Infinitesimal Dipole

By Takuichi Hirano (Tokyo Institute of Technology)

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

c = 3*(10^8);
λ0 = 1;
k0 = ((2 π)/λ)0;
h = λ0/4;

Electric Field

\(\(\(r\&^\) . \(x\&^\) = 
        Sin[θ]\ Cos[φ];\)\)
\(\(\(θ\&^\) . \(x\&^\) = 
        Cos[θ]\ Cos[φ];\)\)
\(\(\(φ\&^\) . \(x\&^\) = 
        Sin[φ];\)
\)
\(\(\(r\&^\) . \(y\&^\) = 
        Sin[θ]\ Sin[φ];\)\)
\(\(\(θ\&^\) . \(y\&^\) = 
        Cos[θ]\ Sin[φ];\)\)
\(\(\(φ\&^\) . \(y\&^\) = 
        Cos[φ];\)
\)
\(\(\(r\&^\) . \(z\&^\) = Cos[θ];\)\)
\(\(\(θ\&^\) . \(z\&^\) = 
        Sin[θ];\)\)
\(\(\(φ\&^\) . \(z\&^\) = 0;\)\)

er[r_, θ_, t_] := 
        2 Cos[θ] (Cos[k0 (c t - r)]/(((k0*r))^2) + 
              Sin[k0 (c t - r)]/(((k0*r))^3));
etheta[r_, θ_, t_] := 
        Sin[θ] (Cos[k0 (c t - r)]/(((k0*r))^2) - ((1/((k0*r))) - (1/(((k0*r))^3))) Sin[
                  k0 (c t - r)]);
ex[r_, θ_, φ_, t_] := 
        Sin[θ] Cos[φ] er[r, θ, t] + 
          Cos[θ] Cos[φ] etheta[r, θ, 
              t];
ey[r_, θ_, φ_, t_] := 
        Sin[θ] Sin[φ] er[r, θ, t] + 
          Cos[θ] Sin[φ] etheta[r, θ, 
              t];
ez[r_, θ_, φ_, t_] := 
        Cos[θ] er[r, θ, t] + 
          Sin[θ] etheta[r, θ, t];
energy[x_, y_, z_, t_] := 
      Module[{r = (Sqrt[(x^2) + (y^2) + (z^2)]), θ = 
            ArcCos[(z/(Sqrt[(x^2) + (y^2) + (z^2)]))], φ = 
            ArcCos[(x/(Sqrt[(x^2) + (y^2)]))]}, 
ex[r, θ, φ, t]^2 + ey[r, θ, φ, t]^2 + 
          ez[r, θ, φ, t]^2
]

Table[
      DensityPlot[energy[0, x, y, t], {x, -2, 2}, {y, -2, 2}, 
        ColorFunction -> (Hue[-0.7*(# - 1), 1, 1] &), 
        Mesh -> False, FrameTicks -> None, PlotPoints -> 100, 
        Epilog -> {White, AbsoluteThickness[2], 
            Arrow[{0, -0.25}, {0, 0.25}]}], {t, 
        0, (π/(c k0)) - 
          0.001*(π/(c k0)), (π/(c k0))/
          8}]