電磁學自學經驗分享(二)-向量分析-球座標圖

On 由predliu撰寫In 電磁學自學經驗分享

向大師致敬.復刻經典!!

回想起以前教授很賣力在黑板講解這些球座標圖,幾乎要把黑板挖一個洞.加上自己只有大猩猩的智慧,根本就"轉不過來".坐標變換過節怎麼讀？想像看說明書駕駛太空梭,又沒有太空梭可以摸.其實這章節不難,因出場的符號太多,認識他們是很乏味很無聊的差事,眼花撩亂,硬讀會想睡.當我寫程式要復刻一下經典,發現大師要用這張球座標圖交待的事情太多,
下圖(執行我的code),把這章節變成虛擬太空梭,輔助你感覺圖上每條虛線,拉近你與大師的距離.

(*From https://qhand.work/2022/07/15/vector/ *)
(*  Wolfram Mathematica Code V13.3 instal see:  \
https://qhand.work/2022/07/30/install/*)
ClearAll["`*"]
trans = CoordinateTransformData["Spherical" -> "Cartesian", "Mapping"];
Manipulate[
 Subscript[R, inner] = Rin;
 Subscript[R, outter] = Rin + d;
 Subscript[\[Phi], L] = \[Phi]L;
 Subscript[\[Phi], R] = Subscript[\[Phi], L] + width;
 Subscript[\[Theta], upper] = \[Theta]UP;
 Subscript[\[Theta], down] = Subscript[\[Theta], upper] + high;
 
 
 trans = 
  CoordinateTransformData["Spherical" -> "Cartesian", "Mapping"];
 
\!\(\*OverscriptBox[
SubscriptBox[\(O2\[Theta]\), \(upper\)], \(_\)]\) = Line[{
    {trans@{Subscript[R, inner], Subscript[\[Theta], upper], 
       Subscript[\[Phi], L]}, {0, 0, 0}},
    {trans@{Subscript[R, inner], Subscript[\[Theta], down], 
       Subscript[\[Phi], L]}, {0, 0, 0}}}
   ];
 
\!\(\*OverscriptBox[
SubscriptBox[\(zV\[Theta]\), \(upper\)], \(_\)]\) = Line[{
    {trans@{Subscript[R, inner], Subscript[\[Theta], upper], 
       Subscript[\[Phi], L]}, 
     trans@{Cos[Subscript[\[Theta], upper]]  Subscript[R, inner], 0, 
       0}},
    {trans@{Subscript[R, inner], Subscript[\[Theta], upper], 
       Subscript[\[Phi], R]}, 
     trans@{Cos[Subscript[\[Theta], upper]]  Subscript[R, inner], 0, 
       0}}}];
 (*Overscript[Subscript[R\[Theta], upper], _]=Line[{
 (*{trans@{Subscript[R, inner],Subscript[\[Theta], upper],Subscript[\
\[Phi], L]},trans@{Subscript[R, inner],0,0}},*)
 {trans@{Subscript[R, inner],Subscript[\[Theta], upper],Subscript[\
\[Phi], R]},trans@{Subscript[R, inner],0,0}}}];*)
 
 
 
 
\!\(\*OverscriptBox[
SubscriptBox[\(zPP\[Theta]\), \(upper\)], \(_\)]\) = Line[{
    {trans@{Sin[Subscript[\[Theta], upper]]  Subscript[R, 
        inner], \[Pi]/2, Subscript[\[Phi], L]}, 
     trans@{Sin[Subscript[\[Theta], upper]]  Subscript[R, 
         inner], \[Pi]/2, Subscript[\[Phi], L]} + 
      trans@{Cos[Subscript[\[Theta], upper]]  Subscript[R, inner], 0, 
        0}},
    {trans@{Sin[Subscript[\[Theta], upper]]  Subscript[R, 
        inner], \[Pi]/2, Subscript[\[Phi], R]}, 
     trans@{Sin[Subscript[\[Theta], upper]]  Subscript[R, 
         inner], \[Pi]/2, Subscript[\[Phi], R]} + 
      trans@{Cos[Subscript[\[Theta], upper]]  Subscript[R, inner], 0, 
        0}}}];
 
\!\(\*OverscriptBox[
SubscriptBox[\(OXY\[Theta]\), \(upper\)], \(_\)]\) = Line[{
    {trans@{Sin[Subscript[\[Theta], upper]]  Subscript[R, 
        inner], \[Pi]/2, Subscript[\[Phi], L]}, {0, 0, 0}},
    {trans@{Sin[Subscript[\[Theta], upper]]  Subscript[R, 
        inner], \[Pi]/2, Subscript[\[Phi], R]}, {0, 0, 0}}}];
 
