Potential and Field in a Shifted
Semi-Infinite Plane Condensor

St. Rosegger, G. Schweitzer, B. Schnizer

Institut fuer Theoretische Physik
Technische Universitaet Graz
A-8010 Graz, Austria, Europe
Email: schnizer@itp.tu-graz.ac.at

2004-12-03

1.  Description

The potential and field distributions for the configuration shown below, a twodimensional semi-infinite plane condensor of width  D=huD, whose upper electrode extends from minus infinity to b=hub,  are computed by conformal maps. The upper electrode is at  Voltage V=huV ;  the lower electrode, which coincides with the x-axis, is grounded. 
The potential and the field are plotted, as is the field on the electrodes. The values of the potential and field are computed at a point which may be selected by the user. The relative current  I/qv  induced on the upper left electrode by a charge q moving with constant velocity  v  on various trajectories parallel to the y-axis is shown as function of y.  The user may also chose the x-coordinate of such a trajectory.

By the map  W =  f(z)  the  upper half of the z = x + i y  -plane comprising the above configuration is mapped to 
a closed infinite plane condensor of width   π   in the   W = U + i V -plane; whose upper electrode  V = π   is at 
potential  huV, while the lower one,  V = 0,  is grounded.   This map consists of three conformal maps:   
		3.		z = (Subscript[f, 3])(t')  =  (D/π)[1+logt'+t' ]
maps the half space y  ≥  0  to the half space s' ≥ 0 of the complex  t' = r' + i s' -plane such that the lines  y = 0  and  
y = D  become the axis  s' = 0. The function   
		1.		W = (Subscript[f, 1]^(-1))(t)  =  ln t   
maps the closed condensor to the  t = r + i s -plane such that the upper electrode V = D becomes the ray  s = 0,  r <  0; the lower electrode V = 0 the ray    s = 0,  r  >  0. The second map
		2.		t' = (Subscript[f, 2])(t)= t
is the identity transformationn. Combining the maps into one gives:
  			z = (Subscript[f, 3])(t')  =  (D/π)[1 + W + (e^W)],  
It is not possible to express the inverse function  (Subscript[f, 3]^(-1))(z) by a single-valued closed expression; as aconsequence,
the same applies to (f^(-1))(z) . So the inverse (f^(-1))(z) must be found numerically. 
Mathematica provides an analytic expression for (f^(-1))(z) comprising the function  ProductLog[]. It gives the 
principal solution of a multi-valued function. In our case this is not approprate for all arguments needed.

The shift  b  has been introduced so that this configuration may be matched with other ones, in particular with the
plane condensor whose upper electrode contains a gap treated in the notebook  SplitCondensor.nb. Such a match
is done in the notebook Comparisons.nb

In the present notebook almost all variables start with the letters  hu...  . An analogous convention is also adopted in the 
other notebooks on conformal maps applied to Resistive Plate Chamber (RPC) models. So several notebooks may be used simultaneously in the same kernel without risk of interference.

The main parameters are the Voltage  huV  applied to the upper electrode, the width of the condensor   D = hugd  and  
the shift  b = hub. Default values are assigned to these as to all other variables. So the user may evaluate the notebook 
just after opening it.  The cells where values are assigned to these parameters have a background in cyan so that the 
user may locate them easily; he may assign other values to these variables and evaluate the notebook again.
The user may play with some other parameters so that he gets more satisfactory pictures. Such recommendations are also
given in cells with cyan background.

Please report errors and difficulties per email to the adress given above in the affiliation.

2.  Input

Needs["Graphics`ComplexMap`"]

Off[General::spell1]

In this notebook many variables have similar names. The ensuing warnings are switched off above in the second line.
When working on the program these  warning may be useful; switch them on by the corresponding command On[...];
this is also implemented in the last line of this notebook.

