Effect of Finite Velocity

By Takuichi Hirano (Tokyo Institute of Technology)

Electric Dipole

a = 1. ;
η = 120. *π;
k = 2*π;
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_, y_, z_] := 
        Module[{r = (Sqrt[(x^2) + (y^2) + (z^2)]), θ = 
              ArcTan[z, (Sqrt[(x^2) + (y^2)])], φ = 
              ArcTan[x, y]}, 
{Er[r, θ]*
                Sin[θ]*Cos[φ] + 
              Eθ[r, θ]*Cos[θ]*
                Cos[φ], 
Er[r, θ]*
                Sin[θ]*Sin[φ] + 
              Eθ[r, θ]*Cos[θ]*Sin[φ], 
            Er[r, θ]*Cos[θ] - 
              Eθ[r, θ]*
                Sin[θ]}
];
h[x_, y_, z_] := 
        Module[{r = (Sqrt[(x^2) + (y^2) + (z^2)]), θ = 
              ArcTan[z, (Sqrt[(x^2) + (y^2)])], φ = 
              ArcTan[x, 
                y]}, 
{-Hφ[r, θ]*
              Sin[φ], Hφ[r, θ]*Cos[φ], 
            0}
];

<< Graphics`PlotField3D`

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]};
\(\(g1[d_, t_] := 
        PlotVectorField3D[
          Re[e[x, y, z]*Exp[I*ω*t]], {x, \(-0.5\), 0.5}, {y, \(-0.5\), 
            0.5}, {z, \(-0.5\), 
            0.5}, 
ImageSize -> {288, 
              288}, 
PlotRange -> {{\(-0.5\), 
                0.5}, {\(-0.5\), 0.5}, {\(-0.5\), 
                0.5}}, 
PlotPoints -> 
            10, 
ScaleFunction -> \((#\^0.3 &)\), 
ScaleFactor -> 
            0.1, 
ColorFunction -> \((RGBColor[1, 0, 
                  0] &)\), 
VectorHeads -> 
            True, 
DisplayFunction -> 
            Identity];\)\)
\(\(g2[d_, t_] := 
        PlotVectorField3D[
          Re[h[x, y, z]*Exp[I*ω*t]], {x, \(-0.5\), 0.5}, {y, \(-0.5\), 
            0.5}, {z, \(-0.5\), 
            0.5}, 
ImageSize -> {288, 
              288}, 
PlotRange -> {{\(-0.5\), 
                0.5}, {\(-0.5\), 0.5}, {\(-0.5\), 
                0.5}}, 
PlotPoints -> 
            10, 
ScaleFunction -> \((#\^0.3 &)\), 
ScaleFactor -> 
            0.1, 
ColorFunction -> \((RGBColor[0, 0, 
                  1] &)\), 
VectorHeads -> 
            True, 
DisplayFunction -> 
            Identity];\)\)
Show[{g1[d, 0], g2[d, 0]}, 
      DisplayFunction -> $DisplayFunction]

Infinitesimal Dipole

Table[
      Show[{g1[d, t], g2[d, t]}, 
        DisplayFunction -> $DisplayFunction], {t, 0, 1.  - 0.001, 
        1. /32}]