1 "B" }{TEXT -1 1 "_" }{TEXT 292 1 "z" }{TEXT -1 4 " at " }{TEXT 293 1 "x" }{TEXT -1 168 "=0 can not va ry too much inside the solenoid, or the field would not be as homogene ous as it appears to be. Let us check by calculating and plotting the \+ two components." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "DL1:=[se q([iz*0.0001,BSOz(0.,iz*0.0001)],iz=-100..100)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "DL2:=[seq([iz*0.0001,BSOz(0.003,iz*0.0001)],i z=-100..100)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "LL1:=[seq ([LL[i],0.],i=1..nops(LL))]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "PL0:=plot(LL1,style=point,color=blue,symbol=circle): PL1:=plot( DL1,color=red): PL2:=plot(DL2,color=green): display(PL0,PL1,PL2,axes=b oxed,title=\"B_z(x0,z) inside the coil\");" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6)-%'CURVESG6&7(7$$!1+++++++D!#=\" \"!7$$!1+++++++:F*F+7$$!1+++++++]!#>F+7$$\"1+++++++]F2F+7$$\"1+++++++: F*F+7$$\"1+++++++DF*F+-%'COLOURG6&%$RGBGF+F+$\"*++++\"!\")-%&STYLEG6#% &POINTG-%'SYMBOLG6#%'CIRCLEG-F$6$7ew7$$!1+++++++5!#<$\"1+++\\pA\"\\(FQ 7$$!1,++++++**F*$\"1*****p=a>o(FQ7$$!1+++++++)*F*$\"1,++D8pyyFQ7$$!1++ +++++(*F*$\"1+++E\"\\;3)FQ7$$!1*************f*F*$\"1+++>b/\"H)FQ7$$!1+ ++++++&*F*$\"1*****\\o0r])FQ7$$!1+++++++%*F*$\"1+++UA1I()FQ7$$!1****** *******H*F*$\"1+++@`:g*)FQ7$$!1+++++++#*F*$\"1+++@Fj(>*FQ7$$!1+++++++ \"*F*$\"1+++I*\\FW*FQ7$$!1***************)F*$\"1*****RBqdp*FQ7$$!1++++ +++*)F*$\"1******HZ'p&**FQ7$$!1,++++++))F*$\"1+++L7mA5!#;7$$!1******** *****p)F*$\"1+++t**\\]5Fjq7$$!1+++++++')F*$\"1+++S@Cz5Fjq7$$!1,++++++& )F*$\"1+++Pz\"*36Fjq7$$!1*************R)F*$\"1+++;$e&R6Fjq7$$!1+++++++ $)F*$\"1+++^\\>r6Fjq7$$!1,++++++#)F*$\"1+++=-'Q?\"Fjq7$$!1+++++++\")F* $\"1+++_reP7Fjq7$$!1+++++++!)F*$\"1+++9%4CF\"Fjq7$$!1,++++++zF*$\"1+++ J7O38Fjq7$$!1+++++++yF*$\"1+++]tZX8Fjq7$$!1+++++++xF*$\"1+++eHz$Q\"Fjq 7$$!1+++++++wF*$\"1+++2OMB9Fjq7$$!1+++++++vF*$\"1+++E^;k9Fjq7$$!1+++++ ++uF*$\"1+++5NH1:Fjq7$$!1+++++++tF*$\"1+++2[w\\:Fjq7$$!1+++++++sF*$\"1 +++%)\\h%f\"Fjq7$$!1+++++++rF*$\"1+++z(z3k\"Fjq7$$!1+++++++qF*$\"1+++O Xf)o\"Fjq7$$!1+++++++pF*$\"1+++ASzPFjq7$$!1+++++++kF*$\"1+++1?@2?Fjq7$$!1+++++++jF*$\" 1+++^**)f1#Fjq7$$!1+++++++iF*$\"1+++.0YE@Fjq7$$!1+++++++hF*$\"1+++5\\k )=#Fjq7$$!1+++++++gF*$\"1+++58c_AFjq7$$!1+++++++fF*$\"1+++^VA=BFjq7$$! 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In other words, the field is not as uniform as suggested by the f ield lines." }}{PARA 0 "" 0 "" {TEXT 298 1 "B" }{TEXT -1 1 "_" }{TEXT 299 1 "z" }{TEXT -1 111 " as calculated off-axis is stronger inside th e coil, and somewhat weaker outside than the on-axis result (red, " } {TEXT 300 1 "x" }{TEXT -1 206 "=0). Presumably for a longer solenoid a homogeneous field can be guaranteed inside the coil. We leave it as a n exercise to increase the number of current loops, and to repeat the \+ calculations for that case." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "How well is " }{TEXT 297 1 "B" }{TEXT -1 1 "_" }{TEXT 296 1 "x" }{TEXT -1 143 " expelled from the inside of the solen oid? Let us calculate it at heights in-between the current loops begin ning near the edge of the solenoid " }}{PARA 0 "" 0 "" {TEXT -1 30 "an d moving towards the center:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "DL1:=[seq([ix*0.0001,BSOx(ix*0.0001,0.002)],ix=-100..100)]:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "DL2:=[seq([ix*0.0001,BSOx(ix *0.0001,0.001)],ix=-100..100)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "DL3:=[seq([ix*0.0001,BSOx(ix*0.0001,-0.001)],ix=-100..100)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 210 "PL0:=plot([[-0.005,0],[0 .005,0]],style=point,color=blue,symbol=circle): PL1:=plot(DL1,color=re d): PL2:=plot(DL2,color=green): PL3:=plot(DL3,color=magenta): display( PL0,PL1,PL2,PL3,axes=boxed,title=\"B_x(x,z0)\");" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6*-%'CURVESG6&7$7$$!1+++++++]!