Electric Lines of Force of Infinitesimal Dipole
微小ダイポールから放射される電磁界の電気力線

By Takuichi Hirano (Tokyo Institute of Technology)

<< Graphics`Graphics`;
<< Graphics`Arrow`;

Infinitesimal Dipole
微小ダイポール

η = 120*π;    
k = 2*π;    
Er[
          r_, θ_] := η (((k^2) I ℓ (E^(-I k r)))/(2 π)) ((1/(((k r))^2)) - 
              (I/(((k r))^3))) Cos[θ];
Eθ[
          r_, θ_] := η (((k^2) I ℓ (E^(-I k r)))/(4 π)) ((I/(k r)) + 
              (1/(((k r))^2)) - 
              (I/(((k r))^3))) Sin[θ];
Hφ[
          r_, θ_] := η (((k^2) I ℓ (E^(-I k r)))/(4 π)) ((I/(k r)) + 
              (1/(((k r))^2))) Sin[θ];

\(TextForm\`𝔼 = \(r\&⋀\)\ E\_r + \(θ\&⋀\)\ E\_θ\)

Prepare Animation
アニメーションの準備

R = k r
T = ω t
K = η  (((k^2) I ℓ)/(4 π))
By the variable transformation above
Er (e^(I T)) = 
      2 K (e^(I (T - R))) ((1/(R^2)) - 
            (I/(R^3))) Cos[θ]
Eθ  (e^(I T)) = 
      K (e^(I (T - R))) ((1/R) + (I/(R^2)) - 
            (I/(R^3))) Sin[θ]
\(TextForm\`Peak\ expression\ of\ the\ electric\ field\ is\ written\ as\)
(Subscript[e, R]) = 
      2 K Cos[θ]  (Cos[T - R]/(R^2) + 
            Sin[T - R]/(R^3))
(Subscript[e, θ]) = 
      K Sin[θ] [
          Cos[T - R]/(R^2) - ((1/R) - (1/(R^3))) Sin[T - R]]
Differential eq .  for the electric field is
(dR/(Subscript[e, R])) = ((R dθ)/(Subscript[e, θ]))
\(TextForm\`\(\((1 - 1\/R\^2)\)\ Sin[T - R] + Cos[T - R]\/R\)\/\(Cos[T - R] + Sin[T - R]\/R\)\ dR = 
      2\ Cos[θ]\/Sin[θ]\ dθ\ \ \ \ \ \ \ \ \((Variables\ separable)\)\)
\(TextForm\`Then, \ 
    integrate\ this\ equation . 
\(-Log[Cos[T - R] + Sin[T - R]\/R]\) = 
      2\ Log[\ Sin[θ]\ ]\  + C'\)
\(TextForm\`Finnary, 
\([Cos[R - T] + 
            Sin[R - T]\/R]\)\ \ \((Sin[θ])\)\^2 = \(\(const\)\(.\)\)\)

Animation of Electric Lines of Force
(ContourPlot)

l = 0.2;
dip = {RGBColor[1, 0, 0], 
          Arrow[{0, -l/2}, {0, l/2}]};
ncell = 8;
k0 = 2*π;
R[x_, y_] := k0 (Sqrt[(x^2) + (y^2)]);
θ[x_, y_] := ArcTan[(x/y)];
contrs = Table[c, {c, -1.7, 1.7, 0.2}];
\(anim = 
      Table[\ ContourPlot[\((Cos[R[x, y] - T] + 
                Sin[R[x, y] - T]\/R[x, y])\)\ \((\ Sin[\ θ[x, y]\ ]\ )\)\^2, {x, \(-0.16\), 0.16}, {y, \(-0.16\), 
            0.16}, 
PlotPoints -> 
            100, 
Contours -> 
            contrs, 
ContourShading -> 
            False, 
ImageSize -> {288, 
              288}, 
FrameTicks -> 
            None, 
Epilog -> dip], 
	{T, 
          0, π - π/ncell, π/ncell}]\)