Potential and Field 
in a shifted Semi-Infinite Plane Condensor 
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

shifted Semi Infinite Plane Condensor

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 is at the x-axis, is grounded.

The main parameters are the Voltage ϕ=huV  applied to the upper electrode, the width of the condensor   D = scgd  and  
the shifting length  b = hub. 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
ShSemInf.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["hu*"]

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

    \(\(huscV = \ 2\) \); \ \ \ \ \ \  (*\ 
      Potential\ on\ the\ upper\ elecrode\ in\ Volt, \ \ 
\ \ \ \ \ \ \ \ \ \ \ elsewhere\ on\ all\ electrodes\ Potential\  = \ 
        0\ *) 
\(\(huscgd = 
        15\) \); \ \ \ \ \  (*\ \(=\)\(\ \)\(D\  = \ 
          width\ of\ condensor\ in\ mm\)\ *) 
\(\(\(huscgh = 20\)\(\ \)\) \); \ \ \ \  (*\ \(=\)\(\ \)\(g/2\  = \ 
          Half\ the\ width\ of\ gap\ in\ upper\ electrode\)\ *) 
\(\(husctcb =  .011\) \); 	 (*\ 
      Relative\ thickness\ of\ lines\ representing\ electrodes\ *) 
\(\(HuCol1 = Hue[0.7]\) \); \ \ \ \ \ \  (*\ 
      Color\ of\ the\ ElectricField - Plots\  - \ 
        StripCondensor*) 
\(\(\(HuCol2 = 
        Hue[0.3]\)\(\ \)\) \); \)\(\ \ \ \ \ \)\( (*\ 
      Color\ of\ the\ ElectricField - Plots\  - \ 
        SplitCondensor*) \)\)\)

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

2.1  Input Data

huV   = huscV;     ;
hugd = 
      huscgd;       ;
hub = -huscgh;           ;
\(hutcb = husctcb; 	 (*\ 
      Relative\ thickness\ of\ lines\ representing\ electrodes\ *) ;\)

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

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]), the length of shifting 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[Im[huzz] - 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]];

2.3  Input Data - Split

scV   =  huscV;     ;
scgd = 
      huscgd ;       ;
\(scgh = 
      huscgh; \ \ \ \  (*\ \(=\)\(\ \)\(g/2\  = \ 
          Half\ the\ width\ of\ gap\ in\ upper\ electrode\)\ *) ;
\)
\(sctcb = husctcb; 	 (*\ 
      Relative\ thickness\ of\ lines\ representing\ electrodes\ *) ;\)

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

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.

huscneq = 
        20  ;

3.1  Variables needed in all pictures - SemInf

hufur[hurgd_, hurb_]  =  
        2.5  +  
          UnitStep[
              hurgd/Abs[hurb]  -  2]  (0.5  hurgd/Abs[hurb]  -  
                1);

hufyr[hurgd_, hurb_]  =  
        3  +  UnitStep[
              3  -  hurgd/Abs[hurb] ]  (1.5  - 
                0.5 hurgd/Abs[hurb] );

huneq = 
        huscneq  ;

huneq  specifies the number of equipotential and field lines;  The range of the variable U(=hudu) is bounded by hulu  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  =  Re[hui[-huplr]];
huru  =  hufur[hugd, Abs[hub]] ;
hudu  =  huru/5;
hufy  =  hufyr[hugd, Abs[hub]];

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 2.2.3  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,-hulu,0},  DisplayFunction -> Identity,PlotStyle -> {HuCol1}];

hup3 = ParametricPlot[{huf[ww  +  I π] // Re, 
              hufe[ww +  I π] // Im},
{ww,0,hulu},  DisplayFunction -> Identity,PlotStyle -> {HuCol1}];

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, -5 huV/
                                    hugd}, {hub, 5  huV/hugd}}]},
AxesLabel -> {"x [mm]", None}, \(PlotLabel\  -> \ 
SequenceForm[Subscript["         ", "y"], "    
          ", Subscript["E", "y"], \ "  [kV/m]"]\)];