\!\(\*OverscriptBox[
SubscriptBox[\(OX\[Theta]\), \(upper\)], \(_\)]\) = Line[
   {trans@{Sin[Subscript[\[Theta], upper]]  Subscript[R, 
       inner], \[Pi]/2, Subscript[\[Phi], 
      L]}, {(trans@{Sin[Subscript[\[Theta], upper]]  Subscript[R, 
          inner], \[Pi]/2, Subscript[\[Phi], L]})[[1]], 0, 0}}];
 
\!\(\*OverscriptBox[
SubscriptBox[\(OY\[Theta]\), \(upper\)], \(_\)]\) = Line[
   {trans@{Sin[Subscript[\[Theta], upper]]  Subscript[R, 
       inner], \[Pi]/2, Subscript[\[Phi], 
      L]}, {0, (trans@{Sin[Subscript[\[Theta], upper]]  Subscript[R, 
          inner], \[Pi]/2, Subscript[\[Phi], L]})[[2]], 0}}];
 
\!\(\*OverscriptBox[
SubscriptBox[\(zPP\[Theta]\), \(upper\)], \(_\)]\) = Line[{
    {trans@{Sin[Subscript[\[Theta], upper]]  Subscript[R, 
        inner], \[Pi]/2, Subscript[\[Phi], L]}, 
     trans@{Sin[Subscript[\[Theta], upper]]  Subscript[R, 
         inner], \[Pi]/2, Subscript[\[Phi], L]} + 
      trans@{Cos[Subscript[\[Theta], upper]]  Subscript[R, inner], 0, 
        0}},
    {trans@{Sin[Subscript[\[Theta], upper]]  Subscript[R, 
        inner], \[Pi]/2, Subscript[\[Phi], R]}, 
     trans@{Sin[Subscript[\[Theta], upper]]  Subscript[R, 
         inner], \[Pi]/2, Subscript[\[Phi], R]} + 
      trans@{Cos[Subscript[\[Theta], upper]]  Subscript[R, inner], 0, 
        0}}}];
 apt = (35 + 3.5);
 axiresarrowheads = {Arrowheads[.08],
   Arrow[{{0, 0, 0}, {apt, 0, 0}}],
   Arrow[{{0, 0, 0}, {0, apt, 0}}],
   Arrow[{{0, 0, 0}, {0, 0, apt}}]
   };
 AllDashLine = {Dashing[{0.044, 0.014`}], 
\!\(\*OverscriptBox[
SubscriptBox[\(O2\[Theta]\), \(upper\)], \(_\)]\), 
\!\(\*OverscriptBox[
SubscriptBox[\(zV\[Theta]\), \(upper\)], \(_\)]\), 
\!\(\*OverscriptBox[
SubscriptBox[\(OXY\[Theta]\), \(upper\)], \(_\)]\), 
\!\(\*OverscriptBox[
SubscriptBox[\(zPP\[Theta]\), \(upper\)], \(_\)]\)  , 
\!\(\*OverscriptBox[
SubscriptBox[\(OX\[Theta]\), \(upper\)], \(_\)]\)  , 
\!\(\*OverscriptBox[
SubscriptBox[\(OY\[Theta]\), \(upper\)], \(_\)]\)(*,
    axiresarrowheads*)} // Graphics3D;
 \[Theta]Curve = ParametricPlot3D[
    Subscript[R, 
    outter] {  
     Sin[\[Theta]] Cos[(Subscript[\[Phi], L] + Subscript[\[Phi], R])/
       2], Sin[\[Theta]] Sin[(
       Subscript[\[Phi], L] + Subscript[\[Phi], R])/2] , 
     Cos[\[Theta]]}, {\[Theta], 0, Subscript[\[Theta], upper]}, 
   Mesh -> None, PlotStyle -> Directive[Red, Opacity[0.3], Dashed]];
 d\[Phi]Project = 
  ParametricPlot3D[{Sin[Subscript[\[Theta], upper]]  Subscript[R, 
     inner] {Cos[u], Sin[u], 0}, 
    Sin[Subscript[\[Theta], down]]  Subscript[R, 
     inner] {Cos[u], Sin[u], 0}}, {u, Subscript[\[Phi], L], 
    Subscript[\[Phi], R]}, PlotStyle -> Pink];
 inoutSide = ParametricPlot3D[{
     Subscript[R, 
     inner] {  Sin[\[Theta]] Cos[\[Phi]], Sin[\[Theta]] Sin[\[Phi]] , 
      Cos[\[Theta]]},
     Subscript[R, 
     outter] {  Sin[\[Theta]] Cos[\[Phi]], Sin[\[Theta]] Sin[\[Phi]] ,
       Cos[\[Theta]]}}, {\[Theta], Subscript[\[Theta], upper], 
    Subscript[\[Theta], down]}, {\[Phi], Subscript[\[Phi], L], 
    Subscript[\[Phi], R]}, Mesh -> None, 
   PlotStyle -> Directive[Pink, Opacity[0.3]]];
 updownSide = ParametricPlot3D[{
    R*{   Sin[Subscript[\[Theta], upper]] Cos[\[Phi]], 
      Sin[Subscript[\[Theta], upper]] Sin[\[Phi]] , 
      Cos[Subscript[\[Theta], upper]]},
