(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 433179, 9779]*) (*NotebookOutlinePosition[ 461115, 10684]*) (* CellTagsIndexPosition[ 461071, 10680]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[StyleBox["Ph196bEM Homework 2 Solutions - 2004", \ "Subtitle"]], "Title"], Cell[CellGroupData[{ Cell["7. Jackson3ed, problem 2.1e", "Section"], Cell[TextData[{ "A point charge ", Cell[BoxData[ \(TraditionalForm\`q\)]], " is brought to a position a distance ", Cell[BoxData[ \(TraditionalForm\`d\)]], " away from an infinite plane conductor held at zero potential. Using the \ method of images, find:\ne) the potential energy between the charge ", Cell[BoxData[ \(TraditionalForm\`q\)]], " and its image [compare the answer to the work necessary to remove the \ charge from its position to infinity, and discuss]." }], "Text"], Cell[TextData[StyleBox["Be sure to carefully handle the singularity due to \ the point charge in this problem. In particular, even when the point charge \ has been removed to infinity (and there is no induced charge on the grounded \ plate), there is still an energy in the field around the point charge, \ exactly equal to the corresponding infinite term in the energy when the point \ charge is near the plate. This energy is formally infinite, but not \ physically so since other interactions than E&M must be taken into account. \ So to relate stored energy to work done in making a change, you must subtract \ the initial and final electrostatically stored energies.", FontFamily->"Times New Roman"]], "Text"], Cell[CellGroupData[{ Cell["Solution", "Subsection"], Cell[TextData[{ "The electric field at any point in space is the sum of two parts, that due \ to the given point charge, ", Cell[BoxData[ \(TraditionalForm\`E\&\[RightVector]\_q\)]], ", and that due to the charge distribution on the infinite grounded plate, \ ", Cell[BoxData[ \(TraditionalForm\`E\&\[RightVector]\_plate\)]], ", in the ", Cell[BoxData[ \(TraditionalForm\`z = 0\)]], " plane. Clearly, by symmetry, ", Cell[BoxData[ \(TraditionalForm\`E\&\[RightVector]\_plate\)]], " at points ", Cell[BoxData[ \(TraditionalForm\`\(\((x, y, z)\)\(\ \ \)\(and\)\(\ \ \)\((x, y, \(-z\))\)\(\ \)\)\)]], "differ only in that the component of the electric field normal to the \ plate has opposite signs. In particular, in ", Cell[BoxData[ \(TraditionalForm\`z > 0\)]], ", ", Cell[BoxData[ \(TraditionalForm\`E\&\[RightVector]\_plate\)]], " is the same as the field due to a point charge ", Cell[BoxData[ \(TraditionalForm\`\(-q\)\)]], " located at ", Cell[BoxData[ \(TraditionalForm\`\((0, 0, \(-d\))\)\)]], " whereas in ", Cell[BoxData[ \(TraditionalForm\`z < 0\)]], ", ", Cell[BoxData[ \(TraditionalForm\`E\&\[RightVector]\_plate\)]], " is the same as the field due to a point charge ", Cell[BoxData[ \(TraditionalForm\`\(-q\)\)]], " located ar ", Cell[BoxData[ \(TraditionalForm\`\((0, 0, \(+d\))\)\)]], " The electric field situation looks like the following." }], "Text"], Cell[CellGroupData[{ Cell["Begin graphics", "Subsubsection"], Cell[BoxData[ \(<< Graphics`Arrow`\)], "Input"], Cell[BoxData[ \(fieldptch[x_, y_, yptch_, ch_] := Module[{r3, scale}, scale = ch .05; r3 = \((\((y - yptch)\)\^2 + x\^2)\)\^\(3/2\); {{x, y}, {x + scale\ x\/r3, y + scale\ \((y - yptch)\)\/r3}}]\)], "Input"], 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oooo/@3oool00?l0oooo/@3oool00?l0oooo/@3oool00001\ \>"], ImageRangeCache->{{{125.813, 412.813}, {433.125, 146.125}} -> {-2.08657, \ 0.0821284, 0.00774775, 0.00774775}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell[TextData[{ "The field due to the positive point charge ", Cell[BoxData[ \(TraditionalForm\`q\)]], " points away from ", Cell[BoxData[ \(TraditionalForm\`q\)]], " everywhere; the field due to the negative induced charge on the plate \ points generally toward the plate. Thus in the region ", Cell[BoxData[ \(TraditionalForm\`\(\(z\)\(\[GreaterEqual]\)\(0\)\(\ \)\)\)]], "there is a net field whereas in ", Cell[BoxData[ \(TraditionalForm\`z \[LessEqual] 0\)]], " the two contributions are equal but oppositely directed so the total \ field is zero.\n\nThe energy stored in the electrostatic system when the \ point charge is near the plate is\n\n", Cell[BoxData[ \(TraditionalForm\`\(\[CurlyEpsilon]\_0\/2\) \(\[Integral]\_\(all\ \ space\)\(\((E\&\[RightVector]\_q + E\&\[RightVector]\_plate)\)\^2\) \[DifferentialD]\^3 r\)\)]], "\n\nand when the charge is moved a great distance from the plate,\n\n", Cell[BoxData[ \(TraditionalForm\`\(\[CurlyEpsilon]\_0\/2\) \(\[Integral]\_\(all\ \ space\)\(\((E\&\[RightVector]\_q)\)\^2\) \[DifferentialD]\^3 r\)\)]], ".\n\nThis last integral represents the energy necessary to build the point \ charge and is formally divergent in our theory (in the real world, other \ interactions in addition to E&M must be invoked to properly express the \ energy tied up in an electron, for example). To find out how much energy is \ required to bring the pre-existing point charge to the vicinity of the plate \ (or to remove it from the vicinity of the plate) we want the difference of \ the above two terms. Thus\n\n", Cell[BoxData[ \(TraditionalForm\`\(\(W\)\(=\)\(\ \)\)\)]], "Energy to remove point charge from vicinity of plate = \n", Cell[BoxData[ \(TraditionalForm\`\(\[CurlyEpsilon]\_0\/2\) \(\[Integral]\_\(all\ \ space\)\(\((E\&\[RightVector]\_q)\)\^2\) \[DifferentialD]\^3 r\)\ - \(\[CurlyEpsilon]\_0\/2\) \(\[Integral]\_\(all\ space\ \)\(\((E\&\[RightVector]\_q + E\&\[RightVector]\_plate)\)\^2\) \ \(\(\[DifferentialD]\^3 r\)\(\ \)\)\)\)]], "\nor \n", Cell[BoxData[ \(TraditionalForm\`W = \(\(-\ \[CurlyEpsilon]\_0\)\/2\) \(\[Integral]\_\ \(all\ space\)\((2 E\&\[RightVector]\_q\[CenterDot]E\&\[RightVector]\_plate \ + \(E\&\[RightVector]\)\_plate\%2)\) \[DifferentialD]\^3 r\)\)]], ".\n\nThis integral has a singularity at the position of the point charge \ in the integrand, but it's integral is well behaved as we will see shortly. " }], "Text"], Cell[TextData[{ "Let the charge be at ", Cell[BoxData[ \(TraditionalForm\`z = d > 0\)]], " on the ", Cell[BoxData[ \(TraditionalForm\`z - \(\(axis\)\(.\)\)\)]], " Then " }], "Text"], Cell[TextData[Cell[BoxData[{ \(TraditionalForm\`W = \(-\(\[CurlyEpsilon]\_0\/2\)\) \((q\/\(4 \[Pi]\ \ \[CurlyEpsilon]\_0\))\)\^2\), "\[IndentingNewLine]", \(TraditionalForm\`\((\[Integral]\_\(z > 0\)\((\(-2\) \(\((r\&\ \[RightVector] - d\ e\&\[RightVector]\_z)\)\[CenterDot]\((r\&\[RightVector] + \ d\ e\&\[RightVector]\_z)\)\)\/\(\(|\)\(\ \)\(r\&\[RightVector] - d\ e\&\ \[RightVector]\_z\)\( | \^3\)\(\ \)\(|\)\(\ \)\(r\&\[RightVector] + d\ e\&\ \[RightVector]\_z\)\( | \^3\)\) + \((r\&\[RightVector] + d\ e\&\[RightVector]\ \_z)\)\^2\/\(\(\ \)\(\(|\)\(\ \)\(r\&\[RightVector] + d\ e\&\[RightVector]\_z\ \)\( | \^6\)\)\))\) \[DifferentialD]\^3 r\[IndentingNewLine] + \[Integral]\_\(z < 0\)\((\(-2\) \((r\&\ \[RightVector] - d\ e\&\[RightVector]\_z)\)\^2\/\(\(|\)\(\ \)\(r\&\ \[RightVector] - d\ e\&\[RightVector]\_z\)\( | \^6\)\) + \((r\&\[RightVector] \ - d\ e\&\[RightVector]\_z)\)\^2\/\(\(\ \)\(\(|\)\(\ \)\(r\&\[RightVector] - d\ \ e\&\[RightVector]\_z\)\( | \^6\)\)\))\) \[DifferentialD]\^3 r)\)\)}]]], "Text"], Cell[TextData[{ "Now it is obvious that ", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_\(z > 0\)\(\((r\&\[RightVector] + d\ \ e\&\[RightVector]\_z)\)\^2\/\(\(\ \)\(\(|\)\(\ \)\(r\&\[RightVector] + d\ e\&\[RightVector]\_z\)\( | \^6\)\)\)\) \ \[DifferentialD]\^3 r\ = \ \[Integral]\_\(z < 0\)\(\((r\&\[RightVector] - d\ e\&\ \[RightVector]\_z)\)\^2\/\(\(\ \)\(\(|\)\(\ \)\(r\&\[RightVector] - d\ e\&\[RightVector]\_z\)\( | \^6\)\)\)\) \ \[DifferentialD]\^3 r\)]], ". Therefore three of the terms in ", Cell[BoxData[ \(TraditionalForm\`W\)]], " cancel against one another leaving\n\n", Cell[BoxData[{ \(TraditionalForm\`W = \(\[CurlyEpsilon]\_0\/2\) \(\((q\/\(4 \[Pi]\ \ \[CurlyEpsilon]\_0\))\)\^2\) \(\[Integral]\_\(z > 0\)2 \(\(\((r\&\ \[RightVector] - d\ e\&\[RightVector]\_z)\)\[CenterDot]\((r\&\[RightVector] + d\ e\&\[RightVector]\_z)\)\)\/\(\(|\)\(\ \)\(r\&\ \[RightVector] - d\ e\&\[RightVector]\_z\)\( | \^3\)\(\ \)\(|\)\(\ \)\(r\ \&\[RightVector] + d\ e\&\[RightVector]\_z\)\( | \^3\)\)\) \ \[DifferentialD]\^3 r\)\), "\[IndentingNewLine]", \(TraditionalForm\`\(\(=\)\(\(\(\(\[CurlyEpsilon]\_0\)( q\/\(4 \[Pi]\ \[CurlyEpsilon]\_0\))\)\^2\) \ \(\[Integral]\_\(z > 0\)\(\(r\^2 - d\^2\)\/\(\(\((x\^2 + y\^2 + \((z - d)\)\^2)\)\^\(3/ 2\)\) \((x\^2 + y\^2 + \((z + d)\)\^2)\)\^\(3/2\)\)\ \) \[DifferentialD]\^3 r\)\)\)\), "\[IndentingNewLine]", \(TraditionalForm\`\(\(=\)\(2 \[Pi]\ \(\(\(\[CurlyEpsilon]\_0\)( q\/\(4 \[Pi]\ \[CurlyEpsilon]\_0\))\)\^2\) \(\[Integral]\_\(\ \[Rho] = 0\)\%\[Infinity]\(\[Integral]\_\(z = 0\)\%\[Infinity]\ \(\(\(\ \)\(\ \[Rho]\^2 + z\^2 - d\^2\)\)\/\(\(\((\[Rho]\^2 + \((z - d)\)\^2)\)\^\(3/ 2\)\) \((\[Rho]\^2 + \((z + d)\)\^2)\)\^\(3/2\)\)\ \) \[Rho]\ \[DifferentialD]\[Rho]\ \[DifferentialD]z\)\)\)\)\), "\[IndentingNewLine]", \(TraditionalForm\`\(\(=\)\(\[Pi]\ \(\(\(\[CurlyEpsilon]\_0\)( q\/\(4 \[Pi]\ \[CurlyEpsilon]\_0\))\)\^2\) \ \(\[Integral]\_\(x = 0\)\%\[Infinity]\(\[Integral]\_\(z = 0\)\%\[Infinity]\ \ \(\(\ \)\(x + z\^2 - d\^2\)\)\/\(\(\((x + \((z - d)\)\^2)\)\^\(3/2\)\) \((x + \ \((z + d)\)\^2)\)\^\(3/2\)\)\ \[DifferentialD]x\ \ \[DifferentialD]z\ \ \ \ \ \ \ \ \ \ \ \ eq\ \((1)\)\)\)\)\)\)}]] }], "Text"], Cell[TextData[{ "For future reference, notice that \n\n", Cell[BoxData[{ \(TraditionalForm\`W = \(\[CurlyEpsilon]\_0\/2\) \(\((q\/\(4 \[Pi]\ \ \[CurlyEpsilon]\_0\))\)\^2\) \(\[Integral]\_\(z > 0\)2 \(\(\((r\&\ \[RightVector] - d\ e\&\[RightVector]\_z)\)\[CenterDot]\((r\&\[RightVector] + d\ e\&\[RightVector]\_z)\)\)\/\(\(|\)\(\ \)\(r\&\ \[RightVector] - d\ e\&\[RightVector]\_z\)\( | \^3\)\(\ \)\(|\)\(\ \)\(r\ \&\[RightVector] + d\ e\&\[RightVector]\_z\)\( | \^3\)\)\) \ \[DifferentialD]\^3 r\)\), "\[IndentingNewLine]", \(TraditionalForm\`\(\(=\)\(\(-\[CurlyEpsilon]\_0\) \(\[Integral]\_\(z \ > 0\)E\&\[RightVector]\_\(due\ to\ q\)\[CenterDot] E\&\[RightVector]\_\(due\ to\ plate\)\ \[DifferentialD]\^3 r\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ e\ q\ \(\((2)\)\(.\)\)\)\)\)\)}]], " \n\nNow do the integral in eq (1)." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[\[Pi]\ \(\[CurlyEpsilon]\_0\) \(\((q\/\(4 \[Pi]\ \ \[CurlyEpsilon]\_0\))\)\^2\) \(\[Integral]\_0\%\[Infinity]\((x + z\^2 - d\^2)\)/\((\(\((x + \((z - d)\)\^2)\)\^\(3/ 2\)\) \((x + \((z + d)\)\^2)\)\^\(3/2\))\) \ \[DifferentialD]x\), {d > 0, z > 0, d \[NotEqual] z}]\)], "Input"], Cell[BoxData[ \(\(q\^2\ \((\(-d\) + z + Abs[d - z])\)\)\/\(32\ \[Pi]\ z\^2\ Abs[d - z]\ \ \[CurlyEpsilon]\_0\)\)], "Output"] }, Open ]], Cell[TextData[{ "So being careful about the absolute values (which arose from ", Cell[BoxData[ \(TraditionalForm\`\@x\^2 = \(\(|\)\(x\)\(|\)\)\)]], " and which makes this term vanish for ", Cell[BoxData[ \(TraditionalForm\`z < d\)]], ") we get" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(W = \[Integral]\_d\%\[Infinity]\(\( q\^2\ \((\(-d\) + z - d + z)\)\)\/\(32\ \[Pi]\ z\^2\ \((z - d)\)\ \[CurlyEpsilon]\_0\)\) \[DifferentialD]z\)], "Input"], Cell[BoxData[ \(q\^2\/\(16\ d\ \[Pi]\ \[CurlyEpsilon]\_0\)\)], "Output"] }, Open ]], Cell[TextData[{ "This, as it must be, is the same result obtained by directly calculating \ the work by using the forces acting on the charge being moved. \n\nThis \ frontal attack on the problem yields the correct answer and ", StyleBox["Mathematica", FontSlant->"Italic"], " is smart enough to know that ", Cell[BoxData[ \(TraditionalForm\`\@x\^2 = \(\(|\)\(x\)\(|\)\)\)]], " but it is very tempting when doing the integrations by hand to make the \ error ", Cell[BoxData[ \(TraditionalForm\`\@x\^2 = x\)]], ", which gives the wrong answer. An alternative approach which does not \ tempt you to fall in this trap for the unwary is to use Gauss's law to undo \ part of the machinery leading to the result that energy is the integral of ", Cell[BoxData[ \(TraditionalForm\`\(\(E\^2\)\(.\)\)\)]], " We had obtained above \n\n", Cell[BoxData[ \(TraditionalForm\`\(\(W\)\(=\)\)\)]], Cell[BoxData[{\(\(-\[CurlyEpsilon]\_0\) \(\[Integral]\_\(z > 0\)\ E\&\[RightVector]\_\(due\ to\ q\)\[CenterDot] E\&\[RightVector]\_\(due\ to\ plate\) \[DifferentialD]\^3 r\)\), "\[IndentingNewLine]", RowBox[{"=", RowBox[{"-", FormBox[\(\(\[CurlyEpsilon]\_0\) \(\[Integral]\_\(z > 0\)\ \[Del]\&\ \[RightVector] \[CapitalPhi]\_\(due\ to\ \ q\)\[CenterDot]\[Del]\&\[RightVector] \[CapitalPhi]\_\(due\ to\ plate\) \ \[DifferentialD]\^3 r\)\), "TraditionalForm"]}]}]}]], "\n\nBy Green's relation we can write\n\n", Cell[BoxData[{ \(TraditionalForm\`\[Del]\&\[RightVector]\(\(\[CenterDot]\)\((\ \[CapitalPhi]\_\(due\ to\ plate\)\ \[Del]\&\[RightVector] \ \[CapitalPhi]\_\(due\ to\ q\))\)\)\), "\[IndentingNewLine]", \(TraditionalForm\`\(\(=\)\(\[Del]\&\[RightVector] \[CapitalPhi]\_\(due\ \ to\ plate\)\[CenterDot]\[Del]\&\[RightVector] \[CapitalPhi]\_\(due\ to\ q\)\ \ + \(\[CapitalPhi]\_\(due\ to\ plate\)\) \[Del]\^2 \[CapitalPhi]\_\(due\ to\ \ q\)\)\)\), "\[IndentingNewLine]", \(TraditionalForm\`\(\(=\)\(\[Del]\&\[RightVector] \[CapitalPhi]\_\(due\ \ to\ q\)\[CenterDot]\[Del]\&\[RightVector] \[CapitalPhi]\_\(due\ to\ plate\)\ \ - \(q\/\[CurlyEpsilon]\_0\) \[CapitalPhi]\_\(due\ to\ plate\)\ \ \(\(\[Delta]\^3\)(r\&\[RightVector] - d\ e\&\[RightVector]\_z)\)\)\)\)}]] }], "Text"], Cell[TextData[{ "So \n\n", Cell[BoxData[{ \(TraditionalForm\`W = \(-\[CurlyEpsilon]\_0\) \(\[Integral]\_\(z > 0\)\ \ \((\[Del]\&\[RightVector]\(\(\[CenterDot]\)\((\[CapitalPhi]\_\(due\ to\ \ plate\)\ \[Del]\&\[RightVector] \[CapitalPhi]\_\(due\ to\ q\))\)\) + \ \(q\/\ \[CurlyEpsilon]\_0\) \[CapitalPhi]\_\(due\ to\ plate\)\ \(\(\[Delta]\^3\)( r\&\[RightVector] - d\ e\&\[RightVector]\_z)\))\) \[DifferentialD]\^3 r\)\), "\[IndentingNewLine]", \(TraditionalForm\`\(\(=\)\(\(\[CurlyEpsilon]\_0\) \(\[Integral]\_\(z = \ 0\)\ \[CapitalPhi]\_\(due\ to\ plate\)\ E\&\[RightVector]\_\(due\ to\ q\)\ \[CenterDot]\[DifferentialD]A\&\[RightVector]\) - \ q\ \(\(\(\[CapitalPhi]\_\(due\ to\ plate\)\)( r\&\[RightVector] = d\ e\&\[RightVector]\_z)\)\(.\)\)\)\)\)}]] }], "Text"], Cell[TextData[{ "Note that the singularity on the point charge has been defanged. So\n\n", Cell[BoxData[ \(TraditionalForm\`\(\(W = \(\[CurlyEpsilon]\_0\) \(\[Integral]\_\(\ \[Rho] = 0\)\%\[Infinity]\((\(\(-q\)\/\(4 \[Pi]\ \[CurlyEpsilon]\_0\)\) 1\/\@\(d\^2 + \[Rho]\^2\))\) \((\(q\/\(4 \[Pi]\ \ \[CurlyEpsilon]\_0\)\) d\/\((d\^2 + \[Rho]\^2)\)\^\(3/2\))\) 2 \[Pi]\ \[Rho] \[DifferentialD]\[Rho]\)\ + q \( q\/\(4 \[Pi]\ \[CurlyEpsilon]\_0\)\) 1\/\(2 d\)\)\(\[IndentingNewLine]\) \)\)]], ". " }], "Text"], Cell[TextData[{ "Let ", Cell[BoxData[ \(TraditionalForm\`x = \(\(\[Rho]\^2\)\(.\)\)\)]], " Then" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(-\[Pi]\)\ \(\[CurlyEpsilon]\_0\) \(\((q\/\(4 \[Pi]\ \ \[CurlyEpsilon]\_0\))\)\^2\) \(\[Integral]\_0\%\[Infinity]\( d\/\((x + d\^2)\)\^2\) \[DifferentialD]x\) + \(q\^2\/\(4 \ \[Pi]\ \[CurlyEpsilon]\_0\)\) 1\/\(2 d\)\)], "Input"], Cell[BoxData[ \(q\^2\/\(16\ d\ \[Pi]\ \[CurlyEpsilon]\_0\)\)], "Output"] }, Open ]], Cell["As we have above, but the integral is now trivial.", "Text"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["8. Jackson3ed, problem 1.15.", "Section"], Cell[TextData[{ "Prove ", StyleBox["Thompson's Theorem: ", FontSlant->"Italic"], "If a number of conducting surfaces are fixed in position and a given total \ charge is placed on each surface, then the electrostatic energy in the region \ bounded by the surfaces is an absolute minimum when the charges are so placed \ that every surface is an equipotential, as happens when they are conductors." }], "Text"], Cell[TextData[{ StyleBox["Note that what is given is the charge ", FontFamily->"Times New Roman"], Cell[BoxData[ \(TraditionalForm\`Q\_i\)]], " ", StyleBox["to be placed on conductor ", FontFamily->"Times New Roman"], Cell[BoxData[ \(TraditionalForm\`i\)]], StyleBox[", not just the total charge ", FontFamily->"Times New Roman"], Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(\(\ \)\(i\)\)Q\_i\)]], " ", StyleBox["to be placed on all of the conductors. \nHint: Consider the \ integral \n", FontFamily->"Times New Roman"], Cell[BoxData[ \(TraditionalForm\`\[Integral]\_V\ \((E\&\[RightVector]\^\(\(\ \ \)\(2\)\) - E\&\[RightVector]\^\(\(\ \)\(\[Prime]\ 2\)\))\)\ \ \[DifferentialD]\^3 r\ = \ \[Integral]\_V\((\ \((E\&\[RightVector]\ - \ E\&\ \[RightVector]\^\(\(\ \)\(\[Prime]\)\))\)\^\(\(\ \)\(2\)\) + 2 E\&\[RightVector]\[CenterDot] E\&\[RightVector]\^\(\(\ \)\(\[Prime]\)\(\ \)\) - 2\ E\&\[RightVector]\^\(\(\ \)\(\[Prime]\ 2\)\))\)\ \ \[DifferentialD]\^3 r\)]], StyleBox["\nwhere ", FontFamily->"Times New Roman"], Cell[BoxData[ \(TraditionalForm\`\(\(E\&\[RightVector]\)\(\ \)\)\)]], StyleBox[" is the field produced by the arbitrarily given distribution of \ potentials on the surfaces and ", FontFamily->"Times New Roman"], Cell[BoxData[ \(TraditionalForm\`E\&\[RightVector]\^\(\(\ \)\(\[Prime]\)\)\)]], StyleBox[", that produced when each surface is an equipotential, i.e., the \ surfaces are conductors. Of course, use the relation \n", FontFamily->"Times New Roman"], Cell[BoxData[ \(TraditionalForm\`\[Del]\&\[RightVector]\(\(\[CenterDot]\)\((\[Psi]\ \ \[Del]\&\[RightVector] \[Phi])\)\)\ = \ \[Del]\&\[RightVector] \[Psi]\ \[CenterDot]\ \[Del]\&\[RightVector] \[Phi]\ + \ \[Psi]\ \ \[Del]\&\ \[RightVector]\^2 \[Phi]\)]], "\n", StyleBox["as appropriate.", FontFamily->"Times New Roman"] }], "Text"], Cell[CellGroupData[{ Cell["Solution", "Subsection"], Cell[TextData[{ "Let ", Cell[BoxData[ \(TraditionalForm\`E\&\[RightVector]\)]], " be the electric field in ", Cell[BoxData[ \(TraditionalForm\`V\)]], " when the charges ", Cell[BoxData[ \(TraditionalForm\`Q\_i\)]], " are placed on the boundary surfaces ", Cell[BoxData[ \(TraditionalForm\`S\_i\)]], " in an arbitrary way and ", Cell[BoxData[ \(TraditionalForm\`E\&\[RightVector]\^\[Prime]\)]], ", that when they are distributed in such a way that each surface is an \ equipotential, i.e., the surfaces are conductors. The geometry could look \ like this." }], "Text"], Cell[CellGroupData[{ Cell["Begin graphics", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(boundary = Graphics[\[IndentingNewLine]{Circle[{0, 0}, 1], \[IndentingNewLine]Circle[{ .3, 0}, .2], \[IndentingNewLine]Circle[{ .0, .5}, .1], \ \[IndentingNewLine]Circle[{\(- .25\), .25}, .15], \ \[IndentingNewLine]Text[\*"\"\<\!\(S\_3\)\>\"", { .175, .5}], \ \[IndentingNewLine]Text[\*"\"\<\!\(S\_2\)\>\"", {\(- .22\), .0}], \ \[IndentingNewLine]Text[\*"\"\<\!\(S\_1\)\>\"", { .55, \(- .14\)}], \ \[IndentingNewLine]Text[\*"\"\<\!\(S\_4\)\>\"", { .49, \(- .73\)}], \ \[IndentingNewLine]Text["\", {\(- .77\), .046}]\[IndentingNewLine]\ \[IndentingNewLine]}]\)], "Input", CellLabel->"In[1]:="], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output", CellLabel->"Out[1]="] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["End graphics", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ 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\[GreaterEqual] 0\)]] }], "Text"], Cell["\<\ when the charges on the surfaces are constrained to be as specified. Use the identity\ \>", "Text"], Cell[TextData[{ " ", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_V\((E\^2\ - \ E\^\[Prime]2)\) \[DifferentialD]\^3 r\ = \[Integral]\_V\((\ \((E\&\[RightVector]\ - \ E\&\ \[RightVector]\^\[Prime])\)\^2 + 2 E\&\[RightVector]\[CenterDot] E\&\[RightVector]\^\[Prime]\ - 2\ E\^\[Prime]2\ )\) \[DifferentialD]\^3 r\)]], "." }], "Text"], Cell[TextData[{ "With the corresponding potentials being ", Cell[BoxData[ \(TraditionalForm\`\[CapitalPhi]\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\[CapitalPhi]\^\[Prime]\)]], ", we have" }], "Text"], Cell[TextData[{ " ", Cell[BoxData[{ \(TraditionalForm\`\[Integral]\_V\((\ E\&\[RightVector]\[CenterDot]E\&\[RightVector]\^\[Prime]\ - E\^\[Prime]2\ )\) \[DifferentialD]\^3 r\), "\[IndentingNewLine]", \(TraditionalForm\`\(\(=\)\(\[Integral]\_V\((\ \[Del]\&\[RightVector] \ \[CapitalPhi]\[CenterDot]\[Del]\&\[RightVector] \[CapitalPhi]\^\[Prime]\ - \ \[Del]\&\[RightVector] \[CapitalPhi]\^\[Prime]\[CenterDot]\[Del]\&\ \[RightVector] \[CapitalPhi]\^\[Prime])\) \[DifferentialD]\^3 r\)\)\), "\[IndentingNewLine]", \(TraditionalForm\`\(\(=\)\(\[Integral]\_V\[Del]\&\[RightVector] \ \[CapitalPhi]\^\[Prime]\ \[CenterDot]\[Del]\&\[RightVector]\((\ \[CapitalPhi] \ - \[CapitalPhi]\^\[Prime])\) \[DifferentialD]\^3 r\)\)\), "\[IndentingNewLine]", \(TraditionalForm\`\(\(=\)\(\[Integral]\_V\((\ \ \[Del]\&\[RightVector]\ \(\(\[CenterDot]\)\((\[CapitalPhi]\^\[Prime]\ \[Del]\&\[RightVector]\((\ \ \[CapitalPhi] - \[CapitalPhi]\^\[Prime])\))\)\) - \(\[CapitalPhi]\^\[Prime]\) \ \[Del]\^2\((\ \[CapitalPhi] - \[CapitalPhi]\^\[Prime])\)\ \ )\) \ \[DifferentialD]\^3 r\)\)\)}]] }], "Text"], Cell[TextData[{ "But in ", Cell[BoxData[ \(TraditionalForm\`V\)]], ", there are no free charges (except on the boundary and we can exclude the \ boundary from ", Cell[BoxData[ \(TraditionalForm\`V\)]], "). So we have ", Cell[BoxData[ \(TraditionalForm\`\[Del]\^2\ \[CapitalPhi] = \(\[Del]\^2 \[CapitalPhi]\ \^\[Prime] = 0\)\)]], " and " }], "Text"], Cell[TextData[{ " ", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_V\((\ E\&\[RightVector]\[CenterDot]E\&\[RightVector]\^\[Prime]\ - E\^\[Prime]2\ )\) \[DifferentialD]\^3 