{VERSION 2 3 "APPLE_PPC_MAC" "2.3" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 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 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 
0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 
0 0 0 0 0 0 -1 0 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 
0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "Autho
r" 0 19 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 
8 8 0 0 0 0 0 0 -1 0 }}
{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 26 "Computer Intensive Physic
s" }}{PARA 18 "" 0 "" {TEXT -1 23 "Physics 212E: Fall 1997" }}}{EXCHG 
{PARA 19 "" 0 "" {TEXT -1 16 "Steven R. Dunbar" }}{PARA 19 "" 0 "" 
{TEXT -1 40 "Department of Mathematics and Statistics" }}{PARA 19 "" 
0 "" {TEXT -1 30 "University of Nebraska-Lincoln" }}{PARA 19 "" 0 "" 
{TEXT -1 23 "Lincoln, NE, USA  68588" }}{PARA 19 "" 0 "" {TEXT -1 20 "
sdunbar@math.unl.edu" }}{PARA 0 "" 0 "" {TEXT -1 32 "http://www.math.u
nl.edu/~sdunbar" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 507 "\nMoving Char
ge in a Magnetic Field\n\n\nKEYWORDS: Magnetic Force, Charges moving i
n a Magnetic Field, Uniform Magnetic Force, Cathode-Ray Tube, Particle
 Beam Deflection,;\n\nOBJECTIVE: Derive the deflection of a electric p
article beam in a cathode-ray tube by a uniform manetic field from a p
air of circular coils (Helmholtz coils). \n\nPREREQUISITE: Know the fo
rce exerted on a particle beam by a uniform magnetic field by using th
e cross product. Know basic mechanics and kinematics.  Know parametric
 equations.\n" }}{PARA 0 "" 0 "" {TEXT -1 44 "LENGTH:  One class perio
d, about 50 minutes." }}{PARA 0 "" 0 "" {TEXT -1 59 "AUDIENCE:  Instru
ctor, student use in computer lab setting." }}{PARA 0 "" 0 "" {TEXT 
-1 21 "SOFTWARE: Maple V R 4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 38 "FUTURE:  Add in a descriptive diagram." }}
{PARA 0 "" 0 "" {TEXT -1 85 "      Need to add in an explanation of th
e parametes used for numerical calculation. " }}{PARA 0 "" 0 "" {TEXT 
-1 159 "      The numerical solution picks up imaginary components (th
rough round-off?)  Needs to be eliminated by counting only the real pa
rt.  Eliminate this kludge." }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }
{METAFILE 435 387 387 1 ":::::::::::::::::::::::::::::::::::::::::::::
::::::::::::::::::::::::::::::::::::::::n]B[:FZ=HZmT>?rySjyyyyyY:B:j:H
e:FZ=@:;B:bA;R:d:<j:F:`S:u@<j:Z:B:ZB;B:DZ;b:<Z:VZBfov[:JbAxyxi=jyyiyM:
@JOeTOeD;b:eTOeTO>Z=J:<:j::<j:Z:c<B:=:?r^DjC;:::::::Z@VNlYsoZ]<:N\\]D:
tiBv;sI=AHvYBjYhyHkK;QBFAx:YB=SBF[Cv;H:HkK;?BFAJymj<<MR<VZ;j?xj;^[I^yy
>x:ARDkKkxO[=E=SJR>r:A:a:siyDZ=YZxa[?::b;YJxWZjy[xAr;B:Grb<Zle;:ORn^aN
Mx;HVd;SRnDj;xmyUGsABI<Ve;j;<JZUG:::^Y;@b<YP:<Ve;:::::::::hJbGZ=>xP=d:
::::::::::::::::YBn;R:FV<muCyrmbr@j;<vu;WrmZ:^[p=:BFgAv;rr@j;nXJA:<J@P