2.1  Input Data

Clear["hu*"]

    
huV   = 1;     ;
hugd = 
          10 ;      ;
hub = -4;           ;
\(\(hutcb =  .011	 (*\ 
        Relative\ thickness\ of\ lines\ representing\ electrodes\ *) ;\)\)

\(Print[
      SequenceForm[
"  Width of condensor:                                  D = ", \ hugd, \ " mm."]]\)
\(Print[
      SequenceForm[
"  position of the right end of the upper electrode :   b = ", \ \ hub, \ " mm."]]\)

2.2 Functions for the conformal map

2.2.1  The function  z  =  f(W)

The three conformal maps (Subscript[f, 1]), (Subscript[f, 2]), (Subscript[f, 3]) and the resulting map f  as described above are implemented below. 
Notations are:  W = huWw,  t = hutt, t' = hutp.

huf1[huWw_] = Exp[huWw];
huf2[hutt_] = hutt;
huf3[hutp_] = 
        hugd/π (1 + Log[hutp] + hutp);
huf[scWw_]  =  huf3[huf2[huf1[scWw]]] + hub;

2.2.2 The inverse function  W = (f^(-1))(z)

It is not possible to express the inverse function (Subscript[f, 3]^(-1))(z)  as a closed analytic expression. So it must be computed 
numerically; this is done by a Newton iteration implemented in the command  FindRoot. This in turn needs an 
appropriate starting value depending on  z = huzz. Mathematica permits one to implement this procedure in such
 a way that the inverse function  W=(f^(-1))(z) = hui may be specified as a function of z.

hui[huzz_] :=  
      hupqq   /.  
        FindRoot[(huf[hupqq] ) - huzz, {hupqq, 
            1.2 Sign[huyp - hugd] + I π/2}]

2.2.3  The complex electric field

The complex electric field   (Subscript["E", "x"])"\:f39f+ i "(Subscript["E", "y"])  at a point   z = x + i y   is given by the function  hufe, which depends on 
W = huWw. The value of W belonging to the given value  z  is found by  hui.
Note: Here MKSA units are used; but D = hugd is given in 
mm, so the electric field given by that formula is in  kV/m.

hufe[
          huWw_] = -I (huV /hugd)*1/
            Conjugate[1 + Exp[huWw]];

3.  Section with Field Graphics

3.1  Variables needed in all pictures

Most evaluations for finding values or graphs are done by using the values of  W = U + i V  belonging to a given z. For 
curves appropriate ranges of U  must be found; this is accomplished by a function hufur depending on D = hugd and 
b= hub.  The range of  V  is almost always  0 ≤ V ≤  π.  In some cases plot ranges must be specified; this is done by 
hufyr.  These two functions have been found empirically.

hufur[hugd_]  =  
        2.5  +  
          UnitStep[hugd/5  -  2]  (0.5  hugd/5  -  1);

hufyr[hugd_]  =  
        3  +  UnitStep[
              3  -  hugd/5 ]  (1.5  - 0.5 hugd/5 );

huneq = 
        20  ;

huneq  specifies the number of equipotential and field lines;  this value in turn specifies the step widths used for the
variables  U ( = hudu) and  V = π/huneq used in hucm. The range of the variable U is bounded by hulu = - 6 and 
by huru. This value and the function hufur have been determined by trial and error. This applies also to the function
hufyr rendering a value hufy needed for the plot range in picture hup23. The horizontal plot range in all pictures is 
determined by huplr.

huplr  =   (hugd  + 
            Abs[ 2 hub]);
huims  =  500;
hulu  =  - 6;   ;
huru  =  hufur[hugd] ;
hudu  =  Abs[huru  - hulu]/huneq;
hufy  =  hufyr[hugd];

3.2 Field graphics

The user may want to improve the pictures drawn below. For this, he may try to obtain better values for 
hulu, huru, hudu, hufy. If he wants to do so it is recommended to remove the semicolons of the last block
at the end of the preceeding subsection, so that the values are printed. Then he may try a slight change of
these values and see what comes out. It is commendable to insert these new values below in the
commands for  hup2, hup3, hup23, hucm, hup1. It is extremely commendable to save this  trial notebook
under a new name beforehand so that the notebook with the orginal specifications remains intact!

hup2 (hup3) gives the electric field at the inside and outside of the upper electrode.
These are then combined to gives figure hup23.  hucm plots the equipotential and electric field lines. 
hup1 shows  the electric field at the bottom electrode. In all four hup pictures the unit of the electric
field strength is  kV/m  since in the definition of the function  hufe in 2.2.4  D = hugd is used, whose value
is defined in mm in  1.1, while we use  MKSA units.

hup2  =  
        ParametricPlot[{huf[ww  +  I π] // Re, 
            hufe[ww +  I π] // Im}, {ww, -10, 
            0},   DisplayFunction -> Identity];

hup3  =  
        ParametricPlot[{huf[ww  +  I π] // Re, 
            hufe[ww +  I π] // Im}, 
{ww, 0, 
            10},   DisplayFunction -> Identity];

