{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 "" -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 "Normal" 
-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):" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 153 "Maple can remind us of the Laplacian in variou
s orthogonal coordinate systems (without explicit computations). We ar
e interested in the radial part only:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 61 "v1 := [r, theta, z]:\nlaplacian(f(r), v1, coords=cyli
ndrical);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "DE:=diff(V(r),
r$2)+diff(V(r),r)/r=0;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "This di
fferential equation has a very simple solution:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 16 "dsolve(DE,V(r));" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 199 "The boundary conditions of zero potential at the filamen
t and potential V0 at the cylinder can be imposed, but the thickness o
f the filament has to be taken into account due to the log-divergence \+
at " }{TEXT 257 1 "r" }{TEXT -1 3 "=0." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "BC:=V(l)=0,V(d)=V0;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "sol:=dsolve(\{DE,BC\},V(r));" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 24 "Vp1:=simplify(rhs(sol));" }}}{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);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 181 "This expression is more natural, \+
as it makes it obvious that the logs are taken of dimensionless number
s. One way to read the equation is to say that we are measuring the di
stances " }{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 immediately that the two express
ions are identical:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "simp
lify(Vp-Vp1);" }}}{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);" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 171 "The electric field decreases in
versely proportional with the radial distance (as expected from the di
vergence of the field lines). Its radial component is calculated from:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "E:=-diff(Vp,r);" }}}
{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.mw
s" }{TEXT -1 149 " , where a charge was distributed linearly on a long
 wire, and the electric field was calculated using cylindrical probe s
urfaces at different radii " }{TEXT 277 1 "r" }{TEXT -1 215 ". The pre
sent result was obtained directly for the electrostatic potential in a
 slightly different setting: we assumed that a given outer cylinder si
ts at a fixed potential difference against the wire in the center." }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "We graph the electric field using
 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);
" }}}{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 negative of " }{TEXT 275 1 "E" }{TEXT -1 
49 " has to be graphed so that the log can be taken):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 67 "loglogplot(subs(l=1,d=100,V0=100,-E),r=1..100,
scaling=constrained);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "The grap
h below shows how the voltage increases as a function of radial distan
ce." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "plot(subs(l=1,d=100,
V0=100,Vp),r=1..100);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "This res
ult generalizes a simple result known from the parallel-plate condense
r: when a voltage " }{TEXT 272 1 "V" }{TEXT -1 596 "0 is applied acros
s two parallel plates, the electric field between the plates is consta
nt (the field lines are parallel, i.e., the field is homogeneous) and \+
the electric potential changes linearly as one goes from one plate to \+
the other. In the case of the cylindrical geometry the situation is di
fferent: the field lines follow radial beams that connect the two circ
ular cross sections, i.e., the distance between them increases as one \+
goes from the inner cylinder to the outer one. Correspondingly, the el
ectric field weakens with radial distance. We note that the electric f
ield falls like 1/" }{TEXT 273 1 "r" }{TEXT -1 133 " (as opposed to be
ing constant), and that the potential grows only logarithmically (rath
er than linearly) in this different geometry." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "The following graph illu
strates the cylinders (filament and anode cylinder) via their cross se
ctions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with(plottools):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "c1 := disk([0,0], 1, co
lor=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*sin(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);" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "If we need to calcula
te the electric field in Cartesian coordinates, we substitute and diff
erentiate:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "VpC:=subs(r=s
qrt(x^2+y^2),Vp);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "[-diff
(VpC,x),-diff(VpC,y)];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "One can
 see that the force is central (in the " }{TEXT 259 1 "x" }{TEXT -1 1 
"-" }{TEXT 258 1 "y" }{TEXT -1 112 " plane): the orientation of the el
ectric field is towards the center, i.e., proportional to the position
 vector " }{TEXT 262 2 "r " }{TEXT -1 3 "= [" }{TEXT 261 1 "x" }{TEXT 
-1 2 ", " }{TEXT 260 1 "y" }{TEXT -1 45 "]. The force on a charged par
ticle 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);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Ey:=-y/(x^2+y^2);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "To understand this assignment dif
ferentiate the expression for the field in cylindrical coordinates " }
{TEXT 19 14 "V(r) = A*ln(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 particle 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:=dt^2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 161 "
We are interested in an initial condition which corresponds to particl
es at rest. We translate this into two subsequent postions to start th
e two-step recursion." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "vx
0:=0; vy0:=0;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 181 "We have to star
t the trajectories at a finite distance (filament thickness or larger)
. We can also choose finite initial velocities (they depend on the tem
perature of the filament)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 60 "Let's measure distances in multiples of the filam
ent radius." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "rx[0]:=1; rx
[1]:=rx[0]+dt*vx0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "ry[0]
:=1; ry[1]:=ry[0]+dt*vy0;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 329 "We \+
construct a procedure that implements a single step of the recursion. \+
The first argument is the index at which the recursion is carried out,
 the second is a parameter that allows one to change the overall stren
gth of the electric field which is proportional to the potential diffe
rence 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:=eval
f(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;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "
for i from 1 to N do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Verlet(i,1
0); od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "nops(convert(rx,
list));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "rx[40],ry[40];" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "plot([seq([rx[i],ry[i]],i
=0..N)],style=point,scaling=constrained);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 47 "Suppose we want to see only a subset of points?" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "RL:=[[rx[0],ry[0]]]: for i f
rom 1 to N by 5 do: RL:=[op(RL),[rx[i],ry[i]]]: od:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 54 "plot(RL,style=point,scaling=constrained,s
ymbol=cross);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 291 "We notice that \+
the charged particle picks up speed quickly (accelerates strongly) ear
ly on at small values of the radius. For later times the spacing betwe
en the points becomes more equidistant, which means that the velocity \+
is approaching a constant due to the decrease in the acceleration." }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "62" 0 }
{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }