Potential and Field in a Plane 
Condensor with a Voltage Strip 
on an Electrode
Long Write-Up

B. Schnizer, St. Rosegger

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

2004-12-01

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 strip of width  w,  are computed by conformal maps. The strip is at  a Voltage   vsV ;  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 relativecurrent  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  vsV, while the lower one, 
V = 0,  is grounded.  This map consists of three nested conformal maps:   

	3.			z = (Subscript[f, 3])(t')  =  (D/π) ln t' ,   t'  =  (Subscript[f, 3]^(-1))(z)  =  (e^(π z/D))
	
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  ,   t = (Subscript[f, 1])(W)  =  (e^W)
	
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  

	2.			t' = (Subscript[f, 2])(t)= ((α^2) t -  1)/(t - (α^2)),  t =  (Subscript[f, 2]^(-1))(t') = (t' (α^2) + 1 )/(t' +  (α^2))
	
(with  α = (e^(π w/(4  D))))  maps the t- on the t'-plane such that the s- becomes the s'-axis.

The total conformal maps are then:

	z  =  f(W) =  (D/π) Log[ 
          Sinh[((π w)/(4 D)) + (W/2)]/
            Sinh[((π w)/(4 D)) - (W/2)]] + (c - w/2),
	
	W = (f^(-1))(z) =  Log[
        Cosh[((π (z -  c +  w))/(2 D))]/
          Cosh[((π (z - c))/(2  D))]].

The inverse function has been found by hand. Mathematica does not succeed to give such a 
simple expression.
The complex potential and the field in the simple plane condensor are:

	Ψ(W) = - i ((Subscript[V, 0])/π)W,     (Subscript[E, W])(W) = (Subscript[E, U])   +  i (Subscript[E, V])= - i ((Subscript[V, 0])/π)  =  - (d(Ψ/dW))^( * ).

The real potential and field in the plane condensor with the Voltage strip as shown above are:

	(Subscript[Φ, 1])(x, 
          y)  =  ((Subscript[V, 0])/π)[arctan( tanh[(π/(2  D)) (- x + c)] tan[((π y)/(2  D))])  +  arctan( tanh[(π/(2  D)) (x - c  +  w)] tan[((π y)/(2  D))]) ].
	
  (Subscript[E, x])(x, y)   =   ((Subscript[V, 0])/(2  D))sin[((π y)/D)]\(\(\([1\/\(cosh[\(π\ \((x\  - \ c)\)\)\/\(2\ D\)]\  + \ \ cos[\(π\ y\)\/D]\) - 
          1\/\(cosh[\(π\ \((x\  - \ c\  + \ w)\)\)\/\(2\ D\)]\  + \ \ cos[\(π\ y\)\/D]\)]\)\(,\)\)\)
	 
  (Subscript[E, y])(x, y)   =  - ((Subscript[V, 0])/(2  D))\(\(\([sinh[\(π\ \((x\  - \ c)\)\)\/\(2\ D\)]\/\(cosh[\(π\ \((x\  - \ c)\)\)\/\(2\ D\)]\  + \ \ cos[\(π\ y\)\/D]\) - 
            sinh[\(π\ \((x\  - \ c\  + \ w)\)\)\/\(2\ D\)]\/\(cosh[\(π\ \((x\  - \ c\  + \ w)\)\)\/\(2\ D\)]\  + \ \ cos[\(π\ y\)\/D]\)]\)\(.\)\)\)
	 
The potential and the field in the condensor with the Voltage strip may be extended to the corresponding 
complex quantities:

	Φ(z) = (Subscript[Φ, 1])(x, y)+ i (Subscript[Φ, 2])(x, y) =   - i ((Subscript[V, 0])/π)W    (with  W = (f^(-1))(z)  as given above);
	
	E(z) =  i ((Subscript[V, 0])/(2  D))[tanh[\(TraditionalForm\`\(π\/\(2\ D\)\) \((\(z\^*\)\  - \ c)\)\)]  -  tanh[\(TraditionalForm\`\(π\/\(2\ D\)\) \((\(z\^*\)\  - \ c\  + \ 
            w)\)\)] ;
	
	E(W) = i \(TraditionalForm\`\(\(V\_0\/D\)\(\ \)\([\)\(cosh[\(W\^*\)]\)\)\) -  cosh[((π w)/(2 D))] ]/sinh[((π w)/(2 D))]  =  i ((Subscript[V, 0])/D)\(TraditionalForm\`\(2\ cosh[\(W\^*\)]\ \ \  - \ \ α\^2\  - \ \ \ α\^\(-2\)\)\/\(α\^2\  - \ \ α\^\(-2\)\)\).

This notebook is the long write-up giving more details of these transformations. These are dicussed in Sect.2. The short version is contained in the program notebook  CondVoSt.nb.

