Potential and Field in a Plane 
   Condensor with One Split Electrode

St. Rossegger, B. Schnizer, G. Schweitzer

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 infinite plane condensor 
of width  D, whose upper electrode has a gap of width  g,  are computed by conformal maps. The upper left electrode 
is at  Voltage   scV ;  the other electrodes are 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  scV, while the lower one,  V = 0,  is grounded.   This map consists of three conformal maps:   
			z = (Subscript[f, 3])(t')  =  (D/π)[ ln t' + ((((1 - p))^2)/p)((1/2)- (1/(1+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   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 
rational function  t' = (Subscript[f, 2])(t)= t/(1 - t) maps the t- on the t'-plane such that the s- becomes the s'-axis.  A longer write-up 
giving more details of these transformations is contained in the notebook  SplitCondLW.nb.

In the notebook just quoted somewhat different notations are used. In the present notebook almost all variables start with 
the letters  sc...  . 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. However, this convention is not observed in the long write-up notebooks whose names end with the 
letters LW.

The main parameters are the Voltage  scV  applied to the upper left electrode, the width of the condensor   D = scgd  and  
the half of the gap  g/2 = scgh. 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; by tests we are quite sure that the program will run without difficulty as long as the conditions   0.5 ≤  D/g ≤ 5   
are observed.
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

\(\(\(\(\(Clear["sc*"]\)\(
\)
    \)

    \(scV\ \  = \ 1;\)\ \  (*\ 
      Potential\ on\ the\ left\ upper\ electrode\ in\ Volt, \ \ 
\ \ \ \ \ \ \ \ \ \ \ elsewhere\ on\ all\ electrodes\ Potential\  = 0\ *) 

    \(\(scgd = 
        13\)\(\ \)\); \ \ \ \  (*\ \(=\)\(\ \)\(D\  = \ 
          width\ of\ condensor\ in\ mm\)\ *) 
\(\(scgh = 
        5\)\(\ \)\); \ \ \ \  (*\ \(=\)\(\ \)\(g/2\  = \ 
          Half\ the\ width\ of\ gap\ in\ upper\ electrode\)\ *) 

\(\(\(sctcb =  .011\)\(\ \)\) \); \)\(	\)\( (*\ 
      Relative\ thickness\ of\ lines\ representing\ electrodes\ *) \)\)\)

\(Print[
      SequenceForm[
"  Width of condensor:               D = ", \ 
        scgd, \ " mm."]]\)
\(Print[
      SequenceForm[
"  Width of gap in upper electrode : g = ", \ 
        2\ scgh, \ " mm."]]\)

2.2 Functions for the conformal map

2.2.1   Finding the value of the parameter p belonging to the dimensions selected in 2.1

The conformal map needs one parameter  p = scpa, whose value depends on the ratio of the two geometric parameters 
D and g;  it is the root of a transcendental equation and is contained in the rule named scsop.

scsop  =   FindRoot[
          Log[scpa]  +  (scpa^2  -  1)/scpa/
              2  -  π scgh/scgd, {scpa, 2}];

2.2.2  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 = scWw,  t = sctt, t' = sctp,  p = scpa.

scf1[scWw_]  =  Exp[scWw] ;
scf2[sctt_]  =  
        sctt/(1  -  sctt) ;
scf3[sctp_]  =  
        scgd/π (Log[
                sctp]  +  (1  -  scpa)^2/
                  scpa (1/2   -  
                    1/(1  +  
                          sctp)))  ;
scf[scWw_]  =  scf3[scf2[scf1[scWw]]]  /.  scsop;

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

The inverse functions to  (Subscript[f, 1])and (Subscript[f, 2]) may be expressed as elementary functions:

sci1[sctt_] =  Log[sctt] ;
sci2[sctp_]  =  sctp/(1  +  sctp) ;

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 = sccz, which is prepared in scstv. Mathematica permits one to implement 
this procedure in such a way that the inverse function  (Subscript[f, 3]^(-1))(z) = sci3 may be specified as a function of z:

\(\(scstv[sccz_]\  := \ 
        Which[Re[sccz] ≥ \ 0\  && \ 
            Im[sccz]\  ≥ \ 
              scgd, \(-\ 0.5\) \((scpa\  + \ 1)\)\  + \ 
            0.2  I, \ 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Re[sccz] < \ 0\  && \ 
            Im[sccz]\  ≥ \ 
              scgd, \(-\ 0.5\) \((1/scpa\  + \ 1)\)\  + \ 
            0.2\ I, 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Re[sccz] ≤ \ 0\  && \ 
            Im[sccz]\  < \ scgd, 
          0.2  I, 
			\ \ \ \ \ \ \ \ Re[sccz] > \ 
              0\  && \ Im[sccz]\  < \ scgd, 1.2  I];\)\)

\(\(scα\  = \((\(-1\) + scpa)\)\^2\/scpa\  /. \ 
          scsop\ ;\)\ \ 
\)
scia3[
          zzz_] = (2 (E^(π zzz/scgd + sc(α/2))) + 8 (E^(2 π zzz/scgd + scα)) scα + (E^(4 π zzz/scgd + 2 scα)) sc(α^2) (-2 + 
                    3 scα) + (E^(3 π zzz/scgd + ((3 scα)/2))) scα (-2 + 11 scα))/(2 + 
              2  (E^(π zzz/scgd + sc(α/2))) scα (5 + (E^(π zzz/scgd + sc(α/2))) (-2 + 
                          scα (9 + (E^(2 π zzz/scgd + scα)) (-2 + 
                                      scα) scα + (E^(π zzz/scgd + sc(α/2))) (-4 + 
                                      6 scα)))));

scit3[sczz_] :=  
      scpqq   /.  
        FindRoot[(scf3[scpqq]  /.  scsop) -  sczz, {scpqq, 
            scstv[sczz]  /.  scsop}]

However. it turns out that this method fails for  values of sczz located within the condensor and having 
a large negative real part. An approximate analytic formula for  (Subscript[f, 3]^(-1))(z)(viz. scia3[z]) has been derived;
the pertinent calculations are done in the long write-up notebook SplitCondLW.nb. So the general 
definition of (Subscript[f, 3]^(-1))(z) used below checks for failure of FindRoot[]; in case of convergence failure the
value of the approximate function is inserted if sczz is located in the domain of validity of the 
approximation scia3[z].

sci3[sczz_]  =  
        Which[0  ≤   Im[sczz]  ≤   scgd  &&  
            Re[sczz]  ≤   (2 scgh/
                    scgd), 
Check[scit3[sczz], 
            scia3[sczz]],  True,  scit3[sczz]];

This all is used to define the total inverse mapping function   W = (f^(-1))(z) = sci[].

sci[sczz_]  :=  sci1[sci2[sci3[sczz]]]  /.  scsop;

2.2.4  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  scfe, which depends on 
W = scWw. The value of W belonging to the given value  z  is found by  sci. The formula for the complex electric field 
is derived in the long write-up notebook and in the paper.  Note: Here MKSA units are used; but D = scgd is given in 
mm, so the electric field given by that formula is in  kV/m.

scfe[
          scWw_] =   I scV  / 
              scgd Conjugate[((-1 + (E^scWw)) scpa )/((E^scWw) (((-1 + scpa))^2) - (E^(2 scWw)) (((-1 + scpa))^2) + scpa)  /.  scsop] // 
          Chop ;

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  scfur depending on D = scrgd and 
g/2 = scrgh.  The range of  V  is almost always  0 ≤ V ≤  π.  In some cases plot ranges must be specified; this is done by 
scfyr.  These two functions have been found empirically.

scfur[scrgd_, scrgh_]  =  
        2.5  +  
          UnitStep[
              scrgd/scrgh  -  2]  (0.5 scrgd/scrgh  -  
                1);

scfyr[scrgd_, scrgh_]  =  
        3  +  UnitStep[
              3  -  scrgd/scrgh ]  (1.5  - 
                0.5 scrgd/scrgh );

scneq = 
        20  ;

scneq  specifies the number of equipotential and field lines;  this value in turn specifies the step widths used for the
variables  U ( = scdu) and  V = π/scneq used in sccm. The range of the variable U is bounded by sclu = - 6 and 
by scru. This value and the function scfur have been determined by trial and error. This applies also to the function
scfyr rendering a value scfy needed for the plot range in picture scp23. The horizontal plot range in all pictures is 
determined by scplr.

