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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$2) = 4*Pi*rho(x,y,z)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 64 "The problem reduces to one dimension in cylindric
al coordinates." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart;
 with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected \+
names norm and trace have been redefined and unprotected\n" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 153 "Maple can remind us of the Laplacian in \+
various orthogonal coordinate systems (without explicit computations).
 We are interested in the radial part only:" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 61 "v1 := [r, theta, z]:\nlaplacian(f(r), v1, coords=c
ylindrical);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&-%%diffG6$-%\"fG6#
%\"rGF+\"\"\"*&F+F,-F&6$F(-%\"$G6$F+\"\"#F,F,F,F+!\"\"" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 557 "We need to make some assumptions about t
he radial distribution of the charge cloud. A heated filament results \+
in a constant flow of charge in and out of the filament to avoid a net
 charge on it. If an electric field is applied between the anode (oute
r cylinder) and cathode (filament), then one has an electron charge de
nsity that extends between both cylinders. We will not attempt to mode
l a self-consistent situation, but rather calculate the electrostatic \+
repulsion originating from an assumed charge cloud. We simply assume a
n exponential distribution:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
8 "rho0:=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%rho0G\"\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "rho:=rho0*exp(-r/10);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$rhoG-%$expG6#,$%\"rG#!\"\"\"#5" }}}
{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));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"
+q%\\1D'!\"(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 443 "This is the tota
l amount of charge. It is controlled by the heating current of the fil
ament. At higher temperatures (more heating current) a larger number o
f electrons will overcome the work function. The electrons are assumed
 to be distributed in an exponential fashion (this means that in princ
iple some of them reach the anode at a distance of 100 filament radii.
 (This results in the diode acting as a battery when the filament is h
eated)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "evalf(2*Pi*int(r
ho*r,r=1..infinity));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+(G(y`i!\"
(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "We formulate Poisson's equat
ion:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "DE:=diff(V(r),r$2)+
diff(V(r),r)/r=rho;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DEG/,&-%%dif
fG6$-%\"VG6#%\"rG-%\"$G6$F-\"\"#\"\"\"*&-F(6$F*F-F2F-!\"\"F2-%$expG6#,
$F-#F6\"#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 215 "This differential \+
equation has a solution that combines the result from the density-free
 case (which is incorporated through the boundary condition which spec
ifies the potential) and the electron-electron repulsion:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dsolve(DE,V(r));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/-%\"VG6#%\"rG,**&%$_C1G\"\"\"-%#lnG6#,$F'#!\"\"\"#5F
+F+*&\"$+\"F+-%$expGF.F+F+*&F4F+-%#EiG6$F+,$F'#F+F2F+F+%$_C2GF+" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 199 "The boundary conditions of zero p
otential at the filament and potential V0 at the cylinder can be impos
ed, 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;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#BCG6$/-%\"VG6#%\"lG\"\"!/-F(6#%\"dG
%#V0G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "sol:=dsolve(\{DE,B
C\},V(r));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$solG/-%\"VG6#%\"rG,**
&*&,,-%$expG6#,$%\"dG#!\"\"\"#5!$+\"*&\"$+\"\"\"\"-%#EiG6$F9,$F2#F9F5F
9F4*&F8F9-F/6#,$%\"lGF3F9F9*&F8F9-F;6$F9,$FCF>F9F9%#V0GF9F9-%#lnG6#,$F
)F3F9F9,&-FJF0F4-FJFAF9F4F4*&F8F9-F/FKF9F9*&F8F9-F;6$F9,$F)F>F9F9*&,,*
&FOF9F.F9F6*(F8F9FOF9F:F9F4*&FOF9FHF9F9*(F8F9F@F9FNF9F9*(F8F9FEF9FNF9F
