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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 16 "Poisson equation" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 407 "We solve
 the Poisson problem in cylindrical coordinates to find the variation \+
of electrostatic force and potential with radial distance between two \+
concentric cylinders in the presence of a charge cloud at the cathode \+
(the inner cylinder which represents a heated filament). For the case \+
withoud the charge cloud we solved Laplace's equation with the boundar
y condition that the potential is held constant (" }{TEXT 264 1 "V" }
{TEXT -1 1 "(" }{TEXT 263 1 "l" }{TEXT -1 61 ")=0 at the inner cylinde
r, i.e, the filament of a diode, and " }{TEXT 262 1 "V" }{TEXT -1 1 "(
" }{TEXT 260 1 "d" }{TEXT -1 2 ")=" }{TEXT 261 1 "V" }{TEXT -1 57 "0 a
t the anode cylinder). This was done in the worksheet " }{TEXT 19 11 "
Laplace.mws" }{TEXT -1 413 ". Our interest here is to take the electro
n plasma in a diode into account, i.e., once the filament is heated an
d a current flows, such that there is a stable equilibrium. The electr
onic charge cloud develops its own potential which can impede the flow
 of current (space-charge-limited region of the current-voltage charac
teristic). We attempt to solve precisely this part of the problem in t
he present worksheet." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 64 "We need to solve the equation given in Cartesian coord
inates as " }}{PARA 0 "" 0 "" {TEXT 19 53 "diff(V,x$2)+diff(V,y$2)+dif
f(V,z$2) = 4*Pi*rho(x,y,z)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 64 "The problem reduces to one dimension in cylindric
al coordinates." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart;
 with(linalg):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 153 "Maple can remi
nd us of the Laplacian in various orthogonal coordinate systems (witho
ut explicit computations). We are interested in the radial part only:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "v1 := [r, theta, z]:\nl
aplacian(f(r), v1, coords=cylindrical);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 557 "We need to make some assumptions about the radial distri
bution of the charge cloud. A heated filament results in a constant fl
ow of charge in and out of the filament to avoid a net charge on it. I
f an electric field is applied between the anode (outer cylinder) and \+
cathode (filament), then one has an electron charge density that exten
ds between both cylinders. We will not attempt to model a self-consist
ent situation, but rather calculate the electrostatic repulsion origin
ating from an assumed charge cloud. We simply assume an exponential di
stribution:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "rho0:=1;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "rho:=rho0*exp(-r/10);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "This is a surface density really, \+
as we do not take into account what happens along the " }{TEXT 265 1 "
z" }{TEXT -1 6 " axis." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "e
valf(2*Pi*int(rho*r,r=1..100));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
443 "This is the total amount of charge. It is controlled by the heati
ng current of the filament. At higher temperatures (more heating curre
nt) a larger number of electrons will overcome the work function. The \+
electrons are assumed to be distributed in an exponential fashion (thi
s means that in principle some of them reach the anode at a distance o
f 100 filament radii. (This results in the diode acting as a battery w
hen the filament is heated)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 37 "evalf(2*Pi*int(rho*r,r=1..infinity));" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 32 "We formulate Poisson's equation:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 38 "DE:=diff(V(r),r$2)+diff(V(r),r)/r=rho;" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 215 "This differential equation has a \+
solution that combines the result from the density-free case (which is
 incorporated through the boundary condition which specifies the poten
tial) and the electron-electron repulsion:" }}}{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 filament and \+
potential V0 at the cylinder can be imposed, but the thickness of 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 23 "Vp:=simplify(rhs(sol));" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 88 "The solution is in terms of the exponential integral. \+
Maple can evaluate it numerically:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "evalf(subs(l=1,d=100,V0=100,r=2,Vp));" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 94 "The above result shows that we need to re
move the imaginary part in order to graph the result:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "plot([subs(l=1,d=100,V0=100,Re(Vp))
,10*r*rho],r=1..100,color=[red,blue]);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 132 "The result shows what happens due to the presence of the
 electron cloud: the potential does not interpolate in a simple way be
tween " }{TEXT 268 1 "V" }{TEXT -1 22 "=0 at the cathode and " }{TEXT 
266 1 "V" }{TEXT -1 1 "=" }{TEXT 267 1 "V" }{TEXT -1 168 "0 at the ano
de. The electrostatic repulsion has the effect that only the electrons
 in the tail region of the density profile experience an attraction to
wards the anode." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 24 "Note that we have shown " }{TEXT 19 5 "r*rho" }{TEXT -1 
166 " (multiplied by an arbitrary scale factor) rather than the densit
y itself. This quantity provides a measure of how much charge is found
 in a circular shell at radius " }{TEXT 276 2 "r " }{TEXT -1 8 "(betwe
en" }{TEXT 278 3 " r " }{TEXT -1 4 "and " }{TEXT 279 4 "r+dr" }{TEXT 
-1 1 ")" }{TEXT 277 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "The r
adial component of the electric field is calculated as follows:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "E:=-diff(Vp,r);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 49 "We graph the radial force acting on an el
ectron (" }{TEXT 271 1 "F" }{TEXT -1 1 "=" }{TEXT 270 2 "qE" }{TEXT 
-1 7 ", with " }{TEXT 269 1 "q" }{TEXT -1 51 "=-1) using as a distance
 scale the filament radius " }{TEXT 258 1 "l" }{TEXT -1 50 ", and assu
me that the anode cylinder has a radius " }{TEXT 259 1 "d" }{TEXT -1 
34 " of 100 times the filament radius." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "plot(subs(l=1,d=100,V0=100,-E),r=1..100);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 358 "The force due to the electron-electron r
epulsion pushes the nearby electrons back into the filament (a stable \+
density can be maintained if the heating mechanism ejects as many elec
trons as are re-absorbed). The electron-electron repulsion potential o
vershadows the electric field due to the applied external voltage for \+
distances out to 10-15 filament radii." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT 273 11 "Exercise 1:" }}{PARA 0 "" 0 "" 
{TEXT -1 81 "Explore what happens when one reduces (increases) the ele
ctron charge by varying " }{TEXT 275 3 "(i)" }{TEXT -1 12 " the factor
 " }{TEXT 19 4 "rho0" }{TEXT -1 2 "; " }{TEXT 274 4 "(ii)" }{TEXT -1 
40 " the factor controlling the fall-off of " }{TEXT 19 6 "rho(r)" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 90 "What are the implication
s for the current-voltage characteristic of a diode (vacuum tube)?" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 272 11 "Exercise
 2:" }}{PARA 0 "" 0 "" {TEXT -1 80 "Repeat the calculation for a Gauss
ian shape of the radial electron distribution." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 156 "Other examples of charge
 densities for which the potential can be readily calculated in terms \+
of elementary functions are bell-shaped functions of the type " }
{TEXT 19 13 "1/(a^2+r^2)^n" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 16 }{VIEWOPTS 1 1 0 1 1 1803 1 
1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }