plotting the shade of a 3D shape on a plain | help

Question

as part of my electromagnetic waves course I want to plot all of the fields in order to better understand the material, but I ran into a few problems in the process. for example:

f1[x] = x^2
f2[y] = -y^2
Coulomb = {x^2, -y^2, 0}
Plot3D[x^2 - y^2, {x, -4, 4}, {y, -4, 4}]
VectorPlot[{x^2, -y^2}, {x, -3, 3}, {y, -3, 3}, PlotLegends -> Automatic]
VectorPlot3D[Evaluate[Coulomb],{x,-1,1},{y,-1,1},{z,-1,1}, VectorPoints -> Coarse, VectorScale -> Medium, 
 VectorStyle -> "LeftArrow3D"]

ECoulomb2 = {$\frac{qd*Cos[\Theta]}{2 \Pi r^3}$, $\frac{qd*Sin[\Theta]}{4 \Pi r^3}$, 0}

Ecartesian2 = TransformedField["Spherical" -> "Cartesian", Ecoulomb2, {r,$\Theta$,$\Phi$} -> {x,y,z}]

all runs smoothly and I'm getting the desired outputs. but when I try to create the following outputs I get empty graphs on the other end:

SliceContourPlot3D[Ecartesian2,"CenterPlanes", {x, -2, 2}, {y, -2, 2}, {z, -2, 2}]
VectorPlot[Ecartesian2, {x,-3,3}, {y,-3,3}, PlotLegends -> Automatic]
VectorPlot3D[Evaluate[Ecartesian2], {x,-1,1}, {y,-1,1}, {z,-1,1}, VectorPoints -> Coarse, 
 VectorScale -> Medium, VectorStyle -> "LeftArrow3D"]

I tried to leave ECoulomb2 in {r,$\Theta$,$\Phi$} coordinates and plot it in a 2 variable plain, depending only on r and $\Theta$ but still getting an empty graph.

what will be the difference between plotting it in {r,$\Theta$,$\Phi$} and {x,y,z} coordinates?

I know the output supposed to looks something like this:

Answer 1

Make sure you write out your input properly. You're using capital Pi $\Pi$ not lowercase $\pi$ i.e 3.1415... so this should be:

ECoulomb2 = {q*d*Cos[Θ] 2 π r^3, q*d*Sin[Θ] 4 π r^3, 0};

And in your conversion you made a typo: Ecoulomb2 should be ECoulomb2.

The plots will still not work because you didn't provide values for d or q, but if I give these values of 1, then I get the VectorPlot3D working.

The 2D VectorPlot doesn't work however and will be all white because your Ecartesian2 has three coordinates in a 2D plot, and you're also plotting the wrong coordinates. Try this instead:

q = 1;
d = 1;

ECoulomb2 = {q*d*Cos[Θ] 2 π r^3, q*d*Sin[Θ] 4 π r^3, 0};

Ecartesian2 = 
 TransformedField["Spherical" -> "Cartesian", ECoulomb2, {r, Θ, Φ} -> {x, y, z}]

(* here I'm taking the x,z coordinates {1,3} only and setting y to 0.25 *)
VectorPlot[Ecartesian2[[{1, 3}]] /. y -> 0.25, {x, -3, 3}, {z, -3, 3},
  PlotLegends -> Automatic]

VectorPlot3D[
 Evaluate[Ecartesian2], {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, 
 VectorPoints -> Coarse, VectorScale -> Medium, 
 VectorStyle -> "LeftArrow3D"]