In the program notebook just quoted somewhat different notations are used; there almost all variables start with the letters  vs...  . An analogous convention is also adopted in the other program notebooks on conformal maps applied to RPC models. So several notebooks  may be used simultaneously in the same kernel without risk of interference.   At the contrary, a long write-up notebook should be used alone in a kernel.

The main parameters are the Voltage  V=vsV  applied to the strip on the upper  electrode, the width of the condensor   D = vsgd  and  the width of the strip  w = vsw.  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 shure that the program will run without difficulty as long as the conditions 

						0.1 < D/w < 15 
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`"]
Needs["Graphics`Graphics`"]

Off[General::spell1]

In this notebook many variables have similar names. The ensuing warnings are switched off above in the third 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["vs*"]

    
vsV   =  1;         ;
vsgd = 
      10;        ;
vsw     = 
      8;           ;
\(vstcb =  .011; 	 (*\ 
      Relative\ thickness\ of\ lines\ representing\ electrodes\ *) ;\)

2.1.1 Graph of Condensor

up  =  {Thickness[ .01], Hue[0.2], 
            Line[{{-2*vsw, vsgd}, { -vsw/2, vsgd}}], Hue[0], 
            Line[{{- vsw/2 , vsgd}, {vsw/2, vsgd}}],  Hue[ .6], 
            Line[{{2  vsw, vsgd}, {vsw/2, vsgd}}]} // 
          Graphics ;

lo  =  {Thickness[ .01], Line[{{0, 0}, {2  vsw, 0}}],  
            GrayLevel[ .7], Line[{{-2*vsw, 0}, {0, 0}}]} // 
          Graphics;

cou  = 
        Show[up, lo,  Axes   ->  True, Ticks  ->  None,  
          AspectRatio  ->  Automatic, ImageSize  ->  550, 
          PlotRange  ->  {-0.2, 1.2 vsgd}, 
          AxesLabel  ->   {"x", "y"}, 
          TextStyle  ->  {FontSize  ->  16} ];

2.2  The conformal maps

2.2.1  Mapping the z-plane (the real condensor) to the t'-plane (Conformal Map f\(TraditionalForm\`\_3\))

The real condensor of width  D = vsgd comprising the Voltage strip  y = D,  -w/2 ≤  x  ≤ w/2 is mapped
to the t'-plane such that the upper (lower) electrode becomes the negative (positive) part of the s'-axis.

fc3[tp_]  =  vsgd/π Log[tp] ;
fi3[z_]  =  Exp[z π/vsgd];

The two transformation are the inverse of each other:

fi3[fc3[ss]] // Simplify

Various important points in the z-plane:

lz3  = {- Infinity, -2 vsw,  0, 2  vsw,  
        Infinity,  -vsw/2  +  I vsgd,  I vsgd,  
        vsw/2  +  I vsgd,  I vsgd/2}

Same points in the t'-plane:

ltp3  = 
      Simplify[fi3[lz3],  vsgd  >  0 &&  vsw  >  0] // Chop

ltra  = Abs[1.5 ltp3[[8]]];

lsp3   =  {Thickness[ .01],  GrayLevel[ .7], 
            Line[{{0, 0}, {1, 
                  0}}],  
              GrayLevel[0], 
            Line[{{ltra, 0}, {1, 
                  0}}], 
               Hue[ .6], Line[{{-ltra, 0}, {ltp3[[8]], 0}}], 
Hue[0], 
            Line[{{ltp3[[8]], 0}, {ltp3[[6]], 
                  0}}], 
Hue[ .2], 
            Line[{{ltp3[[6]], 0}, {0, 0}}]} // Graphics;

trs  = 
        Show[lsp3,  Axes  ->  True,  
          PlotRange  ->  {-1, 1},  
          AspectRatio  ->   .2, 
ImageSize  ->   600, DisplayFunction -> Identity];

Only for a demonstration, a transformation is shown, which maps a homogeneous field in z-plane 
to the t'-plane (upper elektrode at V, lower electrode grounded). This is not used in the program.

lety  =  -0.59;
cmz  =  
        CartesianMap[# &, 2  vsw {-1, 1}, {0, vsgd},  
          DisplayFunction  ->  Identity];
\(\(Show[cmz, \ cou, \ 
        ImageSize\  -> \ \ \ 650, \ \ PlotRange\  -> \ 
          vsgd\ {\(-0.1\), 1.1}, 
\ 
        DisplayFunction\  -> \ $DisplayFunction, 
Epilog\  -> \ {Text["D", { .29, 
                lety}], 
Text["O", {\(-2.05\)\ vsw,  .5\ vsgd}], 
            Text["←", {\(-2.05\)\ vsw,  .46\ vsgd}], 
Text["O'", {2.065\ vsw,  .5\ vsgd}], 
            Text["→", {2.05\ vsw,  .46\ vsgd}], 
            Text["C", {\(-0.5\) vsw, 1.05\ vsgd}], 
            Text["B", {0.5  vsw, 
                1.05\ vsgd}]}, 
TextStyle\  -> \ {FontSize\  -> \ 14, \ FontWeight\  -> \ Bold}, \ 
        Ticks\  -> \ None];\)\)