Print["Equipotential and electric field lines"]
hucm = 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 }},PlotStyle -> {HuCol1},Epilog->{Thickness[hutcb],Hue[0],Line[{{-huplr, hugd},{ hub, hugd}}],
Text[
                          SequenceForm["Φ = ", 
                            huV, " Volt"], {- 0.6 huplr, 
                            1.2  hugd }],GrayLevel[0],Line[{{huplr, 0}, {-huplr, 0}}]}];

Print["Electric field distribution on \
lower electrode"]
hupl = ParametricPlot[{huf[ww] // Re, hufe[ww] // Im},{ww, -hulu, hulu},PlotStyle -> {HuCol1},
ImageSize  ->  huims,Epilog -> { Hue[0], 
                    AbsoluteThickness[
                      2], 
Line[{{hub, -huV/
                            hugd}, {hub, 0}}]}, PlotRange  ->  {huplr {-1, 1}, 
                    huV/hugd {-1, 0}},
AxesLabel -> {"x [mm]", 
                    SequenceForm[
                      Subscript["E", "y"], "  [kV/m]"]}];

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 = 
        huscneq  ;

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[hup23, scp23, 
        DisplayFunction ->  $DisplayFunction];

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

    
Print["Electric field distribution on lower electrode"]
Show[scp1, hupl, 
        DisplayFunction ->  $DisplayFunction];

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.

huscxp  =  -10 ; huscyp  =  
        5 ;

4.1 SemInf

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  =  huscxp ; huyp  = 
        huscyp ;

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 -> {HuCol1, Dashing[{ .01}]}, 
          DisplayFunction -> Identity];
huec = 
        ParametricPlot[
          Evaluate[{Re[huf[Re[huWp] + I s]], Im[huf[Re[huWp] + I s]]}], {s, 
            0, Pi}, DisplayFunction -> Identity, 
          PlotStyle -> {HuCol1}];
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[{{-Max[Abs[huxp], huplr], hugd}, { hub, 
                    hugd}}], 
Text[
                SequenceForm["Φ = ", 
                  huV, " Volt"], {- 0.6 huplr, 1.2  hugd }], 
              GrayLevel[0], 
              Line[{{-Max[Abs[huxp], huplr], 0}, {Max[Abs[huxp], huplr], 
                    0}}], AbsolutePointSize[4], 
              Point[{huxp, huyp}]}];

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  =  huscxp; scyp  = huscyp;

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]"}, 
          DisplayFunction -> $DisplayFunction, 
          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 Semi Infinite ----------"];
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 "]];
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, hupoi, 
        DisplayFunction -> $DisplayFunction];

5.   Current Induced on the Electrode for a Selected Trajectory

huscxs  =   -15;

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

5.1 Trajectory - SemInf

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

huxs  =   huscxs;

Show[Graphics[{Thickness[0.005],  Line[{{huxs, 0}, {huxs, hugd}}]}], 
        Graphics[{Thickness[hutcb], 
            Line[Max[Abs[huxs], 
                  huplr] {{-1, 0}, {1, 0}}], 
Hue[0],  
            Line[{{-Max[Abs[huxp], huplr], hugd}, {hub, 
                  hugd}}]}], 
     Axes  ->  
          True,  ImageSize  ->  500];
hupls  =  
        Plot[- Im[hufe[hui[huxs  +  I huyt]]]/huV, {huyt, 0, 
            hugd},  
PlotStyle  ->  {HuCol1, 
              Thickness[0.005]},  
          ImageSize  ->  
            500, 
PlotLabel  ->   SequenceForm["x \
= ", huxs, " mm"], 
AxesLabel  ->   {"y \
[mm]", "ID/qv"}];

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  =  huscxs;

\(\(\( (*\(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, hupls, 
      DisplayFunction ->  $DisplayFunction]