9F9FMF4F9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Vp:=simplify(r
hs(sol));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#VpG*&,>*&-%$expG6#,$%
\"dG#!\"\"\"#5\"\"\"-%#lnG6#,$%\"rGF.F0\"$+\"*(F6F0-%#EiG6$F0,$F,#F0F/
F0F1F0F0*(F6F0-F)6#,$%\"lGF-F0F1F0F.*(F6F0-F96$F0,$FAF<F0F1F0F.*&%#V0G
F0F1F0F.*(F6F0-F)6#,$F5F-F0-F26#,$F,F.F0F.*(F6F0FIF0-F26#,$FAF.F0F0*(F
6F0-F96$F0,$F5F<F0FLF0F.*(F6F0FTF0FPF0F0*(F6F0F(F0FPF0F.*(F6F0F8F0FPF0
F.*&FGF0FPF0F0*(F6F0F>F0FLF0F0*(F6F0FCF0FLF0F0F0,&FLF.FPF0F." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "The solution is in terms of the ex
ponential 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)
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!+;a.`7!\")$\"+YM)GI\"!#;" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "The above result shows that we ne
ed to remove the imaginary part in order to graph the result:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "plot([subs(l=1,d=100,V0=100,
Re(Vp)),10*r*rho],r=1..100,color=[red,blue]);" }}{PARA 13 "" 1 "" 
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g]$)F`_l$\"3'epGdqnG(>Fg]m7$$\"3#)**\\iO9dg&)F`_l$\"3#R(yu^NVR;Fg]m7$$
\"3e+vVTO!)o()F`_l$\"3+?KT(RKOO\"Fg]m7$$\"3?++]6u9g*)F`_l$\"31o;\"\\sF
2:\"Fg]m7$$\"3Q+]7l*[%z\"*F`_l$\"3A6b?'QSuY*!#>7$$\"3#*******))[fv$*F`
_l$\"314V$3&[[ZzFiam7$$\"3)**\\PWNFZe*F`_l$\"3ax%R)H)>:f'Fiam7$$\"3!3]
i&3Q*[y*F`_l$\"3AIUP<nY3bFiam7$F\\^l$\"3Z%[[i(H**RXFiam-F_^l6&Fa^lF+F+
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10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 132 "The result shows what happens d
ue to the presence of the electron cloud: the potential does not inter
polate in a simple way between " }{TEXT 268 1 "V" }{TEXT -1 22 "=0 at \+
the cathode and " }{TEXT 266 1 "V" }{TEXT -1 1 "=" }{TEXT 267 1 "V" }
{TEXT -1 168 "0 at the anode. The electrostatic repulsion has the effe
ct that only the electrons in the tail region of the density profile e
xperience an attraction towards the anode." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Note that we have shown " }
{TEXT 19 5 "r*rho" }{TEXT -1 166 " (multiplied by an arbitrary scale f
actor) rather than the density itself. This quantity provides a measur
e of how much charge is found in a circular shell at radius " }{TEXT 
276 2 "r " }{TEXT -1 8 "(between" }{TEXT 278 3 " r " }{TEXT -1 4 "and \+
" }{TEXT 279 4 "r+dr" }{TEXT -1 1 ")" }{TEXT 277 1 "." }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 68 "The radial component of the electric fiel
d is calculated as follows:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
15 "E:=-diff(Vp,r);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"EG,$*&,4*&-
%$expG6#,$%\"dG#!\"\"\"#5\"\"\"%\"rGF/\"$+\"*&*&F3F1-%#EiG6$F1,$F-#F1F
0F1F1F2F/F1*&*&F3F1-F*6#,$%\"lGF.F1F1F2F/F/*&*&F3F1-F76$F1,$F@F:F1F1F2
F/F/*&%#V0GF1F2F/F/*(F0F1-F*6#,$F2F.F1-%#lnG6#,$F-F/F1F1*(F0F1FIF1-FM6
#,$F@F/F1F/*&*(F3F1FIF1FLF1F1F2F/F1*&*(F3F1FIF1FQF1F1F2F/F/F1,&FLF/FQF
1F/F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "We graph the radial forc
e acting on an electron (" }{TEXT 271 1 "F" }{TEXT -1 1 "=" }{TEXT 
270 2 "qE" }{TEXT -1 7 ", with " }{TEXT 269 1 "q" }{TEXT -1 51 "=-1) u
sing as a distance scale the filament radius " }{TEXT 258 1 "l" }
{TEXT -1 50 ", and assume that the anode cylinder has a radius " }
{TEXT 259 1 "d" }{TEXT -1 34 " of 100 times the filament radius." }}}
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358 "The force due to the electron-electron repulsion pushes the nearb
y electrons back into the filament (a stable density can be maintained
 if the heating mechanism ejects as many electrons as are re-absorbed)
. The electron-electron repulsion potential overshadows the electric f
ield due to the applied external voltage for distances out to 10-15 fi
lament 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 electron 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 controll
ing the fall-off of " }{TEXT 19 6 "rho(r)" }{TEXT -1 1 "." }}{PARA 0 "
" 0 "" {TEXT -1 90 "What are the implications 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 Gaussian 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 po
tential 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 "" }}}}
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