r = \(\[Integral]\_V\ \ \ \[Del]\&\[RightVector]\(\(\[CenterDot]\)\((\[CapitalPhi]\^\[Prime]\ \[Del]\&\ \[RightVector]\((\ \[CapitalPhi] - \[CapitalPhi]\^\[Prime])\))\)\)\ \ \[DifferentialD]\^3 r = \[Integral]\_\(\(\[PartialD]V\)\(\ \)\)\[CapitalPhi]\^\ \[Prime]\ \[Del]\&\[RightVector]\((\ \[CapitalPhi] - \[CapitalPhi]\^\[Prime])\ \)\[CenterDot]\[DifferentialD]A\&\[RightVector]\)\)]], ". " }], "Text"], Cell[TextData[{ "But the charges were assumed to be arranged in just such a way that ", Cell[BoxData[ \(TraditionalForm\`\[CapitalPhi]\^\[Prime]\)]], " is constant on each part of the bounding surface and furthermore the \ values of the charges to be distributed on each surface are specified so that \ ", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_\(\(S\_i\)\(\ \ \)\)\[Del]\&\[RightVector]\((\ \[CapitalPhi] - \[CapitalPhi]\^\[Prime])\)\ \[CenterDot]\[DifferentialD]A\&\[RightVector] = 0\)]], " since the charge density on the surface is just proportional to ", Cell[BoxData[ \(TraditionalForm\`\(E\_ \[UpTee] \)\)]], ". Thus we get" }], "Text"], Cell[TextData[{ " ", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_V\((\ E\&\[RightVector]\[CenterDot]E\&\[RightVector]\^\[Prime]\ - E\^\[Prime]2\ )\) \[DifferentialD]\^3 r = \(\[Sum]\+i \[CapitalPhi]\_i\%\[Prime]\ \ \(\[Integral]\_\(\(S\_i\)\(\ \)\)\[Del]\&\[RightVector]\((\ \[CapitalPhi] - \ \[CapitalPhi]\^\[Prime])\)\[CenterDot]\[DifferentialD]A\&\[RightVector]\) = 0. \)\)]] }], "Text"], Cell["So finally ", "Text"], Cell[TextData[{ " ", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_V\((E\^2\ - \ E\^\[Prime]2)\) \[DifferentialD]\^3 r\ = \[Integral]\_V\ \(\((E\&\[RightVector]\ - \ E\&\[RightVector]\^\[Prime])\)\^2\) \[DifferentialD]\^3 r\ \[GreaterEqual] 0\)]], "\n\nwith equality if and only if ", Cell[BoxData[ \(TraditionalForm\`E\&\[RightVector]\ = \ E\&\[RightVector]\^\[Prime]\)]], ". So we have Thomson's remarkable result that conductors arrange the \ charges on their surfaces to both make all the surfaces equipotentials, but \ also to absolutely minimize the electrostatically stored energy, for \ specified charges on the conductors." }], "Text"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["9. Field Equation in Cylindrical Coordinates", "Section"], Cell[TextData[{ "Cylindrical coordinates are defined by \n", Cell[BoxData[ \(TraditionalForm\`\(\(x = \[Rho]\ cos\ \[CurlyPhi]\)\(,\)\(\ \)\(y = \ \[Rho]\ sin\ \[CurlyPhi]\)\(,\)\(\ \)\(z = z\)\(,\)\(\ \)\)\)]], "where ", Cell[BoxData[ \(TraditionalForm\`\[Rho] \[Element] \([0, \[Infinity])\), \ \ \[CurlyPhi] \[Element] \([0, 2 \[Pi])\), \ z \[Element] \(\((\(-\[Infinity]\), \[Infinity])\)\(.\)\)\)]], "\nFind the divergence and Laplacian in this coordinate system (answer on \ back cover of Jackson)." }], "Text"], Cell[CellGroupData[{ Cell["Solution", "Subsection"], Cell["\<\ a) Cylindrical coordinates are defined pictorially like this:\ \>", "Text"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier 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From ", Cell[BoxData[ \(TraditionalForm\`x = \[Rho]\ cos\ \[CurlyPhi], \ y = \[Rho]\ sin\ \[CurlyPhi], \ z = z\)]], " we easily get the Jacobian matrix: " }], "Text"], Cell[TextData[{ " ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", GridBox[{ {\(\[PartialD]x\/\[PartialD]\[Rho]\), \(\[PartialD]x\/\ \[PartialD]\[CurlyPhi]\), \(\[PartialD]x\/\[PartialD]z\)}, {\(\[PartialD]y\/\[PartialD]\[Rho]\), \(\[PartialD]y\/\ \[PartialD]\[CurlyPhi]\), \(\[PartialD]y\/\[PartialD]z\)}, {\(\[PartialD]z\/\[PartialD]\[Rho]\), \(\[PartialD]z\/\ \[PartialD]\[CurlyPhi]\), \(\[PartialD]z\/\[PartialD]z\)} }], ")"}], "=", RowBox[{ RowBox[{"(", GridBox[{ {\(cos\ \[CurlyPhi]\), \(\(-\[Rho]\)\ sin\ \[CurlyPhi]\), "0"}, {\(sin\ \[CurlyPhi]\), \(\[Rho]\ cos\ \[CurlyPhi]\), "0"}, {"0", "0", "1"} }], ")"}], "."}]}], TraditionalForm]]] }], "Text"], Cell[TextData[{ "The scale factors are just the lengths of the vectors represented by each \ of the three columns of this matrix. So,\n", Cell[BoxData[ \(TraditionalForm\`h\_\[Rho] = 1, \ h\_\[CurlyPhi] = \[Rho], \ h\_z = 1, \)]], "\nand the gradient of a scalar \[CapitalPhi] is\n", Cell[BoxData[ \(TraditionalForm\`\(\[Del]\& \[RightVector] \)\[CapitalPhi] = \(\ \[PartialD]\[CapitalPhi]\/\[PartialD]\[Rho]\) \(u\& \[RightVector] \)\_\[Rho] \ + \(1\/\[Rho]\) \(\[PartialD]\[CapitalPhi]\/\[PartialD]\[CurlyPhi]\) \(u\& \ \[RightVector] \)\_\[CurlyPhi] + \(\[PartialD]\[CapitalPhi]\/\[PartialD]z\) \ \(u\& \[RightVector] \)\_z\)]], ". \nFrom the Jacobian matrix, the unit vectors can be expressed in \ cartesian coordinates as\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ FormBox[\(\(u\&\[RightVector]\_\[Rho]\)\(=\)\), "TraditionalForm"], "cos", " ", "\[CurlyPhi]", \(\(e\& \[RightVector] \)\_x\)}], "+", \(sin\ \[CurlyPhi] \( e\& \[RightVector] \)\_y\)}], TraditionalForm]]], ",\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ FormBox[\(\(\(u\& \[RightVector] \)\_\[CurlyPhi]\)\(=\)\), "TraditionalForm"], "-", \(sin\ \[CurlyPhi] \( e\& \[RightVector] \)\_x\), "+", FormBox[\(cos\ \[CurlyPhi] \( e\& \[RightVector] \)\_y\), "TraditionalForm"]}], ",", "\[IndentingNewLine]", \(\(u\& \[RightVector] \)\_z = \(\(\(e\& \ \[RightVector] \)\_z\)\(.