\(\(\(
\)\(Print["Electric field distribution on upper electrode"]

    \(hup23\  = \ 
        Show[hup2, 
          hup3, 
DisplayFunction -> \ $DisplayFunction, 
\ 
          PlotRange\  -> \ {huplr\ {\(-1\), 1}, \ 
              5  huV/hugd\ {\(-1\), 1}}, \ ImageSize\  -> \ huims, 
          Epilog -> {\ Hue[0], 
              AbsoluteThickness[
                2], 
Line[{{hub, \(-hub\)}, {hub, 
                    hub}}]}, 
AxesLabel -> {"x [mm]", None}, \ 
          PlotLabel\  -> \ 
SequenceForm[
              Subscript["         ", "y"], "    
          ", Subscript["E", "y"], \ "  [kV/m]"]];\)


    Print["Equipotential and electric field lines"]

    \(hum\  = \ 
        CartesianMap[
          huf, {hulu, huru, hudu}, {$MachineEpsilon, 
            Pi \((1 - $MachineEpsilon)\), π/huneq}, 
\ 
          ImageSize\  -> \ 
            huims\ , 
AxesLabel -> {"x [mm]", "y [mm]"}, PlotRange\  -> {huplr\ {\(-1\), 1}, {\(-0.2\), 2  hugd\ }}, 
          Epilog -> {Thickness[hutcb], Hue[0], 
              Line[{{\(-\ hugd\)\  - Abs[\ 2  hub], hugd}, {\ hub, 
                    hugd}}], 
Text[
                SequenceForm["Φ = ", 
                  huV, " Volt"], {\(-\ 0.6\)\ huplr, 1.2  hugd\ }], 
              GrayLevel[0], 
              Line[{{hugd\  + Abs[\ 2  hub], 
                    0}, {\(-hugd\)\  - \ Abs[2  hub], 
                    0}}]}];\)


    Print["Electric field distribution on lower electrode"]

    \(hup1\  = \ 
        ParametricPlot[{huf[ww] // Re, hufe[ww] // Im}, {ww, \(-10\), 10}, 

          ImageSize\  -> \ huims, 
          Epilog -> {\ Hue[0], 
              AbsoluteThickness[
                2], 
Line[{{hub, \(-hub\)}, {hub, 
                    hub}}]}, \ 
          PlotRange\  -> \ {huplr {\(-1\), 1}, 
              huV/hugd\ {\(-1\), 
                  0}}, 
AxesLabel -> {"x [mm]", 
              SequenceForm[
                Subscript["E", "y"], "  [kV/m]"]}];\)\)\)\)

4.  Field and Potential at a Selected Point

Below the coordinates  (huxp, huyp )  of a point  zp  may be chosen by the user. The values of the 
potential and of the electric field at this point  are computed and printed. 
The equipotential line and the field line passing through that point are shown.

huWp is the point  Wp  corresponding to  zp   computed by the inverse mapping function  
W = (f^(-1))(z) = hui.  This value is used to compute the complex potential in the W-condensor,
which is equal to the potential at zp in the real condensor. Wp is also used to compute the
complex electric field by the formula given in  2.2.3.  hupc (huec) displays the equipotential 
(electric field) line passing through  zp.  The factor  1000  in the second Print command 
(giving the electric field strength)  is  needed since in the definition of the function  hufe in 2.2.3  
D = hugd is used, whose value is defined  in mm in  1.1, while we use  MKSA units.

huxp  =  -4 ; huyp  =  
        5.3 ;

huWp = hui[huxp  + I huyp];
Print[
        SequenceForm["At point (huxp,huyp) = (", huxp, ", ", 
          huyp, ") mm : ", "Potential = ", 
          N[huV/π Im[huWp]] // Chop, " V;"]];
Print[
        SequenceForm["             (", 
          Subscript["E", "x"], ",", 
          Subscript["E", "y"], ") = (", 
          Re[1000.  hufe[huWp]], ", ", 
          Im[1000.  hufe[huWp]],  ") V/m "]];

