{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 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 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 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 0 1 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 "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 }