lets  =  - 0.05;
CartesianMap[fi3[#] &, 
        vsw {-1, 1}, {0, 
          vsgd},  
PlotRange  ->  {ltra {-1, 
                1}, vsw /3 {-0.2, 1}}, 
        AxesLabel  ->  {"r'", "s'"}, 
        DisplayFunction  ->  Identity];
\(\(pltp3\  = 
        Show[%, lsp3, \ ImageSize\  -> \ 650, 
          DisplayFunction\  -> \ $DisplayFunction, 
Epilog\  -> \ {Text["← A", 
                ltra {\(-0.95\)\ , lets}], 
              Text["A'→", 
                ltra {0.92\ , 
                    lets}], 
Text["B", {ltp3[\([8]\)], 
                  ltra\ lets}], 
              Text["C", {ltp3[\([6]\)] -  .2, 
                  ltra\ lets}], 
Text["0", {ltp3[\([1]\)] +  .1, ltra\ lets}], 
              Text["D", {ltp3[\([3]\)], 
                  ltra\ lets}]}, 
\ 
          TextStyle\  -> \ {FontSize\  -> \ 14, \ 
              FontWeight\  -> \ Bold}, \ 
          Ticks\  -> \ None];\)\)

2.2.2  Mapping the W-plane (the infinite plane condensor) to the t-plane (Conformal Map f\(TraditionalForm\`\_1\))

The infinite plane condensor of width  π  is mapped by fc1(W) = (e^W) to the t-plane such that the upper (lower) electrode 
becomes the negative (positive) s-axis.  fi1(t) is the inverse function.

fc1[wW_]  =  Exp[wW] 
fi1[tt_]  =  Log[tt]

vsw

plW1 = 
        CartesianMap[# &, 1/2  vsw {-1, 1}, {0, π},  
          DisplayFunction  ->  Identity];
\(\(cmww1\  = 

        Show[plW1, \ Axes\  -> \ True, 
          AxesLabel\  -> \ {"U", "V"}, \ 
          Ticks\  -> \ {Automatic, \ 
              PiScale}, \ \ ImageSize\  -> \ \ \ 550, \ \ DisplayFunction\  -> \ $DisplayFunction, 
Epilog\  -> \ {Thickness[ .01], 
              Line[vsw {{\(-1\), 0}, {1, 
                      0}}], 
Text["E", {\(-0.525\)\ vsw,  .93\ π}], 
              Text["←", {\(-0.525\)\ vsw,  .89\ π}], 
Text["F", {\(-0.525\)\ vsw,  .08\ π}], 
              Text["←", {\(-0.525\)\ vsw,  .03\ π}], 
Text["G", {\ \ \ 0.525\ vsw,  .93\ π}], 
              Text["→", {\ \ 0.525\ vsw,  .89\ π}], 
Text["H", {\ \ 0.525\ vsw,  .08\ π}], 
              Text["→", \ {\  .525\ vsw,  .03\ π}], 
Hue[0], 
              Line[{{\(-vsw\), π}, {vsw, π}}]}, 
PlotRange\  -> \ {0.55\ vsw {\(-1\), 1.01}, 
              Automatic}, 
\ 
          TextStyle\  -> \ {FontSize\  -> \ 12}];\)\)

CartesianMap[fc1[#] &, vsw/3 {-1, 1}, {0, π},  
        DisplayFunction  ->  Identity];
\(\(pltt1\  = \ 
        Show[%, \ Axes\  -> \ True, 
          AxesLabel\  -> \ {"r", "s"}, \ 
          PlotRange\  -> {ltra {\(-1\), 1}, 
              vsw/3 {\(- .15\), 
                  1}}, 
Epilog\  -> \ {Text["E  F", {0, \(-0.25\)}], 
Text["H →", {\ 
                  0.95\ ltra, \(-0.25\)}], 
Text["← G", {\(-0.95\) 
                    ltra, \(-0.25\)}], 
Thickness[ .01], 
              Line[{{0, 0}, {ltra\ , 0}}], Hue[0], 
              Line[{{\(-ltra\), 0}, {0, 
                    0}}]}, \ \ ImageSize\  -> \ \ \ 650, \ \ DisplayFunction\  -> \ $DisplayFunction, 
\ 
          TextStyle\  -> \ {FontSize\  -> \ 12}, \ 
          Ticks\  -> \ None];\)\)

2.2.3  Mapping the t'-plane to the t-plane  (Conformal Map f\(TraditionalForm\`\_2\))

In the two pictures pltt1 and pltp3 the red parts of the electrodes are at potential  (Subscript[V, 0]), while all other parts are at 
potential 0.  We need a function mapping the red parts into each other;so  BC  of Fig.pltp3 must go over into GE
of Fig.pltt1; CODA'AB   into   FH  of Fig.pltt1. This map is performed by the following functions:

