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

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

ThetaT = 
      ArcTan[Sin[xh]/Sinh[yh]] - ArcTan[Cos[xh]/Sinh[yh]]

Ex = ((2*Vo)/(Pi*d))*ThetaT

ThetaL = 
      Log[((Cosh[yh] + Cos[xh])/(Cosh[yh] - Cos[xh]))*((Cosh[yh] + Sin[xh])/(Cosh[yh] - Sin[xh]))]

Ey = (Vo/(Pi*d))*ThetaL

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);

(Subscript[Γ, x]) = ((π (d^3))/(4 (Vo^2)))*grdE2x;(Subscript[Γ, y]) = ((π (d^3))/(4 (Vo^2)))*grdE2y;

d = 1; 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 -> {-2, 2}]

yh = π/200; 
    Plot[Ey, {xh, xhmn, xhmx}, PlotRange -> {-4, 4}]

yh =. ; ThetaSq = 4*(ThetaT^2) + (ThetaL^2)

yh = π/200; 
    Plot[ThetaSq, {xh, xhmn, xhmx}, PlotRange -> {0, 110}]

yh = π/200; 
    Plot[ThetaSq, {xh, (π/4), ((3*π)/4)}, 
      PlotRange -> {0, 110}]

yh = π/200;-FindMinimum[-ThetaSq, {xh, 1.56}]

yh = π/200;Plot[Exx, {xh, xhmn, xhmx}, 
      PlotRange -> {-70, 70}]

yh = π/200; 
    Plot[Exy, {xh, xhmn, xhmx}, PlotRange -> {-40, 40}]

yh = π/200; 
    Plot[grdE2x, {xh, xhmn, xhmx}, 
      PlotRange -> {-300, 300}]

yh = π/200; 
    Plot[grdE2x, {xh, Pi/2 - Pi/20, Pi/2 + Pi/20}, 
      PlotRange -> {-300, 300}]

yh = π/200;-FindMinimum[-(Subscript[Γ, x]), {xh, 1.56}]

yh = π/200; 
    Plot[grdE2x, {xh, -Pi/100, Pi/100}, GridLines -> Automatic, 
      PlotRange -> {-300, 300}]

yh = π/200; 
    Plot[(π/4) grdE2x, {xh, -Pi/100, Pi/100}, 
      PlotRange -> {-300, 300}]

yh = π/200; 
    Plot[(Subscript[Γ, x]), {xh, -Pi/100, Pi/100}, 
      GridLines -> Automatic, 
      PlotRange -> {-300, 300}]

yh = π/200;FindMinimum[(Subscript[Γ, x]), {xh, 0.01}]

(* conclusion :  
          as anticipated,  \[VerticalSeparator]local maxima\[VerticalSeparator]  =  \[VerticalSeparator]local minima\[VerticalSeparator] *)

yh = π/20000; 
    Plot[grdE2x, {xh, -0.0001, 0.003}, 
      PlotRange -> {-50000, 10000}]

yh = π/200; 
    Plot[grdE2y, {xh, xhmn, xhmx}, 
      PlotRange -> {-450, 1}]

yh = π/200; 
    Plot[grdE2y, {xh, -Pi/20, Pi/20}, 
      PlotRange -> {-450, 1}]

yh = π/200; 
    Plot[(Subscript[Γ, y]), {xh, -Pi/20, Pi/20}, 
      GridLines -> Automatic, 
      PlotRange -> {-450, 1}]

yh = π/200;FindMinimum[(Subscript[Γ, y]), {xh, 0}]

yh = π/200;FindMinimum[(Subscript[Γ, y]), {xh, 3}]

yh = π/20; 
    Plot[Exx*Ey, {xh, xhmn, xhmx}, PlotRange -> {-1, 11}]

yh = π/20; 
    Plot[Eyx*Ex, {xh, xhmn, xhmx}, PlotRange -> {-7, 2}]

yh =. ; d =. ; Vo =. ;

ExS = 
      FullSimplify[
        Series[Ex, {xh, 0, 3}, {yh, 0, 3}] /. {xh -> 0}]

EyS = 
      FullSimplify[
        Series[Ey, {xh, 0, 1}, {yh, 0, 3}] /. {xh -> 0}]

ExxS = 
      FullSimplify[
        Series[Exx, {xh, 0, 3}, {yh, 0, 3}] /. {xh -> 0}]

ExyS = 
      FullSimplify[
        Series[Exy, {xh, 0, 3}, {yh, 0, 4}] /. {xh -> 0}]

EyxS = ExyS;  EyyS = -ExxS

grdE2xS = 
      FullSimplify[
        Series[grdE2x, {xh, 0, 1}, {yh, 0, 2}] /. {xh -> 0}]