    
hupc = 
        ParametricPlot[
          Evaluate[{Re[huf[I Im[huWp] + s]], Im[huf[I Im[huWp] + s]]}], {s, 
            hulu, huru}, PlotStyle -> Dashing[{ .01}], 
          DisplayFunction -> Identity];
huec = 
        ParametricPlot[
          Evaluate[{Re[huf[Re[huWp] + I s]], Im[huf[Re[huWp] + I s]]}], {s, 
            0, Pi}, DisplayFunction -> Identity];
hupoi = 
        Show[hupc, huec, AspectRatio -> Automatic, 
          DisplayFunction -> $DisplayFunction, ImageSize -> 600, 
          TextStyle -> {FontSize -> 16}, 
          AxesLabel -> {"x [mm]", "y [mm]"}, 
          PlotRange -> {Max[Abs[huxp], huplr] {-1, 1}, {- .2, 
                2  hugd }}, 
          Epilog -> {Thickness[hutcb], Hue[0], 
              Line[{{- hugd  - Abs[ 2  hub], hugd}, { hub, 
                    hugd}}], 
Text[
                SequenceForm["Φ = ", 
                  huV, " Volt"], {- 0.6 huplr, 1.2  hugd }], 
              GrayLevel[0], 
              Line[{{hugd  + Abs[ 2  hub], 
                    0}, {-hugd  -  Abs[2  hub], 0}}], 
              AbsolutePointSize[4], Point[{huxp, huyp}]}];

5.   Weighting Field = Current Induced on the Electrode

The current I induced by a charge   q  moving with velocity  (Overscript[v, →])   on an electrode is determined by the weighting field 
 (Overscript[E, →]) generated by the Voltage V applied to that electrode according to the formula  I = - q ((Overscript[v, →]) ·(Overscript[E, →]))/V.  Here the charge
 moves in the vertical direction so that  I = - q v (Subscript[E, y]) /V. The list  huxtl contains the horizontal coordinate (measured in 
 units of  g/2 = hugh) of various vertical trajectories. For these the quantity  I/qv  is displayed as a function of  
 y = huyt  in the pictures.

The user may select the horizontal coordinates  x  of a set of vertical trajectories. These are given in the list
huxtl  in units of  b = hub.

huxtl  =  
        hub  {-2, -1.5, -1, - 0.7, -0.3, -0.1, 
            0.1,  .3,  .7, 1.01, 1.5, 2};

\(\(hupli\  = \ 
        Table[Plot[\(-Im[hufe[hui[huxtl[\([k]\)]\  + \ I\ huyt]]]\)/
              huV, {huyt, 0, 
              hugd}, \ 
PlotStyle\  -> \ {Hue[
                  0.1\ k], \ 
                Thickness[
                  0.005]}, 
PlotLabel\  -> \ \ SequenceForm["x = ", 
                huxtl[\([k]\)], " mm"], 
AxesLabel\  -> \ \ {"y [mm]", "I/qv [1/mm]"}, 
            DisplayFunction -> Identity], {k, 
            Length[huxtl]}];\)\)

In the first picture below the position of the trajectories (various colors) in the condensor are shown; in the  next
one the corresponding current ratios (same colors) are shown as functions of the coordinate y.

hutli  =  
        Table[{Hue[0.1 k], 
              Line[{{huxtl[[k]], 0}, {huxtl[[k]], hugd}}]}, {k, 
              Length[huxtl]}] // Flatten;
huwxl  =  
        Show[Graphics[Prepend[hutli, Thickness[0.005]]], 
          Graphics[{Thickness[hutcb], 
              Line[2 {{hulu, 0}, {-hulu, 0}}], 
Hue[
                0],  Line[{{2  hulu, hugd}, {hub, hugd}}]}], 
          Axes  ->  True,  ImageSize  ->  500];

huwcl  =  
        Show[hupli,  PlotLabel  ->  None,  
          ImageSize  ->  500, 
          DisplayFunction -> $DisplayFunction];

6.   Current Induced on the Electrode for a Selected Trajectory

The user may select the horizontal coordinate  x =huxs  of a vertical trajectory. The first figure shows the position of this 
trajectory in the real condensor. The next one gives the current ratio induced in the upper electrode as a function of 
the vertical position  y = huyt  of the point charge, using W = (f^(-1))(z) = hui  in the function hufe for the complex electric 
field introduced in  2.2.3.

huxs  =   -4;