Clear[vsα]
fc2[
          tt_]  =  - (vsα^2 tt  -  1)/(tt  -  
              vsα^2);
fi2[
          tp_]  =      (tp vsα^2  +  1)/(tp  +  
              vsα^2);

The two transformation given above are the inverse of each other:

Clear[t]
fc2[fi2[t]] // Simplify

The value of the parameter α depends on the values selected in 2.1. The conformal map f\(TraditionalForm\`\(\(\_2\)\(\ \)\)\)needs one parameter α=e\(TraditionalForm\`\^\(\(π\ ω\)\/\(4  D\)\)\) whose value depends on the ratio of the two geometric parameters D and w.

vsα  =  Exp[π vsw/(4 vsgd)];

Show[pltp3,  
        PlotRange  ->  {ltra {-1, 1}, 
            vsw /3 {-0.2, 0.01}},  
        ImageSize  ->  1100];

Show[pltt1, Graphics[Text["O'", {3  fi2[0], 5  lets}]], 
        PlotRange  -> {ltra {-1, 1}, vsw/3 {- .15, 0.01}},  
        ImageSize  ->  1200];

The following points ,{*,B,*,C,0,D = 1},in the t'-plane:

lt2  =  {- 2 vsα^2,  - 
            vsα^2, -( vsα^2  +  1/ vsα^2)/
            2,   -1/ vsα^2,  0, 1} // N

correspond to the following points, {*,-∞,*,0,O',D = 1}, in the t-plane:

fi2[lt2] // Chop

2.2.4  The total map  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 = vsWw,  t = vstt, t' = vstp,  α = vsα.

The total map maps a uniform field  in the W-plane (common infinite condensor, width π)  into the z-plane 
(infinite condensor of width  D = vsgd  with a voltage strip of width   w = vsw   on the upper electrode).

vsf[wW_]    =  fc3[fc2[fc1[wW]]] ;

CartesianMap[# &, 1/2  vsw {-1, 1}, {0, π},  
        DisplayFunction  ->  Identity];
\(\(\(cmww1\  = 

        Show[plW1, \ Axes\  -> \ True, 
          AxesLabel\  -> \ {"U", "V"}, \ 
          Ticks\  -> \ {Automatic, \ PiScale}, 
          Epilog\  -> \ {Thickness[ .01], 
              Line[vsw {{\(-1\), 0}, {1, 0}}], Hue[0], 
              Line[{{\(-vsw\), π}, {vsw, π}}]}, \ \ ImageSize\  -> \ \ \ 550, \ \ DisplayFunction\  -> \ $DisplayFunction, \ 
          TextStyle\  -> \ {FontSize\  -> \ 
                16}];\)\(
\)
    \)\)
el =  {Thickness[ .01], 
          Line[{{-2 vsw, vsgd}, { - vsw/2, vsgd}}], 
          Line[{{2  vsw, vsgd}, {vsw/2, 
                vsgd}}], 
Line[{{-2 vsw, 0}, {2  vsw, 
                0}}], Hue[0], 
          Line[{{- vsw/2 , vsgd}, {vsw/2, 
                vsgd}}]} ;

    
plt  = 
        CartesianMap[vsf[#]  &, 
          vsw {-1, 1,  .05}, {10^(-7), π , π/20},  
          DisplayFunction  ->  Identity];
\(\(Show[%, cou, 
        PlotRange -> {vsw\ {\(-1.5\), 1.5}, {\(-0.1\), vsgd\ 1.05}}, \ 
        Epilog\  -> \ el, \ ImageSize\  -> \ 600, \ 
        AxesLabel\  -> {"x", "y"}, 
\ 
        DisplayFunction\  -> \ \ $DisplayFunction, \ \ TextStyle\  -> \ {FontSize\  -> \ 16}];\)\)

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

The inverse functions to  (Subscript[f, 1]) , (Subscript[f, 2])  and (Subscript[f, 3]) may be expressed as analytic functions as shown in 2.2.1 to 2.2.3.

Therefore the total inverse transform  W = (f^(-1))(z) is:

vsi[zz_]   =  fi1[fi2[fi3[zz]]];

2.2.6 The shifted Voltage strip

Up to now a Voltage strip  y = D, -w/2 < x < w/2  was considered.  Above and also in the next chapters 
all  transformations and formulas  are derived and discussed for this configuration. This location of the
strip is also used in the notebook  CondVoStr.nb.  The shifted strip is at  y = D, c - w < x < c.
This convention is used in the notebook CondShVoStr.nb. The latter is also contained and used in the 
notebook  Comparison.nb .
This shift just changes  x  →  x + c - w/2, which implies  z  →  z + c - w/2. The resulting formulae are listed in 
Chap.1  of  the notebook CondShVoStr.nb.