IsA:<yt;Y:LHARw@j;lYs;JXPi;^[vQX:>roU:[Iy>\\IE:PHAZ<V:siv@Kb=j:>x_MXb:
SJxEiJ:BjeQ;<M:lIZ`mALi<>r:^Y`kaKw?ZFNYHJuCvVZ=JxSAY@VZjB:SRy@JZ=Vvs:c
q;l=saXvSj><MZ=MX^[FVZ:ArrMXLi=>r:>ZvMJxf:>rtMZ=MZ?ZF^[:>rjFY=r:;Xw<j;
tIrAY@oAYJ@^;<VRB:SBw<J@ti::Jw;^[j>:[;[Iw>`xA::NY>@<j;<]:A`tE:Hi:>rx=:
oq>>rj^[:V:>`tE:SBwVZwMxCB[p:>xtU<B=AB:;H:;Pw?Zj^Xlmx?ZvmIZt]xU;A:;BjB
jca:iY:bx>@@J@HJyCjqaIJ:HJx?ZvmHZtM@pi=^[:r:oaj^XlJxCyv\\Vm[Qi:>reU:AB
jcaj^y:>ZvMElG;Xwrt<JXHi:By;JxcBGI:Sr:[CjyksmJryj[Iy;<yuCjQa:aXxjwCy:x
VR<v[yjAlJR<m[mJ@XIlh:v[mMRhI:Qr:SBF;rjAB=SBF[[p]gU:ABvaQ:<vw;Acsh:xm[
IJ@vajYrF>Y?BI<vp?ZvB:H:lICcsh:<v[I:[C:ljvs=syy[q:>rmA:scxZgnVSZjjAlJ@
<MRHfFfYQr:yyy[y:ysy;B:yQxIISbv<JZeIbEGQ@BFyCFaI=Sr=;xvABvwq:xJxui<^;;
Xy@ZyaX<ys?Z=FYlI=SrVZrU:ABvuQ;l:qAgQ:xJsCImA_qKDh;r:mq:^yve:[[uU:vX<M
sCjmAN^[A:sYxrxDjy?rtU:ABvmq:>rm=<JxEiUFUxYvCjr=[ix<JXDi:^[rE:saUNUxmu
Cjkq:xJZii:>rlMR\\i:ZVVU=r=XIXIyCyZ;Jv;ASsLJxCjABy<JrUI;hw>xxm;<YsC=ia
v]`E:Aby@JZIjxs=eA^ZIVZyEI]ajvX:PjAlvA<VZIyxCvgq:HZou:yyyAHj]a:yXjJTh?
VZF>`jVrInY;r=ca:iX?r=yIm]a=nX;By:Z=ny:>ZjaX:<yrC=gA>^`E<AB=yCvHJ:v[mM
Z;;Xv>xpE::BF;rjAZjYX;BjZjQXHjtCyv]`u;AB=[CFAx=;r=_Q:<FxyyyJsC=eQ:jJr;
v[:M:<YsCjgQ<>Zj>Zj=eQ:JG<h:v[kMZC:[iy<jALIABw<Jx;<mrkus:sa:iXlMt;JZAI
XIsCy<JRPI;HwZUFV;r=[Q:>Y=ZvE:Ybw<jAHIsABF]q:HZrE:[ix>rqMRHI:<Vr=J:FY;
r=maj>Y=r:;xvZTNVxjqCjBvmaI>YlJs?ZvMGJRxX:bxvaqMZaI[iv^YhKk?Z:yWVY=r:X
I[cxVZkE:saTVV;r=Wa:M:lYuKwCI]A_A>`h]v]nMZiI;Xv<:hkkCjWAuq:>xnMRlY:Bv<
Jxc@CaIfW<F;BIaaI^y:BF[I:;:sBs>`gMZui;v;_Ar:;Hv;:d;<vpKy?ZjIX<MyC=Yav]
_m;ph>VZjE:yCv<J@ti:>xi=\\klC=Sq;<V:HJZ=IAZ=nWlIgRs>`fe:;x=siuDJZ=JXth
:^YPkl;=rj[YuDJZ?rhMxs>EI:;BjSQ:lYpGZyvygMxEiNnVHJpGZyArfU:y;jDTHSRuDj
Y<JZmH:v\\`MRth=VZ:EjNZyvy:qW;:JZyh>v[vE:;Hu@Zy=F]am;xX:@jY<YyCjUa=vyy
iW:^[jAZ;v;ua:]Fv[yiWlI?Sw^[sM@pi:>`jE:[Yy@ZjiWABv;Bj;hu^YTltWZj>Z:^yw
U:ys:ua:IXxywCjWa=nY<MqCy_QNPiC>`:Y`=JR<MZIVys:_q:vytMZuHscy^[hMxSDeq=
<VRHv;<J:by;h:;B:Arv@jAlyvCjYq:HJZuIS:sabj=v[=VZ:>:tI^[nU:yCvkQ:<F;BIu
Q:HJq;ISwTJZyj;B:pYZE:[Sw@jywX<F<Vx?:Yav=BFA@SRy;h:rw@j;xytCj[Q:xjwkqC
y>?TZ:Y`=:ry;XxDjYvyn]jE:[yx<JRxX:^Ylltg[:Arj^y=r:xM:<v;xYrmIAryyIvcQ;
lJ:<vrs:oajvw::QSw<JRtY:`j;<j;>Z=>Z=vYwQ;x=]Q;l=BIoaFvWjMB:tiA^;SJ:HZ=
^yyEvYI:?B=ysy]Q:<Vws:YQ:jKPISByXJR<J:lJ:HjyWW?B=;Jss=ma=:N]de:AJr;Jr]
I<i:^Y`knGZyutZmmAXI:\\knCmOQ;xM:VX<mu?Zj>x::ibtVZaMZMi;VZ:yXlJr=J:JEd
h::;`:^apM@:X;ljmkts:TIABv;B:b?:J:fX;r::JRXHJrMI;Xv^YZ:yVnXxmskr;kbtBt
@ZjyX=JZII:Tko?Z=nV;BjiQ:lI;B=]An<:\\IYRv^;^Y::<yrC=By:SbsRx<j;<i:VZk=
::lMr?::irt^;bx@:[I::n<>``=<JxuH;hv^yl=:qaIfW<Vs;JRZuMrqh:bv:BjGAsQ:xJ
p?ZjNXjEZaU:<YxCvSI:;BF:g:THpi:v[eE:[;Jps:Ea:]jYW;B=_Q:JE:by<jYdHSZ:ZP
:^Z:AZ:VZ:QWJ:jDlh:^[_u:;H:lyns:_Q:NXcBu;B:b:;pA\\HA:::ABvKQ::V\\fAZ:>
`_U:;HIGAaav=:YHjGA:_Bu;B:B:[Ys>rm=H;nV=ry[Ys:B?jmOZyjl;jDJZ]h=v[:EjEa
jVX:Bt>x:>Z:iV:n<RtvaxMZQH:XKp?:Oq:xMZuIPH::Qq:xJZKZQH::Sq:xJrmi:Rs::f
W=r=hIPH::Wq:xZtMZQH:\\Kp?:Yq:bx@j;fV:v\\gm;xh;Byka:iVljs=:n\\g=<JxYI;
@Zlu?jrGZyyypE::ZP:_Q:xYt;::Hi:vamMZ;^Y\\kpktOZIBjaI:<FZvmFph;vajM;xZj
IX<F:f^vm;Hi;vyvQY?r:nY;BFBjca=J:l:lI:@:sI:?r:lYx?ZjZ:=>ZFZvmiliMVZ:r:
;B:SB:;B=Zyyy:^[ynyyQB[K:<=:Bj<::r:V:::l:YB:;B=^;JR>ZvM?li:VZcxy:bvx?m
t:<j;<ZFJ@:yIyl:[K:<=::;P:::AZ=::J@B=:ZjB:saY^y:>Z=VX=rILI[sv>rnm;B:Sr
v^Y<Lx;xjsCFeQ;l]:aXZFZvMDpX:bsDj[?r_=::::F^vAZ:v;<jYJRj[_ynE:S:sIw@:_
qKli;>`:aXHvsCveAHVtCIcASBw^[vMJhi:BwVZmMR:J@J@JxEi\\^Yj:lJZMI[;sk[MIS
:[Cw@j;lIASyDjA<JZQIlJZQi;>xjVXHVt;<=lI;;JRJ:<MtGZ=VxnM@JrIIsa[J:x=xYZ
Mi:^ymA:Li:vanU:lYtGZI^yle>BjEa:]jVXNXB:PH;@[;:;jlCjBF:B:PH;@[;:;jlCjB
F:iBu<JRPH;Xuv[vVY;js;:RuVZjVYVXJElh::lH;xxrvZR^W<MmGZ=>xg]j::i:PHpHY:
VXJCB:SRs<Z`E:Abv<Jx;;:nV:Z::TH:J:Hjl?:GaFNX;:_Z::THB:b>>:Jm;;:_rt;B:J
ZKRDI:Z:]j::JZKR:Hko=J:<mmG:[is>rl=D;nVnVNX::THDIr?QQ:lJmksGZI^yqm;@i:
^Y::lj;XI:lko;:ZtU:YJw?Zv=VZ`E::B=oq:lZtE:RAQA:xZq=DJ@>ruMx;H:ZF^[q=:^
YNW;BjIQ:<vsk;XI[cxD:::Ia:YX:[;:bAOaj>W=ZmA:Z:QY:jEdHSrsBt<j;`ISJwCy:S
::v;oavMCdX:VZa=^[jE::::ZcE:;@;@[Cv<:Zj>W<F>X::\\H<I:lJnkns:YavMD`X:<J
@dH[;V::JoOZjEjB=YABt;B:hh<>`yvycMRxX:<::J:FW:f\\bMRlX:HJr=]:]:yWJDZe=
`HxH:ljoOZvEjOQ:nW;ByN\\bm;`h:bt^[h=:::rs:;@[cu<:ZjNW<FnW:^[dE:hh:VZgE
:saMvv:r:Qq:HZeM@ph:ZLnv::<F<mp?::[st>RRu:B=QQ:^W;Jp?Zv=VZeE:J@lh:^YfV
Z:]j^WlIJRJZ;^Y:Bj:lIJ@JZqHA:sAr:SAB=QABs<JZKZKRhHr>CI:<MqGZ=Ru>re=D;x
jpKq?ZINWlIHh::;@[ct:>ZjnW<FNW:<jAtHxHYRt^YNV;BjBjBFMA:[su>RRt:r=YA[Q:
xJn?Zv][A:JZKR\\H:<MrCjJn;jA<i:Rvrs^YHKj?:_q:HZkMRXHb>;I:;BF]a:IXvV:<j
A@IDi:v[`E:sAy`jQX<F<Ym;jADi:>rmE:PHsAw`jYX<F<yl;jAHi:>rn]^MxkhCvca:]j
aV;By:YBw>roE:Yrr^Y^U;Bvea:]jYV:v[oMZUIYbr<Jxs>q`jyX=ZpMrEH:=r=;xw<JZY
IySr^YNUxmuCjkq:xZ[=FUxYvCjmq:>xZE:sAk`vIY<VwGZ=rq^ylE?kH:=r:six>ruU:A
JiCyV\\QU:<yw?Z:]viUlYs_\\PAZ;v[vaY<vxGZ=vyVAZ:^YPkd=j?Hjy?xjEj;B:yyys
yp<:lJcs=syykP:ZKe:SrITg:^YZjJe;UbnHJrCj:S:;H:;@R=r:saP=dja;:::::::::Z
E>t:FZ=>xP=d::::::::::::::::::ZE::d::::::::::::::::::ZE::d::::::::::::
:::::Q::N;::::::::::::::^;>X;@R=J@<JZ;l:l::ORn>xNMxs<_p:<Y@LGb<:j?j;^T
J=Hd:^Y:::::::::::::::::J;HS:D:::::B:;5:" }}}{EXCHG {PARA 3 "" 0 "" 
{TEXT -1 35 "Part I:  The Force and Acceleration" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 56 "The magnetic field is uniform in a cy
lindrical volume.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 71 "The circular cross-section is centered at the origin an
d has radius R  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 120 "The magnetic field is along the axial direction of the u
niform magnetic field, which is taken to be the negative z-axis." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 62 "B_field := vector([0,0,-B]);  # uniform, back into the diagram" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "current := vector([q*vx, \+
q*vy, 0]);  # electron moves only in x-y plane" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 38 "force := crossprod( current, B_field);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "acceleration  := scalarmul(f