#=\" \"!7$$\"1+++++++]F*F+-%'COLOURG6&%$RGBGF+F+$\"*++++\"!\")-%&STYLEG6#%& POINTG-%'SYMBOLG6#%'CIRCLEG-F$6$7ew7$$!1+++++++5!#<$!1+++a,&3*HFD7$$!1 ,++++++**F*$!1+++%[#)=6$FD7$$!1+++++++)*F*$!1+++*32*QKFD7$$!1+++++++(* F*$!1+++V'oAP$FD7$$!1*************f*F*$!1+++cRL7NFD7$$!1+++++++&*F*$!1 +++nM\\fOFD7$$!1+++++++%*F*$!1+++()G;9QFD7$$!1*************H*F*$!1+++] \\ywRFD7$$!1+++++++#*F*$!1+++H7$y9%FD7$$!1+++++++\"*F*$!1+++KT!yK%FD7$ $!1***************)F*$!1+++#**Qs^%FD7$$!1+++++++*)F*$!1+++xiq;ZFD7$$!1 ,++++++))F*$!1+++VR\"o#\\FD7$$!1*************p)F*$!1+++q)4#[^FD7$$!1++ +++++')F*$!1+++RWe\"Q&FD7$$!1,++++++&)F*$!1+++$>twi&FD7$$!1*********** **R)F*$!1+++K&fs)eFD7$$!1+++++++$)F*$!1+++gu8cv'FD7$$!1+++++++!)F*$!1+++Hb2yqFD7 $$!1,++++++zF*$!1*****Hvi(=uFD7$$!1+++++++yF*$!1+++'o(zyxFD7$$!1++++++ +xF*$!1******pZOf\")FD7$$!1+++++++wF*$!1+++.!4F ]v7$$!1+++++++eF*$!1+++Y*4s1#F]v7$$!1+++++++dF*$!1+++ie9a@F]v7$$!1++++ +++cF*$!1+++s0:QAF]v7$$!1+++++++bF*$!1+++x:LF]v7$$!1+++++++PF *$!1+++*e&>9>F]v7$$!1+++++++OF*$!1+++bLYO=F]v7$$!1+++++++NF*$!1+++Is&) f)*FD7$$!1+++++++AF*$!1+++q)ovG*FD7$$!1+++++++@F*$!1+++:$*>q ()FD7$$!1+++++++?F*$!1+++LF!oE)FD7$$!1+++++++>F*$!1+++X/dwxFD7$$!1++++ +++=F*$!1+++#G0()H(FD7$$!1+++++++ipP'FD7$$!1+++++++:F*$!1+++A'p:$fFD7$$!1+++++++9F*$!1+++Fk^&\\&FD7 $$!1+++++++8F*$!1+++'3!4o]FD7$$!1+++++++7F*$!1+++Kcf[YFD7$$!1+++++++6F *$!1+++][NOUFD7$$FCF*$!1+++#\\02$QFD7$$!1+++++++!*!#>$!1+++A++JMFD7$$F bsFdil$!1+++HXgOIFD7$$FevFdil$!1+++\\u*ok#FD7$$!1**************fFdil$! 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Our naive argument, where we have taken e qually spaced initial conditions for the field line trajectories, coup led with the observation that they remain parallel has misled us." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 111 "We conclud e the workshet with a further analysis of properties of the magnetic f ield for a simple current loop." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 306 37 "div B and curl B for the current loop" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 180 "We have stated above that the two components of the magn etic field are really the transverse and longitudinal components respe ctively. Thus, in cylindrical coordinates we have the " }{TEXT 303 1 " z" }{TEXT -1 54 "-component as defined. and what was calculated as the " }{TEXT 302 1 "x" }{TEXT -1 241 "-component is really the component \+ along the radial direction. Due to the symmetry of the field (circular current loop) there is no phi-component of the magnetic field. We re- define our previously calculated components, and use the variable " } {TEXT 304 1 "r" }{TEXT -1 189 " to specify the cylindrical radial vari able (often denoted by rho, to set it apart from the spherical radial \+ coordinate). We do not specify the amount of current, or the radius of the loop " }{TEXT 305 1 "R" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "restart; with(linalg): with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 33 "Qp:=(Q+R)^2+a^2: Qm:=(Q-R)^2+a^2:" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 19 "Qr:=2*sqrt(Q*R/Qp);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#QrG,$*$-%%sqrtG6#*&*&%\"QG\"\"\"%\"RGF-\"\"\",&*$),& F,F-F.F-\"\"#F/F-*$)%\"aGF4F/F-!\"\"F/F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "Brex:=i_L*a/(5*10^(6))/(sqrt(Qp)*Q)*((R^2+Q^2+a^2)*El lipticE(Qr)/Qm-EllipticK(Qr));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "B ra:=unapply(Brex,Q,a):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%BrexG,$*& *(%$i_LG\"\"\"%\"aGF),&*&*&,(*$)%\"RG\"\"#\"\"\"F)*$)%\"QGF2F3F)*$)F*F 2F3F)F)-%*EllipticEG6#,$*$-%%sqrtG6#*&*&F6F3F1F3F3,&*$),&F6F)F1F)F2F3F )F7F)!\"\"F3F2F)F3,&*$),&F6F)F1!