2.3  The potential and the electric field

The total conformal map as shown above is :
:
	z   =  f(W)  =  (D/π) Log[ 
        Sinh[((π w)/(4 D)) + (W/2)]/
          Sinh[((π w)/(4 D)) - (W/2)]],
	
	W = (f^(-1))(z) =  Log[
        Cosh[((π (w/2   +  z))/(2 D))]/
          Cosh[((π (- w/2   +   z))/(2  D))]].

The complex potential and the electric field in the simple plane condensor in the W-plane are:

	Ψ(W) = - i ((Subscript[V, 0])/π)W,     (Subscript[E, W])(W) = (Subscript[E, U])   +  i (Subscript[E, V])= - i ((Subscript[V, 0])/π)  =  - (d(Ψ/dW))^( * ).

The real potential in the z-plane condensor with the Voltage strip as shown in the picture above is:

	(Subscript[Φ, 1])(x, 
          y)  =  ((Subscript[V, 0])/π)[arctan( tanh[(π/(2  D)) (w/2  -  x)] tan[((π y)/(2  D))])  +  arctan( tanh[(π/(2  D)) (w/2  +  x)] tan[((π y)/(2  D))]) ].
	
This real potential is extended to a complex potential  Φ(z)  by  choosing (Subscript[Φ, 1])(x,y) := Φ(x,y)  as the 
real part of  Φ(z) and computing the imaginary part  (Subscript[Φ, 2])(x,y)  via the Cauchy-Riemann equations as 
shown below. This complex potential is:

	Φ(z) = - i ((Subscript[V, 0])/π)Log[ cosh[(π z  -  w/2)/2  
                  D]/cosh[(π z  +  w/2)/2  D] ]  = 
	      
	        = - i ((Subscript[V, 0])/π) [ Log(cosh[(π z - w/2)/(2D)]) - Log(cosh[(π z + w/2)/(2D)]) ]
	
The electric field is:
  
(Subscript[E, x]) =     ((Subscript[V, 0])/(2  D))[\(sin[π\ y/D]\/\(cosh[π\ \((x\  + \ w/2)\)/D]\  + \ \ cos[π\ y/D]\)\ \  - \ \ sin[π\ y/D]\/\(cosh[π\ \((x\  - \ w/2)\)/D]\  + \ \ cos[π\ y/D]\)\) ]

(Subscript[E, y]) =   -((Subscript[V, 0])/(2  D))[\(sinh[π \((x\  + \ w/2)\)/D]\/\(cosh[π\ \((x\  + \ \ w/2)\)/D]\  + \ \ cos[π\ y/D]\)\ \  - \ \ sinh[π \((x\  - \ \ w/2)\)/D]\/\(cosh[π\ \((x\  - \ w/2)\)/D]\  + \ \ cos[π\ y/D]\)\) ]
	 
The complex electric field   (Subscript["E", "x"])"+i"(Subscript["E", "y"]) is given by:	

	E(W) = i ((Subscript[V, 0])/D)[  cosh[\(W\^*\)] -  cosh[((π w)/(2 D))] ]/sinh[((π w)/(2 D))]  =  i ((Subscript[V, 0])/D)\(TraditionalForm\`\(2\ cosh[\(W\^*\)]\ \ \  - \ \ α\^2\  - \ \ \ α\^\(-2\)\)\/\(α\^2\  - \ \ α\^\(-2\)\)\).
	
	E(z) = - i ((Subscript[V, 0])/(2  D))[tanh[\(TraditionalForm\`\(π\/\(2\ D\)\) \((\(z\^*\)\  + \ w/2)\)\)] - tanh[\(TraditionalForm\`\(π\/\(2\ D\)\) \((\(z\^*\)\  - \ w/2)\)\)] ;

The complex electric field   (Subscript["E", "x"])"+i"(Subscript["E", "y"])  at a point   z = x + i y   is given by the function  vsfe, which depends on 
W = vsWw, or by the function vsfez, which depends on vszz. The value of W belonging to a given value  z  is 
found by  vsi.
Note: Here MKSA units are used; but D = vsgd is given in mm, so the electric field computed from that formula is in  
kV/m.  (Subscript[E, z])(z) = vsfez[vszz] , (Subscript[E, z])(W) = vsfe[vsWw] .

vsfe[
          vsWw_]  =   I vsV/
            vsgd (2 Cosh[Conjugate[vsWw]]  -  (vsα^2  +  
                    1/vsα^2))/(vsα^2  -  
                1/vsα^2);

vsfez[
          vszz_] =   -I vsV  / vsgd/
              2 * (Tanh[π (Conjugate [vszz] +  
                          vsw/2)/(2 vsgd)]  + 
Tanh[π ( 
                        vsw/2  -  Conjugate [vszz])/(2 vsgd)]) // 
          Chop ;

