Reading Hertz's Own Dipole Theory

Manipulate[Switch[plfun,Plot3D,If[shena===True,Plot3D[(r^2/(r^2+z^2))*(Cos[2.*Pi*(Paramet-Sqrt[r^2+z^2])]+Sin[2.*Pi*(Paramet-Sqrt[r^2+z^2])]/(2.*Pi*Sqrt[r^2+z^2])),{r,-2,2},{z,-2,2},PlotRange->{-h,h},PlotPoints->pp,BaseStyle->Opacity[opacity],Boxed->boxing,Axes->boxing,Mesh->mesh,ColorFunction->shading,ImageSize->{400,400}],Plot3D[(r^2/(r^2+z^2))*(Cos[2.*Pi*(Paramet-Sqrt[r^2+z^2])]+Sin[2.*Pi*(Paramet-Sqrt[r^2+z^2])]/(2.*Pi*Sqrt[r^2+z^2])),{r,-2,2},{z,-2,2},PlotRange->{-h,h},PlotPoints->pp,BaseStyle->Opacity[opacity],Boxed->boxing,Axes->boxing,Mesh->mesh,PlotStyle->None,ImageSize->{400,400}]],ContourPlot,If[shena===True,ContourPlot[(r^2/(r^2+z^2))*(Cos[2.*Pi*(Paramet-Sqrt[r^2+z^2])]+Sin[2.*Pi*(Paramet-Sqrt[r^2+z^2])]/(2.*Pi*Sqrt[r^2+z^2])),{r,-2,2},{z,-2,2},PlotRange->{rr,zz},ContourStyle->cont,ColorFunction->shading,Frame->boxing,PlotPoints->pp,ColorFunction->shading,Contours->ControlActive[5,NumCont],PerformanceGoal->"Speed",MaxRecursion->ControlActive[0,2],PlotPoints->ControlActive[15,30],ImageSize->{400,400}],ContourPlot[(r^2/(r^2+z^2))*(Cos[2.*Pi*(Paramet-Sqrt[r^2+z^2])]+Sin[2.*Pi*(Paramet-Sqrt[r^2+z^2])]/(2.*Pi*Sqrt[r^2+z^2])),{r,-2,2},{z,-2,2},PlotRange->{rr,zz},ContourStyle->cont,ColorFunction->shading,Frame->boxing,PlotPoints->pp,ContourShading->None,Contours->ControlActive[5,NumCont],PerformanceGoal->"Speed",MaxRecursion->ControlActive[0,2],PlotPoints->ControlActive[15,30],ImageSize->{400,400}]],DensityPlot,DensityPlot[(r^2/(r^2+z^2))*(Cos[2.*Pi*(Paramet-Sqrt[r^2+z^2])]+Sin[2.*Pi*(Paramet-Sqrt[r^2+z^2])]/(2.*Pi*Sqrt[r^2+z^2])),{r,-2,2},{z,-2,2},PlotRange->{rr,zz},Frame->boxing,PlotPoints->pp,ColorFunction->shading,PerformanceGoal->"Speed",MaxRecursion->ControlActive[0,2],PlotPoints->ControlActive[15,30],ImageSize->{400,400}]],
{{plfun,ContourPlot,"plot type"},{DensityPlot->"density",ContourPlot->"contour", Plot3D->"3D plot"},ControlType->PopupMenu,ImageSize->Small},
{{Paramet,0,"normalized time"},0,1,.0625,AnimationRate->0.02,ImageSize->Small},
{{pp,20,"plot point density"},5,50,1,ImageSize->Small},
{{NumCont,17,"contour density"},5,30,1,Enabled->(plfun===ContourPlot),ImageSize->Small},
{{h,2,"3D peak height"},0.1,3,0.1,Enabled->(plfun===Plot3D),ImageSize->Small},
{{opacity,1,"3D opacity"},1,0,0.1,Enabled->(plfun===Plot3D),ImageSize->Small},
{{boxing,False,"boxing & axes"},{True,False},ImageSize->Small},
{{shena,False,"shading"},{True,False},Enabled->(plfun===ContourPlot || plfun===Plot3D),ImageSize->Small},
{{shading,Automatic,"shading style"},{Automatic,ColorData["ThermometerColors"]->"Thermometer",ColorData["Aquamarine"]->"Aquamarine",ColorData["AvocadoColors"]->"Avocado",ColorData["NeonColors"]->"Neon",ColorData["StarryNightColors"]->"StarryNight"},Enabled->(shena==True|| plfun===DensityPlot),ImageSize->Small},
{{cont,Automatic,"contour style"},{Automatic->"Automatic",None->"None",Dashed->"Dashed"},ControlType->PopupMenu, Enabled->(plfun===ContourPlot),ImageSize->Small},
{{mesh,Automatic,"mesh style"},{Automatic,Full,None,5,4,3,2,1},Enabled->(plfun===Plot3D),ImageSize->Small},
{{rr,{-2,2},"2D plot range r"},{-2,-2},{2,2},Enabled->(plfun===ContourPlot || plfun===DensityPlot),ImageSize->Small},{{zz,{-2,2},"2D plot range z"},{-2,-2},{2,2},Enabled->(plfun===ContourPlot || plfun===DensityPlot),ImageSize->Small},ControlPlacement->Left,AutorunSequencing->{1,2,10,13}]

CAPTION

The plot depicts the time evolution of the lines of force of the electric field around a Hertzian dipole. It straightforwardly visualizes the function (Subscript[Q, n])=(sin^2)θ(cos Φ+(1/(k R))sin Φ) (see Details for further information).

Thus it recomputes Hertz's famous figures that he had created for several instants of time in the interval [0,(3/8)T], T being the period of short dipole electric oscillations. The lines of force are plotted in the z-r plane in a square of size approximately (3/2)λ×(3/2)λ, where λ is the corresponding wavelength. Note that the reader may reproduce questionable ranges (i.e. at t=0.395T) of Hertz’s diagrams, including the self-intersecting lines of force.

DETAILS

Further information: Reading Hertz's Own Dipole Theory.

THIS NOTEBOOK IS THE SOURCE CODE FROM

"Reading Hertz's Own Dipole Theory" from the Wolfram Demonstrations Project
 http://demonstrations.wolfram.com/ReadingHertzsOwnDipoleTheory/

Contributed by: M V Babovic and V M Babovic

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