Summary

Mathematica 12.0 code

c = 1.; \[Omega] = 5; d = 0.1; \[Epsilon]0 = 1; q1 = 1; q2 = -1; (*Some constants*)

vd[t_] := {0, 0, \[Omega] d Cos[\[Omega] t]} (*the charge oscillates up and down (this is its velocity) *)

normr = Sqrt[x^2 + y^2 + z^2]; (*just the norm of the position to make the equations easier to type*)
(*Two scalar potentials and two vector potentials, with opposite charges and velocities, to make a dipole*)
\[CurlyPhi]1 =   1/(4 \[Pi] \[Epsilon]0) q1/(normr - vd[t - normr/c].{x, y, z}/c);
\[CurlyPhi]2 =   1/(4 \[Pi] \[Epsilon]0) q2/(normr - ((-1)*vd[t - normr/c].{x, y, z})/c);
A1 = vd[t - normr/c]/c^2 \[CurlyPhi]1;
A2 = ((-1)*vd[t - normr/c])/c^2 \[CurlyPhi]2;

vm = {v, 0, 0}; (*The dipole is moving along x at speed v*)
\[Gamma] =1/Sqrt[1 - v^2/c^2];
(*Let's Lorentz-transform all the potentials and the coordinates*)
\[CurlyPhi]1m = \[Gamma] \[CurlyPhi]1;
\[CurlyPhi]2m = \[Gamma] \[CurlyPhi]2;
A1m = ({\[Gamma] (A1[[1]] - v/c^2 \[CurlyPhi]1m), A1[[2]], A1[[3]]});
A2m = ({\[Gamma] (A2[[1]] - v/c^2 \[CurlyPhi]2m), A2[[2]], A2[[3]]});
\[CurlyPhi]1m = \[CurlyPhi]1m /. {t -> \[Gamma] (t - v/c^2 x), x -> \[Gamma] (x - v t)};
\[CurlyPhi]2m = \[CurlyPhi]2m /. {t -> \[Gamma] (t - v/c^2 x), x -> \[Gamma] (x - v t)};
A1m = A1m /. {t -> \[Gamma] (t - v/c^2 x), x -> \[Gamma] (x - v t)};
A2m = A2m /. {t -> \[Gamma] (t - v/c^2 x), x -> \[Gamma] (x - v t)};

(*Calculate the two fields and their sum (and take a cut along y=0 so we can plot it*)
Efield1m = -Grad[\[CurlyPhi]1m, {x, y, z}] - D[A1m, t];
Efield2m = -Grad[\[CurlyPhi]2m, {x, y, z}] - D[A2m, t];
totEm = Norm[Efield1m + Efield2m] /. {y -> 0};

(*Define a smooth ramp to gradually increase the dipole velocity*)
smoothramp[kink1x_, kink2y_, slope_, \[Epsilon]_, x_] := (slope) \[Epsilon] Log[1 + E^((x - kink1x)/\[Epsilon])] - (slope) \[Epsilon] Log[1 + E^((-kink1x + x)/\[Epsilon])/(-1 + E^(kink2y/(slope \[Epsilon])))];
T = (2 \[Pi])/\[Omega]//N; (*Calculate the dipole oscillation period, so we can make sure we sample often enough in time*)

(*Plot everything*)
v =.
frames = Table[
   v = smoothramp[5 T, 0.75, 0.02, 1, \[Tau]];
   Grid[{{
      DensityPlot[
       totEm /. {t -> \[Tau]}, {x, -10 + v*\[Tau], 10 + v*\[Tau]}, {z, -10, 10}, PlotPoints -> 100, PlotRange -> {{-10 + v*\[Tau], 10 + v*\[Tau]}, {-10, 10}, {-0.01, 0.5}}, ColorFunction -> "AvocadoColors", 
       Frame -> False, PlotLegends -> BarLegend["AvocadoColors", LegendLabel -> "|\!\(\*OverscriptBox[\(E\), \(\[Rule]\)]\)|", LabelStyle -> {Black, Bold}, LegendMarkerSize -> 200, Ticks -> None], ImageSize -> 300 ]
      ,
      Plot[
       totEm /. {t -> \[Tau], z -> 0}, {x, -10 + v*\[Tau], 10 + v*\[Tau]}, PlotRange -> {{-10 + v*\[Tau], 10 + v*\[Tau]}, {-0.01, 0.5}}, PlotStyle -> {Thick, Black}, Axes -> False, Frame -> True, ImageSize -> 300, FrameTicks -> None, FrameLabel -> {"x", "|\!\(\*OverscriptBox[\(E\), \(\[Rule]\)]\)|"}, LabelStyle -> {Black, Bold}, PlotLabel -> Text[Style[StringForm["v=``c", NumberForm[v, {3, 2}]], Black, Bold]]]
      }}]
   , {\[Tau], 0., 31 T, 0.1 T}];
ListAnimate[frames, 10]

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