scplr  =   (scgd  +  2 scgh);  
    scims  =  500;
sclu  =  Re[sci[-scplr]];
scru  =  scfur[scgd, scgh] ;
scdu  =  scru/5;
scfy  =  scfyr[scgd, scgh];

3.2 Field graphics

The user may want to improve the pictures drawn below. For this, he may try to obtain better values for 
sclu, scru, scdu, scfy. 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  scp2, scp3, scp23, sccm, scp1. It is extremely commendable to save this  trial notebook
under a new name beforehand so that the notebook with the orginal specifications remains intact!

scp2 (scp3) gives the electric field at the inside and outside of the upper left (right) hand electrode.
These are then combined to gives figure scp23.  sccm plots the equipotential and electric field lines. 
scp1 shows  the electric field at the bottom electrode. In all four scp pictures the unit of the electric
field strength is  kV/m  since in the definition of the function  scfe in 2.2.4  D = scgd is used, whose value
is defined in mm in  1.1, while we use  MKSA units.

scp2  =  
        ParametricPlot[{scf[ww] // Re, -scfe[ww] // Im}, {ww, 0, scru},  
          PlotRange  ->  {-1, 
              1}, 
          DisplayFunction -> Identity];

scp3  =  
        ParametricPlot[{scf[ww  +  I π] // Re, 
            scfe[ww +  I π] // Im}, {ww, sclu, 
            scru}, 
          DisplayFunction -> Identity];

scbl23 =  {Hue[0], 
            Line[{{-scgh, 
                  3  scV/scgd }, {-scgh, -3 scV/scgd }}], 
            Line[{{scgh, 3  scV/scgd }, {scgh, -3 scV/scgd }}]} // 
          Graphics;
scbl1 =  {Hue[0], 
          Line[{{-scgh, scV/scgd }, {-scgh, -scV/scgd }}], 
          Line[{{scgh, scV/scgd }, {scgh, -scV/scgd }}]};

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

    \(scp23\  = \ 
        Show[scp2, scp3, scbl23, 
          DisplayFunction -> \ $DisplayFunction, 
\ 
          PlotRange\  -> \ {scplr\ {\(-1\), 1}, \ 
              3.1  scV/scgd\ {\(-1\), 1}}, \ 
          ImageSize\  -> \ 
            scims, 
AxesLabel -> {"x [mm]", 
              None}, \ PlotLabel\  -> \ 
SequenceForm[
              Subscript["         E", "y"], "              ", 
              Subscript["E", "y"], \ "  [kV/m]"]];\)


    Print["Equipotential and electric field lines"]

    \(sccm\  = \ 
        CartesianMap[
          scf, {sclu, scru, scdu}, {$MachineEpsilon, 
            Pi \((1 - $MachineEpsilon)\), π/scneq}, 
\ 
          ImageSize\  -> \ 
            scims\ , 
AxesLabel -> {"x [mm]", "y [mm]"}, PlotRange\  -> {scplr\ {\(-1\), 1}, {\(-0.2\), scfy\ scgd}}, 
          Epilog -> {Thickness[sctcb], Hue[0], 
              Line[{{\(-\ scgd\)\  - \ 2  scgh, scgd}, {\(-\ scgh\), 
                    scgd}}], 
Text["1 Volt", {\(-\ 
                      0.6\)\ scplr, 1.2  scgd\ }], GrayLevel[0], 
              Line[{{scgd\  + \ 2  scgh, 0}, {\(-scgd\)\  - \ 2  scgh, 0}}], 
              Line[{{scgd\  + \ 2  scgh, scgd}, {scgh, 
                    scgd}}]}];\)


    Print["Electric field distribution on lower electrode"]