2.3.1	Derivation of the complex potential via the Cauchy-Riemann equations

First part:

ff1r  =  
      ArcTan[Tan[((π y)/(2  gd))] Tanh[π ((-x +  w/2)/(2 gd))]]

ff1rx  =  D[ff1r, x] // FullSimplify

ff1ry  =  D[ff1r, y] // FullSimplify

ff1i  =  Integrate[ff1rx, y]

ff1i  =  - Integrate[ff1ry, x]

fz1  =  ff1r  +  I ff1i

Second Part:

ff2r  =   ArcTan[
        Tan[((π y)/(2 gd))] Tanh[π (( x +  w/2)/(2 gd))]]

ff2rx  =  D[ff2r, x] // FullSimplify

ff2ry  =  D[ff2r, y] // FullSimplify

ff2i  =  Integrate[ff2rx, y]

ff2i  =  - Integrate[ff2ry, x]

fz2  =  ff2r  +  I ff2i

Combination of the two parts:

ftz  = (fz1  +  fz2)  // PowerExpand  // 
          Simplify;

gives:

hz  =  
        I*Log[Cosh[(( -(w/2) + π ( x  +   I y))/(gd 2))]]   -  
            Log[Cosh[(((w/2) + π ( x  +   I y))/(gd 2))]]  // 
          ComplexExpand;

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

vsneq = 
        20 ;          

    vsims  =  700;

vsneq  specifies the number of equipotential and field lines;  this value in turn specifies the step widths used for the
variables  U ( = vsdu) and  V = π/vsneq used in vscm. The range of the variable U is bounded by vslu  and 
by vsru. These values have been determined by trial and error. The horizontal plot range in all pictures is 
determined by vslx and vsrx. In some figures the vertical plot range is determined by vsmaE and multiples of it.

vslx  =  
          If[vsgd/vsw < 1, -vsw/3, -vsgd/2];;
vsrx  =  If[vsgd/vsw < 1, vsw/3, vsgd/2];;
vslu  =  6.  vsi[vslx];
vsru  =  6.  vsi[vsrx];
vsdu  =  Abs[vsru  -  vslu]/vsneq;
vsmaE  =  vsfez[0] // Im // Abs;

3.2 Field graphics

The user may want to improve the pictures drawn below. For this, he may try to obtain better values for 
vslu, vsru, vsdu, vsmaE. 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  vsp2, vsp1, vsp23, vscm. It is extremely commendable to save this  trial notebook
under a new name beforehand so that the notebook with the orginal specifications remains intact!

vsp2  computes the electric field at the inside of the upper electrode. The corresponding pictureis shown in figure 
vsp23.  vscm plots the equipotential and electric field lines.  vsp1 shows  the electric field at the bottom electrode. 
In the two vsp pictures the unit of the electric field strength is  kV/m  since in the definition of the functions  vsfe 
and vsfez in 2.2.4  D = vsgd is used, whose value is defined in mm in  1.1, while we use  MKSA units.

vstru  =  { Hue[0], AbsoluteThickness[2], 
          Line[{{-vsw/2, -10 vsmaE}, {-vsw/2, 
                10  vsmaE}}], 
Line[{{vsw/
                  2, -10 vsmaE}, {vsw/2, 
                10  vsmaE}}]};
vstrl  =  { Hue[0], 
          Line[{{-vsw/2, 
                0}, {-vsw/2, -vsmaE}}], 
Line[{{vsw/
                  2, 0}, {vsw/2, -vsmaE}}]};

vsp2  =  
        Plot[{vsfez[xx  +  I vsgd] // Im}, {xx, 2  vslx, 2  vsrx}, 
          DisplayFunction ->  Identity];
vsp1  =  
        Plot[{vsfez[xx ] // Im}, {xx, 2  vslx, 2  vsrx}, 
          DisplayFunction ->  Identity];

Print["Electric field distribution on upper electrode"]
\(\(\(vsp23\  = \ 
        Show[vsp2, DisplayFunction -> \ $DisplayFunction, 
\ 
          ImageSize\  -> \ vsims, 
          Epilog\  -> \ 
            vstru, \ \ TextStyle\  -> \ {FontSize\  -> \ 
                16}, 
AxesLabel -> {"x [mm]", 
              None}, \ 
          PlotLabel\  -> \ 
SequenceForm[\ 
              Subscript["E", "y"], \ "  [kV/m]"]];\)\(
\)
    \)\)
Print["Equipotential and electric field lines"]
\(\(\(vscm\  = \ 
        CartesianMap[
          vsf[#]\  &, {vslu, vsru, 
            vsdu}, {$MachineEpsilon, π \((1 - $MachineEpsilon)\), π\ \((1 - 2  $MachineEpsilon)\)/vsneq}, 
\ 
          ImageSize\  -> \ vsims\ , \ 
          PlotRange\  -> {2 {vslx, vsrx}, {\(-0.1\), 
                vsgd\ 1.3}}, 
AxesLabel -> {"x [mm]", "y [mm]"}, \ \ TextStyle\  -> \ {FontSize\  -> \ 
                16}, 
Epilog -> {Thickness[vstcb], 
              Line[2 {{vslx, 0}, {vsrx, 
                      0}}], 
Line[{{2\ vslx, 
                    vsgd}, {\(-vsw\)/2, vsgd}}], 
              Line[{{2\ vsrx, vsgd}, {vsw/2, vsgd}}], 
Hue[
                0], Line[{{\(-vsw\)/2, vsgd}, {vsw/2, 
                    vsgd}}], 
Text[
                SequenceForm["Φ = ", vsV, " Volt"], {0, 
                  1.15\ vsgd}]}];\)\(
\)
    \)\)
Print["Electric field distribution on lower electrode"]
Show[vsp1, Graphics[vstrl],  
        DisplayFunction ->  $DisplayFunction, 
 