\)\)\)}], TraditionalForm]]], "\nand so we can formally check right handedness:\n", Cell[BoxData[ \(TraditionalForm\`\(\(u\&\[RightVector]\_\[Rho]\[Cross]\(u\& \ \[RightVector] \)\_\[CurlyPhi]\)\(=\)\)\)]], "(", Cell[BoxData[ \(TraditionalForm\`cos\ \[CurlyPhi] \( e\& \[RightVector] \)\_x + sin\ \[CurlyPhi] \( e\& \[RightVector] \)\_y\)]], ")\[Cross](", Cell[BoxData[ FormBox[ RowBox[{\(\(-sin\)\ \[CurlyPhi] \( e\& \[RightVector] \)\_x\), "+", FormBox[\(cos\ \[CurlyPhi] \( e\& \[RightVector] \)\_y\), "TraditionalForm"]}], TraditionalForm]]], ")\n", Cell[BoxData[ \(TraditionalForm\`\(\(=\)\(\((\(cos\^2\) \[CurlyPhi] + \(sin\^2\) \ \[CurlyPhi])\) \(e\& \[RightVector] \)\_x\[Cross]\(e\& \[RightVector] \)\_y\)\ \)\)]], "\n", Cell[BoxData[ FormBox[ RowBox[{"=", RowBox[{ FormBox[\(e\&\[RightVector]\_z\), "TraditionalForm"], "=", \(\(\(u\& \[RightVector] \)\_z\)\(.\)\)}]}], TraditionalForm]]] }], "Text"], Cell["The spatial volume element is", "Text"], Cell[TextData[Cell[BoxData[ \(TraditionalForm\`\[DifferentialD]\^3 r\ = \(\[DifferentialD]x\ \[DifferentialD]y\ \[DifferentialD]z = \(\ \[DifferentialD]\[Rho]\ \ \[Rho]\ \[DifferentialD]\[CurlyPhi]\ \ \ \[DifferentialD]z = \(\[Rho]\ \[DifferentialD]\[Rho]\ \[DifferentialD]\ \[CurlyPhi]\ \ \[DifferentialD]z = \(1\/2\) \[DifferentialD]\[Rho]\^2 \ \[DifferentialD]\[CurlyPhi]\ \ \[DifferentialD]z\)\)\)\)]]], "Text"], Cell["and the action is", "Text"], Cell[TextData[Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalA] = \[Integral]\((\(1\/2\) \(\(\ \[CurlyEpsilon]\_0\)(\((\[PartialD]\[CapitalPhi]\/\[PartialD]\[Rho])\)\^2 + \ \((\(1\/\[Rho]\) \[PartialD]\[CapitalPhi]\/\[PartialD]\[CurlyPhi])\)\^2 + \((\ \[PartialD]\[CapitalPhi]\/\[PartialD]z)\)\^2)\) - \(\[Rho]\_charge\) \ \[CapitalPhi])\) \[Rho]\ \ \ \[DifferentialD]\[Rho] \[DifferentialD]\ \[CurlyPhi] \[DifferentialD]z\)]]], "Text"], Cell["\<\ Thus since in the derivation of the field equations from the action principle \ we used\ \>", "Text"], Cell[TextData[{ " ", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalA] = \[Integral]\(\[ScriptCapitalL](\ \[PartialD]\[CapitalPhi]\/\[PartialD]q\_i\ , \[CapitalPhi], q\_i)\)\ \ \[DifferentialD]q\_1 \[DifferentialD]q\_2 \ \[DifferentialD]q\_3\)]], "\nwe get in the case in hand" }], "Text"], Cell[TextData[{ " ", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalL] = \((\(1\/2\) \ \(\(\[CurlyEpsilon]\_0\)(\((\[PartialD]\[CapitalPhi]\/\[PartialD]\[Rho])\)\^2 \ + \((\(1\/\[Rho]\) \[PartialD]\[CapitalPhi]\/\[PartialD]\[CurlyPhi])\)\^2 + \ \((\[PartialD]\[CapitalPhi]\/\[PartialD]z)\)\^2)\) - \(\[Rho]\_charge\) \ \[CapitalPhi])\) \(\(\[Rho]\)\(\ \ \ \)\(.\)\)\)]] }], "Text"], Cell["\<\ The subtle point is that the action principle is formulated in an integral \ over the parameter space, not geometric space so the Jacobian associated with \ the transformation from cartesian coordinates to new parameters gets \ incorporated into the Lagrangian density.\ \>", "Text"], Cell["Finally, the field equation is", "Text"], Cell[TextData[{ " ", Cell[BoxData[ \(TraditionalForm\`\(\[PartialD]\/\[PartialD]\[Rho]\) \((\(\ \[CurlyEpsilon]\_0\) \[Rho] \[PartialD]\[CapitalPhi]\/\[PartialD]\[Rho])\) + \ \(\[PartialD]\/\[PartialD]\[CurlyPhi]\) \((\(\[CurlyEpsilon]\_0\) \[Rho]\ \(1\ \/\[Rho]\^2\) \[PartialD]\[CapitalPhi]\/\[PartialD]\[CurlyPhi])\) + \(\ \[PartialD]\/\[PartialD]z\) \((\(\[CurlyEpsilon]\_0\) \[Rho]\ \[PartialD]\ \[CapitalPhi]\/\[PartialD]z)\) - \((\(-\[Rho]\)\ \[Rho]\_\(\(charge\)\(\ \ \)\))\) = 0\)]], "," }], "Text"], Cell["or", "Text"], Cell[TextData[{ " ", Cell[BoxData[ \(TraditionalForm\`\(1\/\[Rho]\) \(\[PartialD]\/\[PartialD]\[Rho]\) \((\ \[Rho] \[PartialD]\[CapitalPhi]\/\[PartialD]\[Rho])\) + \(1\/\[Rho]\^2\) \ \[PartialD]\^2 \[CapitalPhi]\/\[PartialD]\[CurlyPhi]\^2 + \[PartialD]\^2 \ \[CapitalPhi]\/\[PartialD]z\^2 = \ \(-\(\[Rho]\_charge\/\[CurlyEpsilon]\_0\)\)\)]], "." }], "Text"], Cell[TextData[{ "This is of course the expression for ", Cell[BoxData[ \(TraditionalForm\`\[Del]\^2 \[CapitalPhi] = \(-\(\[Rho]\_charge\/\ \[CurlyEpsilon]\_0\)\)\)]], " in cylindrical coordinates. You can ", Cell[BoxData[ \(TraditionalForm\`use\ \[Del]\^2 \[CapitalPhi] = \ \[Del]\&\[RightVector]\(\(\[CenterDot]\)\(\[Del]\&\[RightVector] \ \[CapitalPhi]\)\)\)]], " to get the expression for ", Cell[BoxData[ \(TraditionalForm\`\[Del]\&\[RightVector]\(\(\[CenterDot]\)\(A\&\ \[RightVector]\)\)\)]], " in cylindrical coordinates by just replacing the components of ", Cell[BoxData[ \(TraditionalForm\`\[Del]\&\[RightVector] \[CapitalPhi]\)]], ", expressed in the curvilinear coordinate basis at a point (the components \ along ", Cell[BoxData[ \(TraditionalForm\`u\&\[RightVector]\_\(r\_i\)\)]], ") with the components of ", Cell[BoxData[ \(TraditionalForm\`A\&\[RightVector]\)]], " resolved in this basis. The result is easy to see to be" }], "Text"], Cell[TextData[{ " ", Cell[BoxData[ \(TraditionalForm\`\(\(\(\[Del]\& \[RightVector] \ \)\(\(\[CenterDot]\)\(A\& \[RightVector] \)\) = \(1\/\[Rho]\) \(\[PartialD]\/\ \[PartialD]\[Rho]\) \((\[Rho]\ A\_\[Rho])\) + \(1\/\[Rho]\) \[PartialD]A\_\ \[CurlyPhi]\/\[PartialD]\[CurlyPhi] + \ \[PartialD]A\_z\/\[PartialD]z\)\(,\)\)\)]], "\nas agrees with the back of Jackson. You should notice the dimensions in \ each of the terms in these expressions: each term in ", Cell[BoxData[ \(TraditionalForm\`\[Del]\^2 \[CapitalPhi]\)]], " has dimension of ", Cell[BoxData[ \(TraditionalForm\`\[CapitalPhi]/length\^2\)]], " and each component of ", Cell[BoxData[ \(TraditionalForm\`\(\[Del]\& \[RightVector] \)\(\(\[CenterDot]\)\(A\& \ \[RightVector] \)\)\)]], " has units of ", Cell[BoxData[ \(TraditionalForm\`A/length\)]], ". You can use this little check to notice accidental errors in using these \ forms." }], "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["10. Oblate Ellipsoidal Coordinates", "Section"], Cell[TextData[{ "Define oblate ellipsoidal coordinates (\[Mu], \[Nu], \[CurlyPhi]), with \ parameter R > 0, by\nx = R cosh \[Mu] sin \[Nu] cos \[CurlyPhi]\ny = R cosh \ \[Mu] sin \[Nu] sin \[CurlyPhi]\nz = R sinh \[Mu] cos \[Nu]\nwith ", Cell[BoxData[ \(TraditionalForm\`\[Mu] \[Element] {0, \[Infinity]), \ \[Nu] \ \[Element] \([0, \[Pi]]\), \ \[CurlyPhi] \[Element] \(\([0, 2 \[Pi])\)\(.