Show[Graphics[{Thickness[0.005],  Line[{{huxs, 0}, {huxs, hugd}}]}], 
        Graphics[{Thickness[hutcb], 
            Line[2 {{hulu, 0}, {-hulu, 0}}], 
Hue[0],  
            Line[{{2  hulu, hugd}, {hub, 
                  hugd}}]}], 
     Axes  ->  
          True,  ImageSize  ->  500,  
        TextStyle  ->  {FontSize  ->  
              14}];
\(\(scpls\  = \ 
        Plot[\(-\ Im[hufe[hui[huxs\  + \ I\ huyt]]]\)/huV, {huyt, 0, 
            hugd}, \ 
PlotStyle\  -> \ {Thickness[
                0.005]}, \ ImageSize\  -> \ 500, \ 
          TextStyle\  -> \ {FontSize\  -> \ 
                14}, 
PlotLabel\  -> \ \ SequenceForm["x = ", 
              huxs, " mm"], 
AxesLabel\  -> \ \ {"y [mm]", "ID/qv"}];\)\)

7.   Distribution of Electric Field

The distribution of the absolute value of the electric field over that part of the configuration, where the variation 
of the electromagnetic field is largest, is shown in the picture below. The real and imaginary parts of the function  
huf[W] = huf[u + I v] give  the position coordinates  x  and  y;  hufe[u + I v]  the complex electric field 
(s. subsect. 2.2.3). The operator Abs[] applied to this function in the command for the picture huabse  produces 
 |E|;  the operator Im[] in huey leads to a plot for (Subscript[E, y]).  This is the weighting field in a case where the influencing 
 charge is moving along he y-direction as is the case in RPC's.

\(\(huabse = 
        ParametricPlot3D[
          Evaluate[{huf[u\  + \ I\ v] // Re, huf[u\  + \ I\ v] // Im, 
              Abs[1000\ hufe[u\  + \ I\ v]]}], {u, hulu, 
            huru}, \ {v, $MachineEpsilon, Pi \((1 - $MachineEpsilon)\)}, \ 
          BoxRatios\  -> \ {1, 1,  .75}, ImageSize\  -> \ 500, \ 
          TextStyle\  -> \ {FontSize\  -> \ 
                14}, 
PlotPoints\  -> \ 50, 
          AxesLabel\  -> {"x [mm]", "y [mm]", "|E| [V/m]"}];\)\)

The two pointed towers arise from the infinite field values at the edges of the gap.

huey = 
        ParametricPlot3D[
          Evaluate[{huf[u  +  I v] // Re, huf[u  +  I v] // Im, 
              Im[1000 hufe[u  +  I v]]}], {u, hulu, 
            huru},  {v, $MachineEpsilon, Pi (1 - $MachineEpsilon)},  
          BoxRatios  ->  {1, 1,  .75}, ImageSize  ->  500,  
          TextStyle  ->  {FontSize  ->  
                14}, 
PlotPoints  ->  50, 
          AxesLabel  -> {"x [mm]", "y [mm]", 
              SequenceForm[
                Subscript["E", "y"], " [V/m]"]}];

The field component  (Subscript[E, y]) changes sign across the electrodes; this is visible at the upper electrode at left. 
There is no field at all below the lower elctrode. The edge leads to an infinite jump.

On[General::spell1]

10.  References

0.1 Papers, Books and Websites

0.             http://www.itp.tu-graz.ac.at/~schnizer/Theory_of_ Chambers/Plane_Geometries/

1.             Th. Heubrandtner,B. Schnizer,  G. Schweitzer, 
	Simple Models for RPC Weighting Fields and Potentials.
    	Nucl. Instrum. Methods Phys. Res., A : 535 (2004) no.1-2, pp.454-457.
	SchzP-82.pdf

2.              St. Rossegger,B. Schnizer,  G. Schweitzer,
	Comparison of simple models for RPC weighting fields and potentials.
             (in preparation)	

3.             G. Wendt, Static Fields and Stationary Currents (in German).	
             Encyclopedia of Physics (Ed. S. Fluegge), vol.16 , in particular pp.71 - 79 .
              Springer, Berlin 1958.

4.            J.Van Bladel,Electromagnetic Fields,
	McGraw-HillBook Comp.,New York 1964.
	pp. 136 - 144.

5.              G. Schweitzer, Quickly available graphics of static electromagnetic field distributions 
	given by conformal maps using Mathematica. 
             Graz,Techn.Univ.,Dipl.-Thesis.,1994
	Revised   1997.  http://www.itp.tu-graz.ac.at/~schnizer/ Conformal_Maps.html

6.              H. Kober, Dictionary of Conformal Representations.
              Dover, 1957. p.117