    \(scp1\  = \ 
        ParametricPlot[{scf[ww], scfe[ww] // Im}, {ww, sclu, \(- .001\)}, 

          ImageSize\  -> \ scims, \ 
          PlotRange\  -> \ {scplr {\(-1\), 1}, \ 
              Automatic}, 
AxesLabel -> {"x [mm]", 
              SequenceForm[
                Subscript["E", "y"], "  [kV/m]"]}, 
Epilog\  -> \ \ scbl1];\)\)\)\)

4.  Field and Potential at a Selected Point

Below the coordinates  (scxp, scyp )  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.

scWp is the point  Wp  corresponding to  zp   computed by the inverse mapping function  
W = (f^(-1))(z) = sci.  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  2.2.4.  scpc (scec) 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  scfe in 2.2.4  
D = scgd is used, whose value is defined  in mm in  1.1, while we use  MKSA units.

scxp  =  2.  ; scyp  = 
        5.  ;

scWp = sci[scxp + I scyp]; 
Print[
        SequenceForm["At point (scxp,scyp) = (", scxp, ", ", 
          scyp, ") mm : ", "Potential = ", 
          N[Re[-I scV/π scWp]] // Chop, " V;"]];
\(\(\(Print[
        SequenceForm["             (", 
          Subscript["E", "x"], ",", 
          Subscript["E", "y"], ") = (", 
          Re[1000\ scfe[scWp]], ", ", 
          Im[1000\ scfe[scWp]], \ ") V/m "]];\)\(
\)
    \)\)
scpc = 
        ParametricPlot[
          Evaluate[{Re[scf[I Im[scWp] + s]], Im[scf[I Im[scWp] + s]]}], {s, 
            sclu, scru}, PlotStyle -> Dashing[{ .01}], 
          DisplayFunction -> Identity];
scec = 
        ParametricPlot[
          Evaluate[{Re[scf[Re[scWp] + I s]], Im[scf[Re[scWp] + I s]]}], {s, 
            0, Pi}, DisplayFunction -> Identity];
scpoi = 
        Show[scpc, scec, AspectRatio -> Automatic, 
          DisplayFunction -> $DisplayFunction, ImageSize -> 600, 
          TextStyle -> {FontSize -> 16}, 
          AxesLabel -> {"x [mm]", "y [mm]"}, 
          PlotRange -> {Max[Abs[scxp], scplr] {-1, 1}, {- .2, 
                scgd scfy}}, 
          Epilog -> {Thickness[sctcb], Hue[0], 
              Line[{{-scgd - 2  scgh, scgd}, {-scgh, scgd}}], 
              Text["1 Volt", {-0.6 scplr, 1.2  scgd}], GrayLevel[0], 
              Line[{{scgd + 2  scgh, 0}, {-scgd - 2  scgh, 0}}], 
              Line[{{scgd + 2  scgh, scgd}, {scgh, scgd}}], 
              AbsolutePointSize[4], Point[{scxp, scyp}]}];

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  scxtl contains the horizontal coordinate (measured in 
 units of  g/2 = scgh) of various vertical trajectories. For these the quantity  I/qv  is displayed as a function of  
 y = scyt  in the pictures.

The user may select the horizontal coordinates  x  of a set of vertical trajectories. These are given in the list
scxtl  in units of  g/2 = scgh.

scgh

scxtl  =  
        scgh  {-2, -1.5, -1, - 0.7, -0.3, -0.1, 
            0.1,  .3};

\(\(scpli\  = \ 
        Table[Plot[\(-Im[scfe[sci[scxtl[\([k]\)]\  + \ I\ scyt]]]\)/
              scV, {scyt, 0, 
              scgd}, \ 
PlotStyle\  -> \ {Hue[
                  0.1\ k], \ 
                Thickness[
                  0.005]}, 
PlotLabel\  -> \ \ SequenceForm["x = ", 
                scxtl[\([k]\)], " mm"], 
AxesLabel\  -> \ \ {"y [mm]", "I/qv [1/mm]"}], {k, 
            Length[scxtl]}];\)\)

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.

sctli  =  
        Table[{Hue[0.1 k], 
              Line[{{scxtl[[k]], 0}, {scxtl[[k]], scgd}}]}, {k, 
              Length[scxtl]}] // Flatten;
scwxl  =  
        Show[Graphics[Prepend[sctli, Thickness[0.005]]], 
          Graphics[{Thickness[sctcb], Hue[0], 
              Line[{{-scgd - 2  scgh, scgd}, {-scgh, 
                    scgd}}], 
GrayLevel[0], 
              Line[{{scgd + 2  scgh, 0}, {-scgd - 2  scgh, 0}}], 
              Line[{{scgd + 2  scgh, scgd}, {scgh, scgd}}]}], 
          Axes  ->  True,  ImageSize  ->  500];

scwcl  =  
        Show[scpli,  PlotLabel  ->  None,  
          ImageSize  ->  500];

6.   Current Induced on the Electrode for a Selected Trajectory

The user may select the horizontal coordinate  x = scxs  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 left electrode as a function of 
the vertical position  y = scyt  of the point charge, using W = (f^(-1))(z) = sci  in the function scfe for the complex electric 
field introduced in  2.2.4.

scxs  =  0;

Show[Graphics[{Thickness[0.005],  Line[{{scxs, 0}, {scxs, scgd}}]}], 
        Graphics[{Thickness[sctcb], Hue[0], 
            Line[{{-scgd - 2  scgh, scgd}, {-scgh, 
                  scgd}}], 
GrayLevel[0], 
            Line[{{scgd + 2  scgh, 0}, {-scgd - 2  scgh, 
                  0}}], 
Line[{{scgd + 2  scgh, 
                  scgd}, {scgh, 
                  scgd}}]}], 
     Axes  ->  
          True,  ImageSize  ->  500];
scpls  =  
        Plot[- Im[scfe[sci[scxs +  I scyt]]]/scV, {scyt, 0, 
            scgd},  
PlotStyle  ->  {Thickness[
                0.005]},  
          ImageSize  ->  
            500, 
PlotLabel  ->   SequenceForm["x \
= ", scxs, " mm"], 
AxesLabel  ->   {"y \
[mm]", "I/qv [mm]"}];

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  
scf[W] = scf[u + I v] give  the position coordinates  x  and  y;  scfe[u + I v]  the complex electric field 
(s. subsect. 2.2.4). The operator Abs[] applied to this function in the command for the picture scabse  produces 
 |E|;  the operator Im[] in scey 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.

\(\(scabse = 
        ParametricPlot3D[
          Evaluate[{scf[u\  + \ I\ v] // Re, scf[u\  + \ I\ v] // Im, 
              Abs[1000\ scfe[u\  + \ I\ v]]}], {u, sclu, 
            scru}, \ {v, $MachineEpsilon, Pi \((1 - $MachineEpsilon)\)}, \ 
          BoxRatios\  -> \ {1, 1,  .75}, 
          ImageSize\  -> \ 
            600, 
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.

scey = 
        ParametricPlot3D[
          Evaluate[{scf[u  +  I v] // Re, scf[u  +  I v] // Im, 
              Im[1000 scfe[u  +  I v]]}], {u, sclu, 
            scru},  {v, $MachineEpsilon, Pi (1 - $MachineEpsilon)},  
          BoxRatios  ->  {1, 
              1,  .75}, 
PlotPoints  ->  50, 
          ImageSize  ->  600, 
          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 edges bounding the gap lead to infinite jumps.

On[General::spell1]

10.  References

10.1 Papers, Books and Websites

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
            
7.	Berechnen und Zeichnen zweidimensionaler Felder mittels konformer Abbildung.
	§ 13.10  des Skriptums zur Vorlesung: Analytische Methoden der Angewandten
	Theoretischen Physik.

10.2 Related Mathematica Notebooks

CondVoStr.nb, CondVoStrLW.nb ;
SemInfCond.nb ;
SplitCond.nb, SplitCondLW.nb;
Comparison.nb, Animation.nb .