{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 "" -1 256 "" 1 16 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{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 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 31 "Field lines for electric \+ fields" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 248 "We calculate field lines for an electric field produced by a few \+ charges by integrating the trajectories for positive probe charges. Th e fields are calculated by taking the gradient of the sum of the Coulo mb potentials created by the fixed charges." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 343 "Suppose we have a single posit ive fixed charge of two units located at the origin, and two single ne gative charges elsewhere in the x-y plane. We will start the trajector ies on a small circle surrounding the positive charge, and we will ter minate them when they get too close to a negative charge, or when the \+ speed reaches a very large value." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 25 "We work in simple units (" }{TEXT 264 1 " e" }{TEXT -1 4 "=1, " }{TEXT 263 1 "m" }{TEXT -1 3 "=1)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "P1:=[-2,-3]; P2:=[3,1];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "Vpot:=(x,y)->2/sqrt(x^2+y^2)-1/sqrt((P1[1]-x)^2+(P1[2]-y)^2)-1/s qrt((P2[1]-x)^2+(P2[2]-y)^2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 234 "We illustrate the potential by a contour diagram which shows equipote ntial lines (lines of equal potential energy). Think about topographic maps which display the height of mountains and depth of seas/ocenas t o understand the diagram." }}{PARA 0 "" 0 "" {TEXT -1 176 "The field l ines will cross these lines at right angles, as the gradient of the po tential (which gives the electric field vector) is perpendicular to th e equipotential contours." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 209 "PL1:=contourplot(Vpot(x,y),x=-6..6,y=-6..6,axes=boxed,filled=true ,coloring=[blue,red],grid=[40,40],contours=[-1,-0.8,-0.6,-0.4,-0.2,0., 0.2,0.4,0.6,0.8,1.,1.2,1.4,1.6,1.8,2.],scaling=constrained): display(P L1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 386 "When calculating the ele ctric field one has to understand that one differentiates the potentia l first, and then supplies the values of (x,y) at which one calculates the x- and y-components. This can be achieved either by differentiati ng the mapping Vpot, and then evaluating. Alternatively one would have to differentiate the expression Vpot(x,y) and subsequently substitute the values." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 160 "We make use of the operator D which differentiates mappings fo r two reasons: 1) it is more elegant; 2) we want to show how to use D \+ with multivariate functions:" }}{PARA 0 "" 0 "" {TEXT -1 78 "(note tha t the minus sign makes the display of Maple's answer look unpleasant) " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Ex:=-D[1](Vpot);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Ey:=-D[2](Vpot);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 248 "To demonstrate where the electric field \+ is strong we can make a contourplot of the magnitude of the electric f ield. No directional information is contained in this plot, but there \+ should be an indication where we expect the field lines to be dense." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 183 "PL2:=contourplot(sqrt(Ex (x,y)^2+Ey(x,y)^2),x=-6..6,y=-6..6,axes=boxed,filled=true,coloring=[wh ite,red],grid=[40,40],contours=[0.2,0.5,1.,2.,5.,10.],scaling=constrai ned): display(PL2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Which way \+ does the field vector point?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "[Ex(0.2,0.),Ey(0.2,0.)];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 214 "To draw the field lines we step along the direction in which the \+ electric field vector points. We start by choosing initial points on a circle around the positive charge at the center and pick 20 initial p ositions:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "r0:=0.2; r02:= r0^2; Nt:=20;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "phi:=i->(i-1)*2*Pi /Nt;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "Nmax:=1000; #the ma ximum number of points" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "dt :=0.01;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 389 "x_i:=r0*cos(phi(1)): y_ i:=r0*sin(phi(1)): Rt:=[[x_i,y_i]]: for it from 1 to Nmax do: Fx:=Ex(x _i,y_i); Fy:=Ey(x_i,y_i); F_m:=sqrt(Fx^2+Fy^2); if dt*F_m>0.25 then dt :=dt/2 elif dt*F_m<0.1 then dt:=dt*2. fi; x_i:=evalf(x_i+dt*Fx); y_i:= evalf(y_i+dt*Fy); Rt:=[op(Rt),[x_i,y_i]]: if evalf((P1[1]-x_i)^2+(P1[ 2]-y_i)^2)70. th en break fi; od:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "Now that we k now that the calculation for one fieldline worked we can complete the \+ loop:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 449 "for j from 2 to N t do: dt:=0.01; x_i:=evalf(r0*cos(phi(j))): y_i:=evalf(r0*sin(phi(j))) : Rt:=[op(Rt),[x_i,y_i]]: for it from 1 to Nmax do: Fx:=Ex(x_i,y_i); F y:=Ey(x_i,y_i); F_m:=sqrt(Fx^2+Fy^2); if dt*F_m>0.25 then dt:=dt*0.5 e lif dt*F_m<0.1 then dt:=dt*2. fi; x_i:=evalf(x_i+dt*Fx); y_i:=evalf(y_ i+dt*Fy); Rt:=[op(Rt),[x_i,y_i]]: if evalf((P1[1]-x_i)^2+(P1[2]-y_i)^ 2)70. then break fi; od: od:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "We can check the \+ total number of points assembled into the list (all fieldlines in one \+ list):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "nops(Rt);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "PL3:=plot(Rt,style=point,sy mbol=cross,view=[-6..6,-6..6],axes=boxed,scaling=constrained,color=bla ck): display(PL3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "displ ay(PL1,PL3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "display(PL2 ,PL3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "We note that the criter ia to stop field lines are somewhat touchy:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 266 "If we step too quickly towards a charge where the field line is supposed to stop (i.e., dt is so lar ge that we jump over the charge), we can create artificial lines that \+ stop in 'mid-air'. We tried hard to recognize the vicinity of a strong increase in the field by:" }}{PARA 0 "" 0 "" {TEXT -1 63 "a) adjustin g the 'time'-step according to the size of the field" }}{PARA 0 "" 0 " " {TEXT -1 89 "b) terminating if the field strength becomes large in a ddition to the vicinity criterion." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 87 "In a correct field line plot we have the \+ following properties for electric field lines:" }}{PARA 0 "" 0 "" {TEXT -1 76 "a) field lines originate in positive charges and terminat e in negative ones;" }}{PARA 0 "" 0 "" {TEXT -1 58 "b) field lines are perpendicular to equipotential contours" }}{PARA 0 "" 0 "" {TEXT -1 73 "c) the density of field lines is indicative of the strength of the field." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 147 "Our plot has one deficiency (and increasing the number of points \+ Nmax to 10,000 has resulted in excessive calculations without fixing t he problem):" }}{PARA 0 "" 0 "" {TEXT -1 150 "there is a field line es caping to 'infinity' in the top left of the graph, and apparently an i ncoming line is missing in the bottom left of the graph." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 283 "We are interested in demonstrating some \+ real trajectories. The field lines indicate the acceleration that a pa rticle will experience as it moves in the electric field. However, its actual motion depends very much also on the speed which it picks up ( actually on the velocity vector)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "t:='t';" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "The su perposition principle tells us that the " }{TEXT 262 1 "x" }{TEXT -1 6 "- and " }{TEXT 261 1 "y" }{TEXT -1 163 "-components of the position vector are calculated by the integration of the respective Newton equ ation components. However, as is evident from the expressions for " } {TEXT 260 2 "Ex" }{TEXT -1 5 " and " }{TEXT 259 2 "Ey" }{TEXT -1 17 ", the motions in " }{TEXT 258 1 "x" }{TEXT -1 5 " and " }{TEXT 257 1 "y " }{TEXT -1 13 " are coupled." }}{PARA 0 "" 0 "" {TEXT -1 49 "We choos e particles of unit mass and unit charge:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 11 "m:=1; q:=1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "NEx:=m*diff(x(t),t$2)=q*Ex(x(t),y(t));" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 38 "NEy:=m*diff(y(t),t$2)=q*Ey(x(t),y(t));" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 381 "To specify the initial conditions we have to make a choice of radius and pick the number of trajectorie s to emerge from the vicinity of the positve charge. We pick a larger \+ value of r0 than before, so that the acceleration due to the positive \+ charge at the center does not completely dominate the motion. In other words: we do not wish the probe particle to pick up too much speed." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "r0:=0.5; t:='t':" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "IC:=i->[x(0)=r0*cos(phi(i)), D(x)(0)=0,y(0)=r0*sin(phi(i)),D(y)(0)=0];" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 74 "sol:=dsolve(\{NEx,NEy,op(IC(1))\},\{x(t),y(t)\},num eric,output=listprocedure);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "xt:=subs(sol,x(t)): yt:=subs(sol,y(t)):" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 119 "We check that the numerical integration works, i.e., i t gives numerical answers for the position as a function of time:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "xt(1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "We pick a time interval at which the trajectory