(* Near field @ edge ⟹ 0 < y ≤ d/100 or 0 < \(y\&^\) ≤ π/200 planar electrode array surface *)

d =. ; Vo =. ; d2 = 2*d; 
      xh =. ; yh =. ;

Ex = ((2*Vo)/(Pi*d))*(ArcTan[Sin[xh]/Sinh[yh]] - 
            ArcTan[Cos[xh]/Sinh[yh]])

Ey = 
      (Vo/(Pi*d))*
        Log[((Cosh[yh] + Cos[xh])/(Cosh[yh] - Cos[xh]))*((Cosh[yh] + Sin[xh])/(Cosh[yh] - Sin[xh]))]

Exx = ((2*Vo*
              Sinh[yh])/(d^2)) (Cos[xh]/(Cosh[2*yh] - Cos[2*xh]) + 
            Sin[xh]/(Cosh[2*yh] + Cos[2*xh]))

Exy = ((2*Vo*
              Cosh[yh])/(d^2)) (Cos[xh]/(Cosh[2*yh] + Cos[2*xh]) - 
            Sin[xh]/(Cosh[2*yh] - Cos[2*xh]))

Eyx = Exy; Eyy = -Exx;

grdE2x = 2*(Ex*Exx + Ey*Eyx);grdE2y = 2*(Ex*Exy + Ey*Eyy);

d = 10; Vo = 1; d2 = 2*d; y = 0.01*d ; xhmn = 0; 
    xhmx = 2*Pi;

yh = Pi*y/(2*d)

yh = π/200;  
    Plot[Ex, {xh, xhmn, xhmx}, PlotRange -> {-0.2, 0.2}]

yh = π/200; 
    Plot[Ey, {xh, xhmn, xhmx}, PlotRange -> {-0.4, 0.4}]

yh = π/200 ;Plot[Exx, {xh, xhmn, xhmx}, 
      PlotRange -> {-0.70, 
          0.70}]

yh = π/200; 
    Plot[Exy, {xh, xhmn, xhmx}, 
      PlotRange -> {-0.40, 0.40}]

yh = π/200; 
    Plot[grdE2x, {xh, xhmn, xhmx}, 
      PlotRange -> {-0.300, 0.300}]

yh = 2*π/200; 
    Plot[grdE2x, {xh, xhmn, xhmx}, 
      PlotRange -> {-0.150, 0.150}]

d = 10; Vo = 1; yh = π/200; 
    Plot[grdE2x, {xh, Pi/2 - Pi/20, Pi/2 + Pi/20}, 
      PlotRange -> {-0.300, 0.300}]

yh = 2*π/200; 
    Plot[grdE2x, {xh, Pi/2 - Pi/20, Pi/2 + Pi/20}, 
      PlotRange -> {-0.05, 0.12}]

yh = π/100;-FindMinimum[-grdE2x, {xh, 1.56}]

yh = π/100;FindMinimum[grdE2x, {xh, 1.61}]

yh = π/200; 
    Plot[grdE2y, {xh, xhmn, xhmx}, PlotPoints -> 200, 
      PlotRange -> {-0.450, 0.01}]

yh = π/100; 
    Plot[grdE2y, {xh, (π/2) - Pi/20, (π/2) + Pi/20}, 
      PlotRange -> {-0.20, 0.01}]

yh = π/100;FindMinimum[grdE2y, {xh, 1.5}]

(* NB :  
            The above value concurs with FlexPDE output for PhD thesis Fig  .6  .2  
              c in dyE2i wordpad file with minimum (∂_y) (E^2)= -0.176244 @ x=5.05 (or xh = 1.5786) & y=0.2 *)

Plx = 
      Plot3D[grdE2x, {xh, 0, 2*Pi}, {yh, Pi/200, Pi/20}, 
        PlotPoints -> {200, 40}]

y = d/100; d = 10; 
    Plot[grdE2y, {x, 0, 4*d}, PlotPoints -> 100, 
      PlotRange -> {-0.45, 0.010}]

Show[Plx, PlotRange -> {-0.3, 0.3} ]

Show[Plx, ViewPoint -> {1.2, 1.2, 1.2}, 
        PlotRange -> {-0.3, 0.3}];

<<Graphics`

<<Graphics`PlotField`

PlotVectorField[{Sin[x], Cos[y]},
             {x, 0, Pi}, {y, 0, Pi}]

yh =. ;xh =. ;Vo = 1;d = 10;PlotVectorField[{grdE2x, grdE2y}, {xh, (Pi/4), 
          (Pi/2) + (Pi/4)}, {yh, Pi/200, Pi/2}]
(* \(when\ \(x\&^\)\  = \ π\/4\), \(x\  = \ 0; \ \(x\&^\)\  = π\/2 + π\/4\), x  =  d so transverse plot from x  =  
                0 to d, y = (d/100) to d ☺ *)

xh =  π*x/(2*d)  +  π/4;   yh =  π*
        y/(2*d);

y = d/100; d = 10; Vo = 1; 
    Plot[grdE2x, {x, 0, 4*d}, 
      PlotRange -> {-0.300, 0.300}]

d = 10; Vo = 1; y =. ; 
    Plot3D[grdE2x, {x, 0, 4*d}, {y, d/100, d/10}, 
      PlotPoints -> {60, 40}, 
      PlotRange -> {-0.3, 0.3} ]

d = 10; Vo = 1; 
    Plot3D[grdE2y, {x, 0, 4*d}, {y, d/100, d/10}, PlotPoints -> 60, 
      PlotRange -> {-0.45, 0.01} ]