\)\)\)]], "\na) Make a sketch of the lines of constant \[Mu] and \[Nu] in a fixed \ \[CurlyPhi] plane.\nb) Show that these coordinates are orthogonal and right \ handed in the order\n(\[Mu], \[Nu], \[CurlyPhi]), and find the scale factors \ ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{\(h\_\[Mu]\), ",", FormBox[\(h\_\[Nu]\), "TraditionalForm"], ",", FormBox[\(h\_\[CurlyPhi]\), "TraditionalForm"]}], ")"}], TraditionalForm]]], ". Identify the singular points of the transformation.\nc) Finally, write \ out the gradient and use the electrostatics variational principle\nto \ calculate the Laplacian and, from it and the gradient, get the divergence\n\ operator in this coordinate system." }], "Text"], Cell[CellGroupData[{ Cell["Solution", "Subsection"], Cell[TextData[{ "b) Oblate ellipsoidal coordinates are defined by\n\n", Cell[BoxData[{ \(TraditionalForm\`x = R\ cosh\ \[Mu]\ \ \ sin\ v\ \ cos\ \[CurlyPhi]\), "\[IndentingNewLine]", \(TraditionalForm\`y = R\ cosh\ \[Mu]\ \ \ sin\ v\ \ sin\ \[CurlyPhi]\), "\[IndentingNewLine]", \(TraditionalForm\`\ z = R\ sinh\ \[Mu]\ \ cos\ v\)}]], "\n\nwith ", Cell[BoxData[ \(TraditionalForm\`R > 0\)]], " a given parameter, ", Cell[BoxData[ \(TraditionalForm\`\[Mu] \[GreaterEqual] 0, \ \ 0 \[LessEqual] v \[LessEqual] \[Pi], \ 0 \[LessEqual] \[CurlyPhi]\ < 2 \( \(\[Pi]\)\(.\)\)\)]], " Note that as in spherical coordinates where ", Cell[BoxData[ \(TraditionalForm\`sin\ \[Theta] \[GreaterEqual] 0\)]], ", here we have ", Cell[BoxData[ \(TraditionalForm\`sin\ v\ \[GreaterEqual] \(\(0\)\(\ \)\(.\)\)\)]], "\nFirst, the name of these coordinates is oblate spheroidal (or \ ellipsoidal); an oblate spheroid is sort of the shape of a basketball when \ you sit on it. A prolate spheroid is like a football. The ellipsoids in the \ coordinates given here are like a flattened sphere; interchange the \"cosh\" \ and \"sinh\" to get the prolate case. First the range of \[Mu] is 0 to \ \[Infinity], that of \[Nu] is 0 to \[Pi], and that of \[CurlyPhi] is 0 to 2 \ \[Pi]. Draw the coordinates for the plane ", Cell[BoxData[ \(TraditionalForm\`\[CurlyPhi] = 0, \ so\ x = R\ cosh\ \[Mu]\ \ \ sin\ \[Nu], \ z = R\ sinh\ \[Mu]\ \ cos\ \[Nu]\)]], ". 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L .70534 .95831 L .70236 .94464 L .69868 .93145 L .69383 .9173 L .68273 .89197 L .66888 .86735 L .65122 .84202 L .61178 .79831 L .55965 .75529 L .50046 .71799 L .42894 .68345 L .35289 .65574 L .27434 .63471 L .23214 .62612 L .18639 .61875 L .16568 .61605 L .14323 .61356 L .12372 .61174 L .10232 .61014 L .08254 .609 L .06411 .60823 L .0539 .60793 L .04462 .60773 L .03948 .60765 L .03677 .60762 L .03391 .60759 L .0314 .60758 L .02865 .60756 L .02615 .60755 L .02381 .60755 L Mfstroke .02381 1.45333 m .02655 1.45333 L .02904 1.45332 L .03192 1.4533 L .03464 1.45328 L .03948 1.45322 L .04474 1.45313 L .05052 1.45301 L .05665 1.45284 L .06764 1.45246 L .07897 1.45195 L .08957 1.45136 L .11365 1.44965 L .13811 1.44736 L .16037 1.44479 L .20993 1.43733 L .25224 1.42901 L .2974 1.418 L .38207 1.39079 L .45753 1.35797 L .52978 1.31657 L .59042 1.27105 L .63945 1.22278 L .66055 1.19687 L .68067 1.1675 L .69692 1.13844 L .70903 1.11137 L .71878 1.08288 L .72292 1.06704 L .72603 1.05196 L .72826 1.03759 L .72911 1.03032 L .72983 1.02229 L .73028 1.01546 L .73046 1.01162 L .73053 1.00979 L .73058 1.00809 L .73062 1.00647 L .73065 1.00499 L .73067 1.00335 L .73069 1.0016 L .73069 .99991 L .73068 .99808 L .73067 .99635 L .73064 .99476 L .7306 .99288 L .73055 .9911 L .73043 .98777 L .73026 .98418 L .73002 .98027 L Mistroke .72946 .97324 L .72878 .96673 L .72669 .95184 L .72362 .93605 L .71982 .92082 L .71482 .90447 L .70338 .87521 L .68909 .84678 L .67088 .81751 L .6302 .76703 L .57644 .71733 L .51539 .67424 L .44164 .63434 L .3632 .60234 L .28219 .57804 L .23866 .56812 L .19149 .55961 L .17013 .55649 L .14698 .55361 L .12685 .55151 L .10478 .54965 L .08438 .54834 L .06537 .54746 L .05484 .54711 L .04527 .54688 L .03997 .54679 L .03718 .54675 L .03423 .54672 L .03164 .5467 L .0288 .54668 L .02622 .54667 L .02381 .54667 L Mfstroke .02381 1.51623 m .02665 1.51622 L .02923 1.51621 L .03221 1.51619 L .03502 1.51617 L .04003 1.5161 L .04547 1.516 L .05145 1.51586 L .05779 1.51567 L .06916 1.51523 L .08089 1.51465 L .09186 1.51399 L .11678 1.51204 L .14209 1.50943 L .16512 1.5065 L .21641 1.49801 L .2602 1.48853 L .30693 1.47599 L .39455 1.44501 L .47263 1.40763 L .5474 1.36049 L .61015 1.30866 L .66089 1.25369 L .68272 1.22419 L .70354 1.19074 L .72037 1.15764 L .73289 1.12682 L .74298 1.09438 