\"\"F2F3F)F7F)FGF)-%*EllipticKGF;FLF)F 3*&-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)*(Elli pticE(Qr)*(R^2-Q^2-a^2)/Qm+EllipticK(Qr));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Bza:=unapply(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 -1 244 "The first property we wish to dem onstrate is the fact that there are no magnetic charges, i.e., that th e B field is divergence free. The divergence in cylindrical coordinate s is given as follows [it is part of the linalg package loaded before] :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "B := vector([Br(r,z), 0, Bz(r,z)]): v := vector([r, phi, z]):\ndivB:=diverge(B, v, coords= cylindrical);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%divBG*&,(*&-%%diff G6$-%#BrG6$%\"rG%\"zGF.\"\"\"F.F0F0F+F0*&-F)6$-%#BzGF-F/F0F.\"\"\"F0F6 F.!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "Here we have made use \+ of the fact that there is no phi-component of the magnetic field." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "Let us s ee what the calculation tells us about the divergence of the magnetic \+ field for the current loop." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "divBcl:=simplify(subs(Br(r,z)=Bra(r,z),Bz(r,z)=Bza(r,z),divB));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'divBclG\"\"!" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 190 "Indeed, the magnetic field for the current loop i s divergence-free. The student is encouraged to look at the result bef ore the simplify command is applied to see how the cancellation occurs ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 386 "We \+ can continue with a calculation of the curl. According to Maxwell's eq uations the curl of the vector field B should be related to the rate o f change of electric fields, and the current density. If we assume tha t we have no electric field present, and if we stay away from the curr ent loop (assumed to be of infinitesimal extent), then we should calcu late that the field is curl-free." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "B := vector([Br(r,z), 0, Bz(r,z)]): v := [r, phi, z] :\nrotB:=curl(B, v, coords=cylindrical);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%rotBG-%'vectorG6#7%\"\"!,&-%%diffG6$-%#BrG6$%\"rG%\"zGF2\"\" \"-F,6$-%#BzGF0F1!\"\"F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "The \+ reader is encouraged to look at the expression in the general case wit h non-vanishing phi-component of the field." }}{PARA 0 "" 0 "" {TEXT -1 150 "For our special case of the current loop we have the following [note that the subs command has to substitute into a vector quantity, thus the op()!] :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "rotBc l:=simplify(subs(Br(r,z)=Bra(r,z),Bz(r,z)=Bza(r,z),op(rotB)));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'rotBclG-%'vectorG6#7%\"\"!F)F)" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "It is worthwhile to look at the in dividual parts that get cancelled below." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "simplify(diff(Bra(r,z),z))-simplify(diff(Bza(r,z),r)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 476 "We can arrive at a more general conclusion: whenever the situation is as simple as discussed here from a symmetry point of vie w (vanishing phi-component), the two statements (divergence-free, and \+ curl-free properties of the magnetic field; the first statement is alw ays true, the second requires no time-varying electric fields, and no \+ current density in the spatial region considered) can be combined to r elate variations in the radial and transverse components of the field. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 179 "We finish with the remark that the analytic express ions for the magnetic field components of the current loop are used in MagBottle.mws to calculate charged-particle trajectories." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "105 0 0" 179 } {VIEWOPTS 1 1 0 1 1 1803 }