        ImageSize  ->  
          vsims,   TextStyle  ->  {FontSize  ->  
              16}, 
AxesLabel -> {"x [mm]", 
            None},  PlotLabel  ->  
SequenceForm[ 
            Subscript["E", "y"],  "  [kV/m]"], 
        Epilog  ->  vstrl];

4.  Field and Potential at a Selected Point

Below the coordinates  (vsxp, vsyp )  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.

vsWp is the point  Wp  corresponding to  zp   computed by the inverse mapping function  W = (f^(-1))(z) = vsi.  
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.  
vspc (vsec) 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  vsfe in 2.2.4  
D = scgd is used, whose value is defined  in mm in 1.1, while we use  MKSA units.

vsxp  =  2 ; vsyp  = 8 ;

vsWp = vsi[vsxp + I vsyp]; 
Print[
        SequenceForm["At point (vsxp,vsyp) = (", vsxp, ", ", 
          vsyp, ") mm : ", "Potential = ", 
          N[Re[-I vsV/π vsWp]] // Chop, " V;"]];
Print[
        SequenceForm["             (", 
          Subscript["E", "x"], ",", 
          Subscript["E", "y"], ") = (", 
          N[Re[1000 vsfe[vsWp]]], ", ", 
          N[Im[1000 vsfe[vsWp]]] ") V/m "]];

    
vspc = 
        ParametricPlot[
          Evaluate[{Re[vsf[I Im[vsWp] + s]], 
              Im[vsf[I Im[vsWp] + s]]}], {s, -2, 4}, 
          PlotStyle -> Dashing[{ .01}], 
          DisplayFunction -> Identity];
vsec = 
        ParametricPlot[
          Evaluate[{Re[vsf[Re[vsWp] + I s]], Im[vsf[Re[vsWp] + I s]]}], {s, 
            0, Pi}, DisplayFunction -> Identity];
\(\(vspoi = 
        Show[vspc, vsec, AspectRatio -> Automatic, 
          DisplayFunction -> $DisplayFunction, ImageSize -> 600, 
          TextStyle -> {FontSize -> 16}, 
          AxesLabel -> {"x [mm]", "y [mm]"}, 
\ PlotRange\  -> {2 {vslx, vsrx}, {\(-0.1\), vsgd\ 1.3}}, 
          Epilog -> {Thickness[vstcb], 
              Line[2 {{vslx, 0}, {vsrx, 
                      0}}], 
Line[{{2\ vslx, 
                    vsgd}, {\(-vsw\)/2, vsgd}}], 
              Line[{{2\ vsrx, vsgd}, {vsw/2, vsgd}}], 
Hue[
                0], Line[{{\(-vsw\)/2, vsgd}, {vsw/2, 
                    vsgd}}], 
Text[
                SequenceForm["Φ = ", vsV, " Volt"], {0, 
                  1.16  vsgd}], AbsolutePointSize[4], 
              Point[{vsxp, vsyp}]}];\)\)

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  vsxtl contains the horizontal coordinate  of various 
 vertical trajectories. For these the quantity  I/qv  is displayed as a function of  y = vsyt  in the pictures.

The user may select the horizontal coordinates  x  of a set of vertical trajectories. These are given in the list
vsxtl  in units of w  =  vsw.

vsxtl  =  
        vsw  {0,  .25,   .5,   .75,  1,  1.25,  1.5};

\(\(vspli\  = \ 
        Table[Plot[\(-Im[vsfe[vsi[vsxtl[\([k]\)]\  + \ I\ vsyt]]]\)/
              vsV, {vsyt, 0, 
              vsgd}, \ 
PlotStyle\  -> \ {Hue[
                  0.1\ k], \ 
                Thickness[
                  0.005]}, 
PlotLabel\  -> \ \ SequenceForm["x = ", 
                vsxtl[\([k]\)], " mm"], 
AxesLabel\  -> \ \ {"y [mm]", "I/qv [1/mm]"}], {k, 
            Length[vsxtl]}];\)\)

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.

