Effect of Finite Velocity

By Takuichi Hirano (Tokyo Institute of Technology)

Electric Dipole

a = 1. ;
η = 120. *π;
k = 2*π;
φ = 0. ;
Er[
          r_, θ_] := η*(((k^2)*a*(E^(-I*k*r)))/(2*π))*((1/(((k*r))^2)) - (I/(((k*r))^3)))*
          Cos[θ];
Eθ[
          r_, θ_] := η*(((k^2)*a*(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)*a*(E^(-I*k*r)))/(4*π))*((I/(k*r)) + (1/(((k*r))^2)))*
          Sin[θ];
e[x_, z_] := 
        Module[{r = (Sqrt[(x^2) + (z^2)]), θ = 
              ArcTan[z, x]}, 
{Er[r, θ]*
                Sin[θ]*Cos[φ] + 
              Eθ[r, θ]*Cos[θ]*Cos[φ], 
            Er[r, θ]*Cos[θ] - 
              Eθ[r, θ]*
                Sin[θ]}
];

<< Graphics`PlotField`

d = 0.1;
λ = 1. ;
freq = 1. ;
ω = 2*π*freq;
k = ((2*π)/λ);
colfun[x_] := 
        Hue[-0.7*(x - 1), 1, 1];
charges[
          d_] := {AbsoluteThickness[2], {{Red, Disk[{0, d/2}, 0.1]}, 
            Circle[{0, d/2}, 0.1]}, 
{{Blue, 
              Disk[{0, -d/2}, 0.1]}}, 
          Circle[{0, -d/2}, 0.1]};
\(\(g[d_, t_] := 
        PlotVectorField[
          Re[e[x, y]*Exp[I*ω*t]], {x, \(-2\), 2}, {y, \(-2\), 
            2}, 
ImageSize -> {288, 
              288}, 
PlotRange -> {{\(-2\), 
                2}, {\(-2\), 2}}, 
PlotPoints -> 
            24, 
ScaleFunction -> \((#\^0.3 &)\), 
ScaleFactor -> 
            0.3, 
ColorFunction -> \((RGBColor[0, 0, 
                  0] &)\), 
Epilog -> {charges[
                d]}, 
DisplayFunction -> 
            Identity];\)\)
Show[g[d, 0], DisplayFunction -> $DisplayFunction]

Electric Dipole

Map[
      Show[g[#[[1]], #[[2]]], 
          Graphics[{RGBColor[0, 1, 1], 
              Table[Circle[{0, 0}, r], {r, 
                  Mod[#[[2]] + λ/4, λ/2], 
                  4*λ, λ/
                    2}]}], 
DisplayFunction -> $DisplayFunction] &, 
      Table[{-0.2*
            Cos[x - π/7], (x - π/7)/(2*π)}, {x, π/
            7, π/7 + 2*π - 0.001, π/8}]]