I am trying to plot the electric field intensity and vector directionality of a simple line charge. I tried plotting with StreamDensityPlot, however, the intensity (that is, the density component of the StreamDensityPlot) comes back as single color, while the stream lines are faint and difficult to see. I have tried changing the color of the density map and the streamlines but nothing changes the fact that the density plot comes back as a single uniform color. Is this a glitch with StreamDensityPlot?
Here is the code that generates the electric field of the line charge:
ClearAll;
(* Constants *)
qe = 1.6023*(10^-19) (* [C] *);
hbar = 1.055*(10^-34) (* [J s] *);
me = 9.109*(10^-31) (* [J s^2 m^-2] *);
\[Epsilon]0 = 8.854*(10^-12) (* [C V^-1] *);
ke = (4*\[Pi]*\[Epsilon]0)^-1 (* [V C^-1] *);
(* Electric Fields: Theory *)
Fprimex[x_, y_, z_, \[Lambda]_, L_] =
ke*(x*\[Lambda])/(x^2 + y^2 + (z - zprime)^2)^(3/2);
Fprimey[x_, y_, z_, \[Lambda]_, L_] =
ke*(y*\[Lambda])/(x^2 + y^2 + (z - zprime)^2)^(3/2);
Fprimez[x_, y_, z_, \[Lambda]_, L_] =
ke*((z - zprime)*\[Lambda])/(x^2 + y^2 + (z - zprime)^2)^(3/2);
Fx[x_, y_, z_, \[Lambda]_, L_] = Integrate[
Fprimex[x, y, z, \[Lambda], L],
{zprime, -(L/2), L/2},
Assumptions -> {
-(L/2) <= zprime <= L/2,
\[Epsilon] > 0,
L > 0,
x \[Element] Reals,
y \[Element] Reals,
z > Abs[L/2],
\[Lambda] \[Element] Reals
}
];
Fy[x_, y_, z_, \[Lambda]_, L_] = Integrate[
Fprimey[x, y, z, \[Lambda], L],
{zprime, -(L/2), L/2},
Assumptions -> {
-(L/2) <= zprime <= L/2,
\[Epsilon] > 0,
L > 0,
x \[Element] Reals,
y \[Element] Reals,
z > Abs[L/2],
\[Lambda] \[Element] Reals
}
];
Fz[x_, y_, z_, \[Lambda]_, L_] = Integrate[
Fprimez[x, y, z, \[Lambda], L],
{zprime, -(L/2), L/2},
Assumptions -> {
-(L/2) <= zprime <= L/2,
\[Epsilon] > 0,
L > 0,
x \[Element] Reals,
y \[Element] Reals,
z > Abs[L/2],
\[Lambda] \[Element] Reals
}
];
(* Longitudinal Electric Field *)
Ez[\[Rho]_, z_, \[Lambda]_, L_] =
Simplify[
Fz[x, y, z, \[Lambda], L] /. { x -> \[Rho]*Cos[\[Phi]],
y -> \[Rho]*Sin[\[Phi]] } ];
(* Radial Electric Field *)
E\[Rho][\[Rho]_, z_, \[Lambda]_, L_] =
Simplify[
Sqrt[Fx[x, y, z, \[Lambda], L]^2 + Fy[x, y, z, \[Lambda], L]^2] /. {
x -> \[Rho]*Cos[\[Phi]], y -> \[Rho]*Sin[\[Phi]] } ];
Here is the StreamDensityPlot code:
Manipulate[
Show[
(* Probabilities *)
StreamDensityPlot[
{
Evaluate[ E\[Rho][\[Rho], z, \[Lambda], L] ],
Evaluate[ Ez[\[Rho], z, \[Lambda], L] ]
},
{ \[Rho], 10^-12, R },
{ z, -Z, Z },
PlotRange -> { {-R, R}, {-Z, Z} },
ImageSize -> Large,
Frame -> True,
FrameLabel -> {
Style["Radial Position [cm]", 20, FontFamily -> "Times"],
Style["Longitudinal Position [cm]", 20, FontFamily -> "Times"] },
BaseStyle -> {FontFamily -> "Times", FontSize -> 20},
AxesOrigin -> {0, 0},
Axes -> True,
PerformanceGoal -> "Quality",
ColorFunction -> "TemperatureMap",
StreamColorFunction -> Directive[ RGBColor[1, 1, 1] ],
MaxRecursion -> 4
],
StreamDensityPlot[
{
-E\[Rho][\[Rho], z, \[Lambda], L], Ez[\[Rho], z, \[Lambda], L]
},
{ \[Rho], -R, -10^-12 },
{ z, -Z, Z },
PlotRange -> { {-R, -10^-12}, {-Z, Z} },
ColorFunction -> "TemperatureMap"
],
Graphics[{
{
AbsoluteThickness[7],
RGBColor[1, 0, 0],
Opacity[0.95],
Line[{{0, -L/2}, {0, L/2}}]
}
(* Graphics *)}]
(* Show *)],
"",
(* Function Parameters *)
"",
Style["Linear Charge Density", Bold, 12, FontFamily -> "Times"],
{{\[Lambda], 5*10^-9,
Style["\[Lambda] [\!\(\*FractionBox[\(C\), \(cm\)]\)]",
FontSize -> 16, FontFamily -> "Times"]}, 0.1*10^-9, 10*10^-9,
0.1*10^-9, ControlType -> {Slider, PopupMenu}, ImageSize -> Medium,
Appearance -> "Labeled"},
"",
Style["Charge Length", Bold, FontSize -> 14, FontFamily -> "Times"],
{{L, 1, Style["L [cm]", FontSize -> 16, FontFamily -> "Times"]}, 0.1,
100, ControlType -> {Slider, PopupMenu}, ImageSize -> Medium,
Appearance -> "Labeled" },
"",
"",
(* Axes *)
"",
"",
{{R, 1, Style["\!\(\*SubscriptBox[\(\[Rho]\), \(max\)]\)",
FontSize -> 16, FontFamily -> "Times"]}, 0.000001, 2,
ControlType -> {Slider, PopupMenu}, ImageSize -> Medium,
Appearance -> "Labeled" },
{{Z, 1, Style["\!\(\*SubscriptBox[\(z\), \(max\)]\)", FontSize -> 16,
FontFamily -> "Times"]}, 0.000001, 2,
ControlType -> {Slider, PopupMenu}, ImageSize -> Medium,
Appearance -> "Labeled" },
ControlPlacement -> Left,
SaveDefinitions -> False,
TrackedSymbols :> {\[Lambda], L, R, Z}
(* Manipulate *)]
One work-around I've noticed somewhat works is to use DensityPlot and StreamPlot separately, connected with Show[]. However, this also produces a gltich. While the density plot shows contours now, there is a large white area near the line charge that cannot be plotted. I have tried adjusting EVERYTHING in the DensityPlot but nothing gets rid of the white blob in the center of the plot.