orce, 1/m);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 83 "Initially, the electron has no component of velocity in
 the y-direction, so vy = 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 91 "Therefore, there is initially no component of f
orce and no acceleration in the x-direction." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 246 "Assume that at any time the co
mponent of velocity in the y-direction is small compared to the compon
ent of velocity in the x-direction.  Then we can assume that the accel
eration is all in the y-direction.  This will simplify all the calcula
tions." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "a := (q/m)*vx*B;" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 
25 "Part II :  The Trajectory" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "
The circular field is centered at the origin and has radius R." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "The parti
cle enters the circular field at (x0,y0) on the circle so that x0^2 + \+
y0^2 = R^2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 44 "The particle velocity along the x-axis is v." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "x := x0 +
 vx*t;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "y := y0 + (1/2)*a
*t^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "intercept_eqn_1 :=
 x^2 + y^2 = R^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "initia
l_condition := \{x0^2 + y0^2 = R^2\};" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 73 "intercept_eqn_2 := simplify( expand(intercept_eqn_1),
 initial_condition);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "int
ercept_time := solve(intercept_eqn_2, t); " }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 64 "exit_time := intercept_time[2];  # check index, you
rs may differ" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "x0 := sqrt(R^2 - y0^2);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "exit_time;" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 93 "test := subs( R=0.06, vx=1*10^7, B = 7*10^(-3), \+
q = 1.6*10^(-19), m = 9*10^(-31), exit_time);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 12 "evalf(test);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "evalf( subs( y0 = 0, test));" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "plot( Re
(test), y0=0..0.6);" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 24 "Part III: \+
The Deflection" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "tantheta \+
:= a*exit_time/vx;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "angl
e_test := Re( evalf(subs( R=0.06, vx=1*10^7, B = 7*10^(-3), q = 1.6*10
^(-19), m = 9*10^(-31), y0 = 0, tantheta)));" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "2 0 0" 1 }{VIEWOPTS 1 1 0 1 1 1803 }