    R*{   Sin[Subscript[\[Theta], down]] Cos[\[Phi]], 
      Sin[Subscript[\[Theta], down]] Sin[\[Phi]] , 
      Cos[Subscript[\[Theta], down]]}}, {R, Subscript[R, inner],  
    Subscript[R, outter]}, {\[Phi], Subscript[\[Phi], L], 
    Subscript[\[Phi], R]}, Mesh -> None, 
   PlotStyle -> Directive[Pink, Opacity[0.3]]];
 RLSide = ParametricPlot3D[Evaluate@{
     R*{  Sin[\[Theta]] Cos[Subscript[\[Phi], L]], 
       Sin[\[Theta]] Sin[Subscript[\[Phi], L]] , Cos[\[Theta]]},
     R*{  Sin[\[Theta]] Cos[Subscript[\[Phi], R]], 
       Sin[\[Theta]] Sin[Subscript[\[Phi], R]] , Cos[\[Theta]]}}, {R, 
    Subscript[R, inner],  Subscript[R, outter]}, {\[Theta], 
    Subscript[\[Theta], upper], Subscript[\[Theta], down]}, 
   Mesh -> None, PlotStyle -> Directive[Pink, Opacity[0.3]]];
 updownLine = ParametricPlot3D[{
     Subscript[R, 
     inner] {   Sin[Subscript[\[Theta], upper]] Cos[\[Phi]], 
      Sin[Subscript[\[Theta], upper]] Sin[\[Phi]] ,   
      Cos[Subscript[\[Theta], upper]]},
     Subscript[R, 
     inner] {   Sin[Subscript[\[Theta], down]] Cos[\[Phi]], 
      Sin[Subscript[\[Theta], down]] Sin[\[Phi]] ,    
      Cos[Subscript[\[Theta], down]]},
    Subscript[R, 
     outter] {   Sin[Subscript[\[Theta], upper]] Cos[\[Phi]], 
      Sin[Subscript[\[Theta], upper]] Sin[\[Phi]] ,   
      Cos[Subscript[\[Theta], upper]]},
    Subscript[R, 
     outter] {   Sin[Subscript[\[Theta], down]] Cos[\[Phi]], 
      Sin[Subscript[\[Theta], down]] Sin[\[Phi]] ,    
      Cos[Subscript[\[Theta], down]]}}, {\[Phi], Subscript[\[Phi], L],
     Subscript[\[Phi], R]}, Mesh -> None, 
   PlotStyle -> Directive[Blue, Opacity[0.3], Dashed]];
 RLLine = ParametricPlot3D[{
     Subscript[R, 
     inner] {  Sin[\[Theta]] Cos[Subscript[\[Phi], L]], 
      Sin[\[Theta]] Sin[Subscript[\[Phi], L]] , Cos[\[Theta]]},
     Subscript[R, 
     inner] {   Sin[\[Theta]] Cos[Subscript[\[Phi], R]], 
      Sin[\[Theta]] Sin[Subscript[\[Phi], R]] , Cos[\[Theta]]},
    Subscript[R, 
     outter] {  Sin[\[Theta]] Cos[Subscript[\[Phi], L]], 
      Sin[\[Theta]] Sin[Subscript[\[Phi], L]] , Cos[\[Theta]]},
    Subscript[R, 
     outter] {   Sin[\[Theta]] Cos[Subscript[\[Phi], R]], 
      Sin[\[Theta]] Sin[Subscript[\[Phi], R]] , 
      Cos[\[Theta]]}}, {\[Theta], Subscript[\[Theta], upper], 
    Subscript[\[Theta], down]}, Mesh -> None, 
   PlotStyle -> Directive[Red, Opacity[0.3], Dashed]];
 final = 
  Show[d\[Phi]Project, AllDashLine, inoutSide, updownSide, RLSide, 
   updownLine, RLLine, \[Theta]Curve, 
   PlotRange -> {{0.0001, 40}, {0.0001, 40}, {0.0001, 40}}, 
   Axes -> True, AxesOrigin -> {0, 0, 0}, AxesStyle -> Thick, 
   AxesEdge -> {{-1, -1}, {-1, -1}, {-1, -1}}, Mesh -> None, 
   Ticks -> None, Boxed -> False, 
   AxesLabel -> {Style["x", Black, Large, Bold, Italic, 
      FontFamily -> "Times New Roman"], 
     Style["y", Black, Bold, Large, Italic, 
      FontFamily -> "Times New Roman"], 
     Style["z", Black, Bold, Large, Italic, 
      FontFamily -> "Times New Roman"]},
   ViewPoint -> {viewX, viewY, viewZ}, ImageSize -> 350] ,
 {{Rin, 32}, 20, 50},
 {{d, 8}, 4, 25},
 {{\[Phi]L, 35 °}, 25 °, 55 °},
 {{width, 25 °}, 10 °, 80 °},
 {{\[Theta]UP, 40 °}, 20 °, 60 °},
 {{high, 14 °}, 4 °, 34 °},
 {{viewX, -43}, -100, 100, 15},
 {{viewY, -17}, -100, 100, 15},
 {{viewZ, 17}, -100, 100, 15}]