will be recorded:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "dt:=0.1;" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "r02:=r0^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "Rt:=[[r0*cos(phi(1)),r0*sin(phi(1) )]]: for it from 1 to 100 do: t:=it*dt; Xt:=xt(t); Yt:=yt(t); Rt:=[op( Rt),[Xt,Yt]]: if Xt^2+Yt^2> 200 then break fi; od:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "One trajectory worked, so we complete the loop:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "for j from 2 to Nt do:" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "t:='t': sol:=dsolve(\{NEx,NEy,op(I C(j))\},\{x(t),y(t)\},numeric,output=listprocedure):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "xt:=subs(sol,x(t)): yt:=subs(sol,y(t)):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "Rt:=[op(Rt),[r0*cos(phi(j)),r0*sin (phi(j))]]: for it from 1 to 100 do: ti:=it*dt; Xt:=xt(ti); Yt:=yt(ti) ; Rt:=[op(Rt),[Xt,Yt]]: if Xt^2+Yt^2> 200 then break fi; od: od:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "PL4:=plot(Rt,style=point,sy mbol=cross,color=green,view=[-6..6,-6..6],axes=boxed,scaling=constrain ed): display(PL1,PL4);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 248 "We obs erve that the attraction to the negative charges located at P1 and P2 \+ influences appreciably only those trajectories that pass by closely. W e may want to observe what happens to probe particles that are started at rest from a larger distance." }}{PARA 0 "" 0 "" {TEXT -1 147 "Also note that the trajectories for the cases where the fixed negative cha rges are almost hit by the probe particle may be calculated inaccurate ly." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "t:='t':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "r0:=1.5;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 69 "IC:=i->[x(0)=r0*cos(phi(i)),D(x)(0)=0,y(0)=r0*sin(p hi(i)),D(y)(0)=0];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "sol:= dsolve(\{NEx,NEy,op(IC(1))\},\{x(t),y(t)\},numeric,output=listprocedur e);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "xt:=subs(sol,x(t)): \+ yt:=subs(sol,y(t)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "xt(1) ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "We pick a time interval at w hich the trajectory will be recorded:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "dt:=0.1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "r02:=r0^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "Rt:=[[r0*co s(phi(1)),r0*sin(phi(1))]]: for it from 1 to 100 do: ti:=it*dt; Xt:=xt (ti); Yt:=yt(ti); Rt:=[op(Rt),[Xt,Yt]]: if Xt^2+Yt^2> 200 then break f i; od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "for j from 2 to N t do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "t:='t': sol:=dsolve(\{NEx, NEy,op(IC(j))\},\{x(t),y(t)\},numeric,output=listprocedure):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "xt:=subs(sol,x(t)): yt:=subs(sol,y(t)):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "Rt:=[op(Rt),[r0*cos(phi(j)),r0*s in(phi(j))]]: for it from 1 to 100 do: t:=it*dt; Xt:=xt(t); Yt:=yt(t); Rt:=[op(Rt),[Xt,Yt]]: if Xt^2+Yt^2> 200 then break fi; od: od:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "PL4:=plot(Rt,style=point,sy mbol=cross,color=green,view=[-6..6,-6..6],axes=boxed,scaling=constrain ed): display(PL1,PL4);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 368 "We obs erve that for the particles which were started at a large distance awa y from the central charge the deflection by the two other charges are \+ more noticeable. This is so as the particles have a lower kinetic ener gy as they encounter the negative fixed charges. They were started fur ther away from the central charge, and therefore have a lower potentia l energy at " }{TEXT 265 1 "t" }{TEXT -1 121 "=0. As a result the majo rity of particles no longer appear to follow simple almost straight li nes as was the case before." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 11 "Exercise 1:" }}{PARA 0 "" 0 "" {TEXT -1 305 "P ick your own locations of three charges in the plane in addition to a \+ strong central charge. Graph the potential, construct the field lines, and graph real charged-particle trajectories that start in the vicini ty of the central charge. Make observations about the field lines as w ell as the trajectories." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 267 11 "Exercise 2:" }}{PARA 0 "" 0 "" {TEXT -1 363 "Pick locations and strengths for a few static charges in the plane. Solve \+ the Newton equation for probe charge trajectories starting at the left of the chosen contourplot window for the potential moving to the righ t. Vary the initial speed for the probe charges and comment on the obs erved trajectories. Repeat the analysis for probe charges of the oppos ite sign." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 \+ 0 0" 30 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }