Potential and Field 
in a Plane Condensor with a shifted Voltage Strip 
in comparison with
a Plane Condensor with One Split Electrode

B. Schnizer, Th. Heubrandtner, St. Rossegger

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

1.  Description

Plane Condensor with a shifted Voltage Strip

The potential and field distributions for the configuration shown below, a twodimensional infinite plane condensor 
of width  D, whose upper electrode has a strip of width  w,  are computed by conformal maps. The strip is at  
a Voltage  V= cvV and can be shifted along the x-axis with the value c;  the other electrodes are grounded.

The main parameters are the Voltage  cvV  applied to the Voltage strip, the width of the condensor   D = cvgd,
the width of the strip  w  =  cvw  and the length of shifting c=cvc. Default values are assigned to these as to all other variables. 

More Information about the Conformal maps and the formulas to derive the field ar given in the notebook
CondShCoStr.nb.

Plane Condensor with One Split Electrode

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 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. 

More Information about the Conformal maps and the formulas to derive the field ar given in the notebook
SplitCondLW.nb.

Comparison of the two configurations described above

The plots of the potential and the field are overlapped for configurations.

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.

The cells where values are assigned to the important parameters have a 
background in red (they will be allocated to the specific variables of the two configurations - Background 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. 
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.

Global input Data

\(\(\(Clear["cvsc*"]

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

    \(cvscV = \ 1;\)\ \ \ \ \ \  (*\ 
      Potential\ on\ the\ upper\ elecrode\ in\ Volt, \ \ 
\ \ \ \ \ \ \ \ \ \ \ elsewhere\ on\ all\ electrodes\ Potential\  = \ 
        0\ *) 

    \(cvscgd = 
        10;\)\ \ \ \ \  (*\ \(=\)\(\ \)\(D\  = \ 
          width\ of\ condensor\ in\ mm\)\ *) 

    \(cvscgh = 
        10\ ;\)\ \ \ \  (*\ \(=\)\(\ \)\(g/2\  = \ 
          Half\ the\ width\ of\ gap\ in\ upper\ electrode\)\ *) 

    \(\(cvscw = 
      30\)\(\ \)\); \ \ \ \  (*\ \(=\)\(\ \)\(w\  = \ \ width\ of\ strip\ on\ upper\ electrode\)\ *) 
cvsctcb =  .011; 	 (*\ 
      Relative\ thickness\ of\ lines\ representing\ electrodes\ *) 
HuCol1 = Hue[0.4]; \ \ \ \ \ \  (*\ 
      Color\ of\ the\ ElectricField - Plots\  - \ 
        StripCondensor*) 
\(\(HuCol2 = 
      Hue[0.7]\)\(\ \)\);\)\(\ \ \ \ \ \)\( (*\ 
      Color\ of\ the\ ElectricField - Plots\  - \ 
        SplitCondensor*) \)\)\)

These Input Data are given as inputs to the two diffrent arangements

2.1  Input Data - Strip

\(\(\(cvV\ \  = \ cvscV // N; \ \ \ \ \ \  (*\ 
      Potential\ on\ the\ strip\ in\ Volt, \ \ 
\ \ \ \ \ \ \ \ \ \ \ elsewhere\ on\ all\ electrodes\ Potential\  = \ 
        0\ *) 
cvgd = 
      cvscgd // N; \ \ \ \  (*\ \(=\)\(\ \)\(D\  = \ 
          width\ of\ condensor\ in\ mm\)\ *) 
cvw\ \ \ \  = \ cvscw\  // 
        N; \ \  (*\ \(=\)\(\ \)\(w\  = \ \ width\ of\ strip\ on\ upper\ electrode\)\ *) 
cvc\  = \ \(-cvscgh\)\  // 
        N; \ \ \ \ \ \ \ \ \  (*\ \(=\)\(\ \)\(c\  = \ 
          position\ of\ the\ right\ end\ of\ the\ strip\ on\ the\ x - 
            axis\)\ *) 
cvtcb = cvsctcb; 	 (*\ 
      Relative\ thickness\ of\ lines\ representing\ electrodes\ *) ;\)\(
\)
    \)\)

\(Print[
      SequenceForm[
"  Width of condensor:                        D = ", \ cvgd, \ " mm."]]\)
\(Print[
      SequenceForm[
"  Width of strip at upper electrode :        w = ", \ cvw, \ " mm."]]\)
Print[
      SequenceForm[
"  Position of the right end of the \
strip:    c = ", cvc ,  " mm."]]
cvmc = cvc - cvw/2;
Print[
      SequenceForm["  Middle Postition of the strip:             mc = ", 
        cvmc ,  " 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  α = cva = (e^(((π w)/(4 D)))),whose value depends on the ratio of the two geometric
parameters D and w.

cva  =  Exp[(π cvw)/(4 cvgd)];

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 = cvWw,  t = cvtt, t' = cvtp,  α  = cva ; z = f(W) = cvf[cvWw].

cvf1[cvWw_]  =  Exp[cvWw] ;
cvf2[
          cvtt_]  =  - (cvtt cva^2   -  
                1)/(cvtt   -   cva^2) ;
cvf3[cvtp_]  =  
        cvgd/π  Log[cvtp]  + cvmc;
cvf[cvWw_]      =   cvf3[cvf2[cvf1[cvWw]]];

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

The inverse functions to  (Subscript[f, 1])to (Subscript[f, 3]) may be expressed as analytic functions;  W = (f^(-1))(z)= cvi[sczz] :

cvi1[cvtt_] =  Log[cvtt] ;
cvi2[
          cvtp_] =  (cvtp  cva^2   +  
              1)/(cvtp  +   cva^2)  ;
cvi3[
          cvzz_] =   Exp[(cvzz  - cvmc) π/
              cvgd];
cvi[cvzz_]    =  cvi1[cvi2[cvi3[cvzz]]];

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  cvfe, which depends on 
W =cvWw, or by the function cvfez, which depends on cvzz. The value of W belonging to a given value  z  is 
found by  cvi. 
Note: Here MKSA units are used; but D = cvgd is given in mm, so the electric field computed from that formula is in  
kV/m.  (Subscript[E, z])(z) = cvfez[vszz] , (Subscript[E, z])(W) = cvfe[vsWw] .

cvfez[
          cvzz_] =   -I cvV  / cvgd/
              2 * (Tanh[π (Conjugate [cvzz - cvmc] + 
                          cvw/2)/(2 cvgd)]  + 
Tanh[π ( 
                        cvw/2  -  
                          Conjugate [cvzz - cvmc])/(2 cvgd)]) // 
          Chop ;

cvfe[
          cvWw_]  =   I cvV/
            cvgd (2 Cosh[Conjugate[cvWw]]  -  (cva^2  +  
                    1/cva^2))/(cva^2  -  1/cva^2);

2.3  Input Data - Split

scV   =  cvscV;     ;
scgd = 
      cvscgd ;       ;
\(scgh = 
      cvscgh; \ \ \ \  (*\ \(=\)\(\ \)\(g/2\  = \ 
          Half\ the\ width\ of\ gap\ in\ upper\ electrode\)\ *) ;
\)
\(sctcb = cvsctcb; 	 (*\ 
      Relative\ thickness\ of\ lines\ representing\ electrodes\ *) ;\)

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

2.4 Functions for the conformal map - Split

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; and this may 
be used to define the total inverse mapping function   W = (f^(-1))(z) = sci.

\(\(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];\)\)
sci3[sczz_] :=  
      scpqq   /.  
        FindRoot[(scf3[scpqq]  /.  scsop) -  sczz, {scpqq, 
            scstv[sczz]  /.  scsop}]

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

Approximation to the inverse function

The approximate formula leading to fat[ ] is only valid for large negative Re(zzz) and   0 < Im(zzz) < D.

scα  = ((((-1 + scpa))^2)/scpa)  /.  
          scsop ;

2nd order

fat2[
          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)))));

Findroot failed - Approximation 2nd order:

sci[sczz_] := sci1[sci2[Check[sci3[sczz], fat2[sczz]]]]

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

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.The range of V is almost always 0≤V≤π.

cvscneq = 
        20  ;

3.1  Variables needed in the pictures - Strip

cvneq = 
        cvscneq  ;

cvneq  specifies the number of equipotential and field lines;  this value in turn specifies the step widths used for the
variables  U ( = cvdu) and  V = π/cvneq used in cvcm. The range of the variable U is bounded by cvlu  and 
by cvru. These values have been determined by trial and error. The horizontal plot range in all pictures is 
determined by cvlx and cvrx. In some figures the vertical plot range is determined by cvmaE and multiples of it.

cvmaE  =  cvV/cvgd ;
cvplr  =  (cvgd  +  
            Abs[2 cvc]);
cvims  =  500;
cvlu  =  -cvi[cvplr];
cvru  =   cvi[cvplr];
cvdu  =  cvru/5;

The user may want to improve the pictures drawn below. For this, he may try to obtain better values for 
cvlu, cvru, cvdu, cvmaE. 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  cvp2, cvp1, cvp23, cvcm. It is extremely commendable to save this  trial notebook
under a new name beforehand so that the notebook with the orginal specifications remains intact!

cvp2  computes the electric field at the inside of the upper electrode (cvp1 of the lower) .The corresponding pictureis shown in figure cvp23.  cvcm plots the equipotential and electric field lines.  cvp1 shows  the electric field at the bottom electrode. 
In the two cvp pictures the unit of the electric field strength is  kV/m  since in the definition of the functions  cvfe 
and cvfez in 2.2.4  D = cvgd is used, whose value is defined in mm in  1.1, while we use  MKSA units.

cvtru  =  { Hue[0], AbsoluteThickness[2], 
          Line[{{-cvw/2 + cvmc, - cvmaE}, {-cvw/2 + cvmc, 
                cvmaE}}], 
Line[{{cvw/2  + cvmc, - 
                  cvmaE}, {cvw/2 + cvmc, 
                cvmaE}}]};
cvtrl  =  { Hue[0], 
          Line[{{-cvw/2 + cvmc, 
                0}, {-cvw/2 + 
                  cvmc, -cvmaE}}], 
Line[{{cvw/2 + 
                  cvmc, 0}, {cvw/2 + cvmc, -cvmaE}}]};

cvp2 = Plot[{cvfez[xx + I cvgd]//Im},{xx, -cvplr, cvplr},DisplayFunction ->  Identity,PlotStyle -> {HuCol1}];
cvp1 = Plot[{cvfez[xx ] // Im},{xx, -cvplr, cvplr},DisplayFunction ->  Identity,PlotStyle -> {HuCol1}];

cvp23  =  
                Show[cvp2,  ImageSize  ->  cvims, 
                  Epilog  ->  
                    cvtru, 
AxesLabel -> {"x \
[mm]", None},  PlotLabel  ->  
SequenceForm[ 
                      Subscript["E", "y"],  "  [kV/m]"]];

  
cvcm = CartesianMap[cvf,{5  cvlu, 5  cvru, cvdu},{$MachineEpsilon, π (1 - $MachineEpsilon), π (1 - 2  $MachineEpsilon)/cvneq},
 ImageSize  ->  cvims ,PlotStyle -> {HuCol1},
 PlotRange  -> {cvplr {-1, 1}, {-0.1, 
                        3  cvgd }},
AxesLabel -> {"x [mm]", "\
y [mm]"},
Epilog->{Thickness[cvtcb],Line[2 {{-cvplr, 0}, {cvplr, 0}}],
Line[{{-cvplr, 
                              cvgd}, {cvc - cvw, cvgd}}],Line[{{ cvplr, cvgd}, {cvc, cvgd}}],
Hue[0],Line[{{cvc, cvgd}, {cvc - cvw, cvgd}}],
Text[SequenceForm["Φ = ", 
                              cvV, " Volt"],{cvmc,1.15 cvgd}]}];

  
Show[cvp1, Graphics[cvtrl], 
 
            ImageSize  -> 
              cvims, 
AxesLabel -> {"x [mm]", 
                None},  
            PlotLabel  ->  
SequenceForm[ 
                Subscript["E", "y"],  "  [kV/m]"], 
            Epilog  ->  cvtrl];

3.2  Variables needed in the pictures - Split

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 = 
        cvscneq  ;

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];

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,PlotStyle -> {HuCol2}];

scp3 = ParametricPlot[{scf[ww  +  I π] // Re, 
              scfe[ww +  I π] // Im},{ww, sclu, scru},
          DisplayFunction -> Identity,PlotStyle -> {HuCol2}];

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

  
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}},PlotStyle -> {HuCol2},\(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}}]}\)];

  
scp1 = ParametricPlot[{scf[ww], scfe[ww] // Im},{ww, sclu, - .001},DisplayFunction ->  Identity,
ImageSize  ->  scims,PlotStyle -> {HuCol2}, PlotRange  ->  {scplr {-1, 1},  
                    Automatic},
AxesLabel -> {"x [mm]", 
                    SequenceForm[
                      Subscript["E", "y"], "  [kV/m]"]}];

3.3  Comparison of the described plane condensors

Print["Electric field distribution on upper electrodes"]
Show[cvp23, scp23, 
        DisplayFunction ->  $DisplayFunction];

    
Print["Equipotential and electric field lines"]
Show[cvcm, sccm, 
        DisplayFunction ->  $DisplayFunction];

    
Print["Electric field distribution on lower electrode"]
Show[scp1, cvp1, Graphics[cvtrl],  
        DisplayFunction ->  $DisplayFunction, 
 
        ImageSize  -> 
          cvims, 
AxesLabel -> {"x [mm]", None},  
        PlotLabel  ->  
SequenceForm[ 
            Subscript["E", "y"],  "  [kV/m]"], 
        Epilog  ->  cvtrl];

4.  Field and Potential at a Selected Point

Below the coordinates (xp,yp ) 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.

cvscxp  =  -15 ; cvscyp  =  
        5 ;

4.1  Strip

cvWp is the point  Wp  corresponding to  zp   computed by the inverse mapping function  W = (f^(-1))(z) = cvi.  
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.  
cvpc (cvec) 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  cvfe in 2.2.4  
D = cvgd is used, whose value is defined  in mm in 1.1, while we use  MKSA units.

cvxp  =  cvscxp ; cvyp  =  
        cvscyp ;

cvWp = Evaluate[cvi[cvxp   + I cvyp]]

  

cvpc = 
        ParametricPlot[
          Evaluate[{Re[cvf[I Im[cvWp] + s]], 
              Im[cvf[I Im[cvWp] + s]]}], {s, -20, 40}, 
          PlotStyle -> {HuCol1, Dashing[{ .01}]}, 
          DisplayFunction -> Identity];
cvec=ParametricPlot[Evaluate[{Re[cvf[Re[cvWp] + I s]], 
                Im[cvf[Re[cvWp] + I s]]}],{s, 0, Pi},PlotStyle -> {HuCol1},DisplayFunction -> Identity];
cvpoi=Show[cvpc,cvec,AspectRatio -> Automatic,ImageSize -> 600,TextStyle -> {FontSize -> 16},AxesLabel -> {"x [mm]", "y [mm]"},
 PlotRange  -> {cvplr {-1, 1}, {-0.1, 
                  cvgd 1.3}},Epilog->{Thickness[cvtcb],Line[2 {{-cvplr, 0}, {cvplr, 0}}],
Line[{{-cvplr, 
                        cvgd}, {cvc - cvw, cvgd}}],Line[{{ cvplr, cvgd}, {cvc, cvgd}}],
Hue[0],Line[{{cvc, cvgd}, {cvc - cvw, cvgd}}],
Text[SequenceForm["Φ = ", 
                        cvV, " Volt"],{cvmc,1.15 cvgd}],
AbsolutePointSize[4],Point[{cvxp, cvyp}]}];

4.2  Split

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.  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   
D = scgd is used, whose value is defined  in mm in  1.1, while we use  MKSA units.

scxp  =  cvscxp ; scyp  = 
        cvscyp  ;

scWp = Evaluate[sci[scxp + I scyp]]

  

scpc = 
          ParametricPlot[
            Evaluate[{Re[scf[I Im[scWp] + s]], 
                Im[scf[I Im[scWp] + s]]}], {s, sclu, scru}, 
            PlotStyle -> {HuCol2, Dashing[{ .01}]}, 
            DisplayFunction -> Identity];

scec=ParametricPlot[Evaluate[{Re[scf[Re[scWp] + I s]], 
                  Im[scf[Re[scWp] + I s]]}],{s, 0, Pi},PlotStyle -> {HuCol2},DisplayFunction -> 
                Identity];

scpoi = 
        Show[scpc, scec, AspectRatio -> Automatic, 
          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}]}];

4.3  Comparison

Print["----------- Shifted Voltage Strip ----------"];
Print[
        SequenceForm["At point (cvxp,cvyp) = (", cvxp, ", ", 
          cvyp, ") mm : ", "Potential = ", 
          N[Re[-I cvV/π cvWp]] // Chop, " V;"]];
Print[
        SequenceForm["             (", 
          Subscript["E", "x"], ",", 
          Subscript["E", "y"], ") = (", 
          N[Re[1000 cvfe[cvWp]]], ", ", 
          N[Im[1000 cvfe[
                    cvWp]]] ") V/m "]];
Print["------------ Splited electrode --------------"];
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 "]];

    
Show[scpoi, cvpoi, 
        DisplayFunction -> $DisplayFunction];

5.   Current Induced on the Electrode for a Selected Trajectory

cvscxs  =   -20;

The user may select the horizontal coordinate  x = cvvsxs  of a vertical trajectory. The first figure shows the position of this 
trajectory in the real condensor.

5.1 Trajectory - Strip

The next one gives the current ratio induced in the upper left electrode as a function of 
the vertical position  y = cvyt  of the point charge, using W = (f^(-1))(z) =cvi  in the function cvfe for the complex electric field.

cvxs  =   cvscxs;

\(\(\( (*\(Show[
          Graphics[{Thickness[0.005], \ Line[{{cvxs, 0}, {cvxs, cvgd}}]}], 
          Graphics[{Thickness[cvtcb], 
              Line[cvplr {{\(-1\), 0}, {1, 
                      0}}], 
Line[{{\(-cvplr\), 
                    cvgd}, {cvc - cvw, cvgd}}], 
              Line[{{\ cvplr, cvgd}, {cvc, cvgd}}], 
Hue[
                0], Line[{{cvc, cvgd}, {cvc - cvw, 
                    cvgd}}]}], 
Axes\  -> \ True, \ 
          ImageSize\  -> \ 
            500];\)*) \)\(
\)\(cvpls\  = \ 
        Plot[\(-\ Im[cvfe[cvi[cvxs + \ I\ cvyt]]]\)/cvV, {cvyt, 0, 
            cvgd}, \ 
PlotStyle\  -> \ {HuCol1, 
              Thickness[0.005]}, \ ImageSize\  -> \ 500, 
          DisplayFunction -> \ 
            Identity, 
PlotLabel\  -> \ \ SequenceForm["x = ", 
              cvxs, " mm"], 
AxesLabel\  -> \ \ {"y [mm]", "I/qv [1/mm]"}];\)\)\)

5.2 Trajectory - Split

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.

scxs  =  cvscxs;

\(\(\( (*\(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\  -> \ {HuCol2, 
              Thickness[0.005]}, \ ImageSize\  -> \ 500, 
          DisplayFunction -> \ 
            Identity, 
PlotLabel\  -> \ \ SequenceForm["x = ", 
              scxs, " mm"], 
AxesLabel\  -> \ \ {"y [mm]", "ID/qv"}];\)\)\)

5.3 Comparison

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];
Show[scpls, cvpls, 
        DisplayFunction ->  $DisplayFunction];