vstli  =  
        Table[{Hue[0.1 k], 
              Line[{{vsxtl[[k]], 0}, {vsxtl[[k]], vsgd}}]}, {k, 
              Length[vsxtl]}] // Flatten;
vswxl  =  
        Show[Graphics[Prepend[vstli, Thickness[0.005]]], 
          Graphics[{Thickness[vstcb], 
              Line[2 {{vslx, 0}, {vsw, 
                      0}}], 
Line[{{2 vslx, 
                    vsgd}, {-vsw/2, vsgd}}], 
              Line[{{2 vsw, vsgd}, {vsw/2, vsgd}}], 
Hue[
                0], Line[{{-vsw/2, vsgd}, {vsw/2, 
                    vsgd}}]}], 
Axes  ->  True,  
          ImageSize  ->  
            500,   TextStyle  ->  {FontSize  ->  
                16}];

vswcl  =  
        Show[vspli,  PlotLabel  ->  None,  
          ImageSize  ->  
            500,   TextStyle  ->  {FontSize  ->  
                16}];

6.   Current Induced on the Electrode for a Selected Trajectory

The user may select the horizontal coordinate  x = vsxs  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 function of the 
vertical position  y = vsyt  of the point charge, using W = (f^(-1))(z) = vsi  in the function vsfe for the complex electric field
introduced in  2.2.4.

vsxs  =  1;

Show[Graphics[{Thickness[0.005],  Line[{{vsxs, 0}, {vsxs, vsgd}}]}], 
        Graphics[{Thickness[vstcb], 
            Line[2 {{vslx, 0}, {vsrx, 
                    0}}], 
Line[{{2 vslx, 
                  vsgd}, {-vsw/2, vsgd}}], 
            Line[{{2 vsrx, vsgd}, {vsw/2, vsgd}}], 
Hue[
              0], Line[{{-vsw/2, vsgd}, {vsw/2, 
                  vsgd}}]}], 
Axes  ->  True,  
        ImageSize  ->  
          500,   TextStyle  ->  {FontSize  ->  
              16}];
vspls  =  
        Plot[- Im[vsfe[vsi[vsxs +  I vsyt]]]/vsV, {vsyt, 0, 
            vsgd},  
PlotStyle  ->  {Thickness[
                0.005]},  
          ImageSize  ->  
            500, 
PlotLabel  ->   SequenceForm["x \
= ", vsxs, " mm"], 
AxesLabel  ->   {"y \
[mm]", "I/qv [1/mm]"},   TextStyle  ->  {FontSize  ->  
                16}];

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 
electric field is largest, is shown in the picture below. The real and imaginary parts of the function  vsf[W] = 
vsf[u + I u] give  the position coordinates  x  and  y;  vsfe[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 vsabse  produces  |E|;  the operator Im[] in 
vscey leads to a plot for (Subscript[E, y]).  This is the weighting field in a case where the influencing charge is moving along the 
y-direction as is the case in RPC's.

vsabse = 
        ParametricPlot3D[
          Evaluate[{vsf[u  +  I v] // Re, vsf[u  +  I v] // Im, 
              Abs[1000 vsfe[u  +  I v]]}], {u, vslu, 
            vsru},  {v, $MachineEpsilon, Pi (1 - $MachineEpsilon)},  
          BoxRatios  ->  {1, 1,  .75}, 
          PlotRange  ->  {0, 
              0.15 10^5  vsmaE}, 
PlotPoints  ->  
            50, AxesLabel  -> {"x [mm]", "y [mm]", "|E| [V/m]\
"}, 
  TextStyle  ->  {FontSize  ->  
                16},  ImageSize  ->   700];

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

\(\(vsey = 
        ParametricPlot3D[
          Evaluate[{vsf[u\  + \ I\ v] // Re, vsf[u\  + \ I\ v] // Im, 
              Im[1000\ vsfe[u\  + \ I\ v]]}], {u, vslu, 
            vsru}, \ {v, $MachineEpsilon, Pi \((1 - $MachineEpsilon)\)}, \ 
          BoxRatios\  -> \ {1, 
              1,  .75}, 
PlotPoints\  -> \ 50, 
          AxesLabel\  -> {"x [mm]", "y [mm]", 
              SequenceForm[
                Subscript["E", "y"], " [V/m]"]}, 
\ \ TextStyle\  -> \ {FontSize\  -> \ 16}, \ 
          ImageSize\  -> \ 700];\)\)

The edges bounding the Voltage strip lead to infinite jumps of the electric field.

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.             Th.Heubrandtner,B. Schnizer, C.Lippmann,W.Riegler,
	Static electric fields in an infinite plane condensor with one or three homogeneous layers.
             Nucl.Instrum.Methods Phys.Res.,A:489 (2002) no.1-3,pp.439-443
                  (=CERN-EP-2002-004;Geneva:CERN,8 Jan 2002)
	(= SchzP80.pdf)

3a.          Th.Heubrandtner,B. Schnizer, C.Lippmann,W.Riegler,
	Static electric fields in an infinite plane condensor with one or three homogeneous layers.
           Report CERN-OPEN-2001-074,31 Oct.2001
	(= SchzP78.pdf)

4.             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.

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

6.              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

7.              H. Kober, Dictionary of Conformal Representations.
              Dover, 1957.