Show[ DensityPlot[], StreamPlot[] ] code here:
Manipulate[
Show[
(* Probabilities *)
StreamDensityPlot[
{
},
{ \[Rho], -R, R },
{ z, -Z, Z },
PlotRange -> { {-R, R}, {-Z, Z} },
ImageSize -> Large,
Frame -> True,
FrameLabel -> {
Style["Radial Position [cm]", 20, FontFamily -> "Times"],
Style["Longitudinal Position [cm]", 20, FontFamily -> "Times"] },
BaseStyle -> {FontFamily -> "Times", FontSize -> 20},
AxesOrigin -> {0, 0},
Axes -> True
],
DensityPlot[
{
E\[Rho][\[Rho], z, \[Lambda], L], Ez[\[Rho], z, \[Lambda], L]
},
{ \[Rho], 10^-12, R },
{ z, -Z, Z },
PlotRange -> { {10^-12, R}, {-Z, Z} },
PlotLegends ->
Placed[
BarLegend[
Automatic,
LegendLabel ->
Style["Electric Field [V \!\(\*SuperscriptBox[\(cm\), \
\(-1\)]\)]", 20, FontFamily -> "Times"],
LabelStyle -> {FontSize -> 20},
TicksStyle -> {FontFamily -> "Times"}
],
Right],
ColorFunction -> "TemperatureMap"
],
StreamPlot[
{
E\[Rho][\[Rho], z, \[Lambda], L], Ez[\[Rho], z, \[Lambda], L]
},
{ \[Rho], 10^-12, R },
{ z, -Z, Z },
PlotRange -> { {10^-12, R}, {-Z, Z} },
(* StreamPoints\[Rule]20, *)
StreamStyle ->
Directive[RGBColor[0, 0, 0], AbsoluteThickness[1], Dashed,
Opacity[0.5]]
],
DensityPlot[
{
E\[Rho][\[Rho], z, \[Lambda], L], Ez[\[Rho], z, \[Lambda], L]
},
{ \[Rho], -R, -10^-12 },
{ z, -Z, Z },
PlotRange -> { {-R, -10^-12}, {-Z, Z} },
ColorFunction -> "TemperatureMap"
],
StreamPlot[
{
-E\[Rho][\[Rho], z, \[Lambda], L], Ez[\[Rho], z, \[Lambda], L]
},
{ \[Rho], -R, -10^-12 },
{ z, -Z, Z },
PlotRange -> { {-R, -10^-12}, {-Z, Z} },
(* StreamPoints\[Rule]20, *)
StreamStyle ->
Directive[RGBColor[0, 0, 0], AbsoluteThickness[1], Dashed,
Opacity[0.5]]
(* ColorFunction\[Rule]"" *)
],
Graphics[{
{
AbsoluteThickness[7],
RGBColor[1, 0, 0],
Opacity[0.95],
Line[{{0, -L/2}, {0, L/2}}]
}
(* Graphics *)}]
(* Show *)],
"",
(* Function Parameters *)
"",
Style["Linear Charge Density", Bold, 12, FontFamily -> "Times"],
{{\[Lambda], 5*10^-9,
Style["\[Lambda] [\!\(\*FractionBox[\(C\), \(cm\)]\)]",
FontSize -> 16, FontFamily -> "Times"]}, 0.1*10^-9, 10*10^-9,
0.1*10^-9, ControlType -> {Slider, PopupMenu}, ImageSize -> Medium,
Appearance -> "Labeled"},
"",
Style["Charge Length", Bold, FontSize -> 14, FontFamily -> "Times"],
{{L, 1, Style["L [cm]", FontSize -> 16, FontFamily -> "Times"]}, 0.1,
100, ControlType -> {Slider, PopupMenu}, ImageSize -> Medium,
Appearance -> "Labeled" },
"",
"",
(* Axes *)
"",
"",
{{R, 1, Style["\!\(\*SubscriptBox[\(\[Rho]\), \(max\)]\)",
FontSize -> 16, FontFamily -> "Times"]}, 0.000001, 2,
ControlType -> {Slider, PopupMenu}, ImageSize -> Medium,
Appearance -> "Labeled" },
{{Z, 1, Style["\!\(\*SubscriptBox[\(z\), \(max\)]\)", FontSize -> 16,
FontFamily -> "Times"]}, 0.000001, 2,
ControlType -> {Slider, PopupMenu}, ImageSize -> Medium,
Appearance -> "Labeled" },
ControlPlacement -> Left,
SaveDefinitions -> False,
TrackedSymbols :> {\[Lambda], L, R, Z}
(* Manipulate *)]
Does anyone have ANY experience using StreamDensityPlot or DensityPlot with StreamPlot? I cannot figure out why neither one of them is working.