Vo =. ;d =. ;yh =. ;grdE2xS1 = -((2 (Vo^2))/((d^3) yh))

grdE2xS2 = -((2 (Vo^2))/((d^3) yh)) + ((2 (Vo^2) (2 + Log[4] - 2 Log[yh]))/((d^3) π))

xh = 0;Vo = 1;d = 1;Plot[{grdE2x,grdE2xS1},{yh, Pi/2000, Pi/200},PlotRange -> {-1500, -100},PlotStyle -> {{RGBColor[1, 0, 0]}, {RGBColor[0, 0, 
                  1]} }]

yh = π/2000; 
    Plot[grdE2x, {xh, -0.001, 0.003}, 
      PlotRange -> {-4000, 100}]

(* grdE2xS2 approxmtn is NOT quoted in Thesis ch6  .3; the plot ↓ shows it overlaps grdE2x @ xh=0 *)

xh = 0;Vo = 1;d = 1;Plot[{grdE2x,grdE2xS1,grdE2xS2},{yh, Pi/2000, Pi/20},PlotRange -> {-1500, -10},PlotStyle -> {{RGBColor[1, 0, 0]}, {RGBColor[0, 0, 
                  1]} , {RGBColor[0, 1, 0]} }]

grdE2yS = 
      FullSimplify[
        Series[grdE2y, {xh, 0, 1}, {yh, 0, 1}] /. {xh -> 0}]

yh =. ; 
    grdE2yS1 = ((4 (Vo^2) (Log[yh/2]))/(((d^3) π) yh))

xh = 0;Vo = 1;d = 1;Plot[{grdE2y,grdE2yS1},{yh, Pi/2000, Pi/200},PlotRange -> {-6000, -100},PlotStyle -> {{RGBColor[1, 0, 0]}, {RGBColor[0, 0, 
                  1]} }]

yh = π/2000; 
    Plot[grdE2y, {xh, -0.001, 0.003}, 
      PlotRange -> {-10000, 100}]

xh =. ;  Vo =. ; d =. ; E2 = Ex^2 + Ey^2

xh = 0;  Vo =. ; d =. ; E2x0 = Ex^2 + Ey^2

E2S = FullSimplify[ExS^2  +  EyS^2]

E2S1 = ((((Vo^2) (1 + ((((Log[4] - 2 Log[yh]))^2)/(π^2))))/(d^2)) - ((4 (Vo^2) yh)/((d^2) π)))

xh = 0;  Vo =. ; d =. ; yh =. ; 
    E2S2 = ((((Vo^2) (1 + ((4 Log[2/yh]^2)/(π^2))))/(d^2)))

xh = 0;Vo = 1;d = 1;Plot[{E2,E2S2},{yh, Pi/2000, Pi/20},PlotRange -> {0, 25},PlotStyle -> {{RGBColor[1, 0, 0]}, {RGBColor[0, 0, 
                  1]} }]

yh = π/200; 
    Plot[E2, {xh, -Pi/100, Pi/100}, PlotRange -> {0, 25}]

yh = π/2000; 
    Plot[E2, {xh, -Pi/400, Pi/400}, PlotRange -> {0, 25}]

yh = π/2000; -FindMinimum[-E2, {xh, 0}]

yh = π/2000; 
    Plot[E2, {xh, -Pi/4, Pi/4}, PlotRange -> {0, 25}]

yh = Pi/200; Vo =. ; 
    FullSimplify[(6.73*10^-10)*E2S2 /. 
        d -> 10^-5]

d = 1; Vo = 1; 
    Plot3D[E2, {xh, -Pi/4, Pi/4}, {yh, Pi/2000, Pi/2}, 
      PlotPoints -> 61, PlotRange -> {0, 25} ]

Vo =. ; xh =. ; yh =. ;

pdf = Exp[(6.73/1)*E2]

(* assumes d  =  10 microns,  
        kf/kBT  =  6.73*10^-10 see Book III p    *)

xh = 0;  yh = Pi/200; Vo = 1; pdf = Exp[(6.73/1)*E2]

xh = 0;  yh = Pi/2; Vo = 1; pdf = Exp[(6.73/1)*E2]

xh =. ; yh =. ; Vo = 0.2; 
    Plot3D[pdf, {xh, -Pi/4, Pi/4}, {yh, Pi/200, Pi/2}, 
      PlotPoints -> 21, PlotRange -> {0, 2200000} ]

Exp[71*Vo^2]

d = 1; Vo = 1; 
    Plot3D[E2, {xh, -Pi/4, Pi/4}, {yh, Pi/200, Pi/2}, 
      PlotPoints -> 61, PlotRange -> {0, 11} ]