球座標相關應用,如下:

一. (柱座標)電場E=k r3 ar ,反求電荷密度

單電荷散度
(*From https://qhand.work/2022/07/15/vector/ *)
(*  Wolfram Mathematica Code V13.3 instal see:  \
https://qhand.work/2022/07/30/install/*)
ClearAll["Global`*"];
field = {k*r^3, 0, 0};
diversionOfEfieldCyl = Div[field, {r, \[Theta], z}, "Cylindrical"]
4 k r^2
\[Rho]r = Solve[diversionOfEfieldCyl == \[Rho]/\[Epsilon], \[Rho]]
{{\[Rho] -> 4 k r^2 \[Epsilon]}}
fieldXYZ = 
  TransformedField["Cylindrical" -> "Cartesian", 
   field, {r, \[Theta], zc} -> {x, y, z}];
\[Rho]rToXYZ = 
  TransformedField[
   "Cylindrical" -> 
    "Cartesian", \[Rho]r[[1, 1, 2]], {r, \[Theta], zz} -> {x, y, z}];
k = 1; \[Epsilon] = 1;
plot\[Rho]r = DensityPlot[\[Rho]rToXYZ, {x, -1, 1}, {y, -1, 1}];
vectorE = VectorPlot[fieldXYZ // Most, {x, -1, 1}, {y, -1, 1}];
Show[plot\[Rho]r, vectorE]

二. (改成球座標)電場E=k r3 ar ,再反求電荷密度

把我的程式碼,複製去執行看看.

(*From https://qhand.work/2022/07/15/vector/ *)
(*  Wolfram Mathematica Code V13.3 instal see:  \
https://qhand.work/2022/07/30/install/*)
ClearAll["Global`*"];
field = {k*R^3, 0, 0};
diversionOfEfieldSph = Div[field, {R, \[Theta], \[Phi]}, "Spherical"]
5 k R^2
\[Rho]R = Solve[diversionOfEfieldSph == \[Rho]/\[Epsilon], \[Rho]]
{{\[Rho] -> 5 k R^2 \[Epsilon]}}
EfieldSphToXYZ = 
  TransformedField["Spherical" -> "Cartesian", 
   field, {R, \[Theta], \[CurlyPhi]} -> {x, y, z}];
\[Rho]RToXYZ = 
  TransformedField[
   "Spherical" -> 
    "Cartesian", \[Rho]R[[1, 1, 2]], {R, \[Theta], \[CurlyPhi]} -> {x,
      y, z}];
k = 1; \[Epsilon] = 1;
plot\[Rho]R = 
  DensityPlot3D[\[Rho]RToXYZ, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, 
   OpacityFunction -> 0.03, ColorFunction -> Hue];
vectorE3D = 
  VectorPlot3D[EfieldSphToXYZ, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, 
   VectorMarkers -> "Tube", VectorSizes -> Small];
Show[plot\[Rho]R, vectorE3D]
(*StreamE=StreamPlot3D[EfieldSphToXYZ,{x,-1,1},{y,-1,1},{z,-1,1}];
Show[plot\[Rho]R,StreamE]*)

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