L .74726 1.07634 L .75049 1.05917 L .75279 1.04281 L .75367 1.03453 L .75442 1.02538 L .75488 1.01761 L .75507 1.01323 L .75514 1.01115 L .75519 1.00922 L .75523 1.00736 L .75526 1.00568 L .75529 1.00381 L .7553 1.00182 L .75531 .99989 L .7553 .99781 L .75528 .99584 L .75526 .99403 L .75522 .99189 L .75517 .98987 L .75504 .98607 L .75486 .98198 L .75461 .97753 L Mistroke .75403 .96952 L .75333 .96211 L .75117 .94516 L .74799 .92717 L .74406 .90983 L .73888 .89122 L .72704 .85789 L .71226 .82552 L .69342 .79219 L .65132 .7347 L .59569 .67811 L .53251 .62904 L .45619 .58361 L .37502 .54717 L .29119 .51949 L .24615 .5082 L .19733 .49851 L .17522 .49495 L .15126 .49167 L .13044 .48929 L .10759 .48717 L .08648 .48567 L .06682 .48467 L .05592 .48427 L .04602 .48401 L .04053 .48391 L .03764 .48387 L .03459 .48383 L .03192 .48381 L .02898 .48379 L .02631 .48378 L .02381 .48377 L Mfstroke .02381 1.58142 m .02675 1.58141 L .02943 1.5814 L .03253 1.58138 L .03545 1.58135 L .04065 1.58127 L .04629 1.58116 L .0525 1.581 L .05909 1.58079 L .07089 1.5803 L .08306 1.57965 L .09445 1.5789 L .12033 1.5767 L .1466 1.57377 L .17051 1.57047 L .22375 1.5609 L .2692 1.55022 L .31772 1.5361 L .40868 1.50121 L .48973 1.45911 L .56735 1.40602 L .63249 1.34764 L .68516 1.28572 L .70783 1.2525 L .72944 1.21483 L .7469 1.17755 L .75991 1.14284 L .77038 1.1063 L .77483 1.08598 L .77817 1.06664 L .78056 1.04822 L .78148 1.03889 L .78226 1.02859 L .78274 1.01983 L .78293 1.0149 L .783 1.01256 L .78306 1.01038 L .7831 1.00829 L .78313 1.0064 L .78316 1.00429 L .78317 1.00205 L .78318 .99988 L .78317 .99753 L .78315 .99532 L .78313 .99327 L .78308 .99087 L .78303 .98859 L .7829 .98431 L .78272 .97971 L .78246 .97469 L Mistroke .78185 .96568 L .78113 .95733 L .77888 .93823 L .77558 .91798 L .77151 .89845 L .76613 .87748 L .75384 .83994 L .73849 .80348 L .71893 .76594 L .67523 .7012 L .61748 .63746 L .5519 .5822 L .47266 .53102 L .3884 .48998 L .30137 .45881 L .25462 .44609 L .20394 .43518 L .18099 .43117 L .15612 .42748 L .13451 .42479 L .11079 .42241 L .08887 .42072 L .06846 .41959 L .05714 .41914 L .04687 .41885 L .04117 .41873 L .03817 .41869 L .035 .41865 L .03222 .41862 L .02917 .4186 L .0264 .41859 L .02381 .41858 L Mfstroke .02381 1.64919 m .02688 1.64919 L .02967 1.64918 L .03288 1.64915 L .03593 1.64912 L .04134 1.64904 L .04722 1.64891 L .05368 1.64873 L .06054 1.64849 L .07283 1.64795 L .0855 1.64722 L .09736 1.64638 L .1243 1.64393 L .15165 1.64065 L .17654 1.63697 L .23197 1.62629 L .2793 1.61436 L .32981 1.5986 L .42451 1.55964 L .50891 1.51263 L .58972 1.45335 L .65754 1.38816 L .71237 1.31903 L .73598 1.28194 L .75847 1.23988 L .77666 1.19825 L .79019 1.15949 L .8011 1.11869 L .80573 1.09601 L .80921 1.07441 L .8117 1.05384 L .81265 1.04343 L .81347 1.03192 L .81397 1.02214 L .81417 1.01664 L .81424 1.01402 L .8143 1.01159 L .81434 1.00926 L .81438 1.00715 L .8144 1.0048 L .81442 1.00228 L .81442 .99987 L .81442 .99725 L .8144 .99477 L .81437 .99249 L .81433 .98981 L .81427 .98726 L .81414 .98248 L .81394 .97734 L .81368 .97174 L Mistroke .81305 .96167 L .81229 .95235 L .80995 .93103 L .80652 .90841 L .80227 .88661 L .79667 .8632 L .78388 .82129 L .7679 .78058 L .74753 .73866 L .70203 .66637 L .64191 .5952 L .57363 .53349 L .49113 .47635 L .4034 .43053 L .31279 .39573 L .26411 .38152 L .21135 .36933 L .18746 .36486 L .16156 .36074 L .13906 .35774 L .11437 .35508 L .09155 .35319 L .0703 .35193 L .05852 .35143 L .04782 .3511 L .04188 .35097 L .03876 .35092 L .03546 .35088 L .03257 .35085 L .0294 .35082 L .02651 .35081 L .02381 .35081 L Mfstroke .02381 1.71986 m .02701 1.71985 L .02992 1.71984 L .03328 1.71981 L .03646 1.71977 L .04211 1.71968 L .04824 1.71954 L .055 1.71934 L .06215 1.71908 L .07498 1.71847 L .08821 1.71766 L .10059 1.71674 L .12872 1.71402 L .15727 1.71038 L .18326 1.7063 L .24113 1.69446 L .29053 1.68124 L .34326 1.66375 L .44213 1.62055 L .53024 1.56843 L .6146 1.5027 L .6854 1.43041 L .74265 1.35376 L .76729 1.31262 L .79078 1.26599 L .80976 1.21983 L .82389 1.17685 L .83528 1.13161 L .84011 1.10646 L .84375 1.08251 L .84634 1.0597 L .84734 1.04815 L .84819 1.0354 L .84871 1.02455 L .84892 1.01845 L .84899 1.01555 L .84906 1.01285 L .8491 1.01027 L .84914 1.00792 L .84916 1.00532 L .84918 1.00253 L .84919 .99985 L .84918 .99695 L .84916 .9942 L .84913 .99167 L .84908 .9887 L .84903 .98587 L .84889 .98058 L .84868 .97488 L .8484 .96866 L Mistroke .84775 .9575 L .84696 .94716 L .84452 .92352 L .84093 .89845 L .8365 .87427 L .83065 .84831 L .8173 .80183 L .80061 .75669 L .77935 .71021 L .73185 .63005 L .66908 .55114 L .5978 .48271 L .51168 .41936 L .42009 .36854 L .3255 .32995 L .27468 .3142 L .2196 .30069 L .19465 .29573 L .16762 .29115 L .14413 .28783 L .11835 .28488 L .09453 .28279 L .07234 .28139 L .06004 .28084 L .04887 .28047 L .04268 .28033 L .03942 .28027 L .03597 .28022 L .03296 .28019 L .02964 .28016 L .02663 .28015 L .02381 .28014 L Mfstroke .02381 1.79372 m .02716 1.79372 L .03021 1.7937 L .03372 1.79367 L .03705 1.79363 L .04296 1.79353 L .04938 1.79337 L .05645 1.79316 L .06394 1.79287 L .07737 1.7922 L .09121 1.7913 L .10417 1.79028 L .1336 1.78728 L .16349 1.78328 L .19068 1.77877 L .25124 1.76572 L .30295 1.75114 L .35814 1.73186 L .46161 1.68423 L .55382 1.62676 L .64211 1.55428 L .71621 1.47457 L .77612 1.39006 L .80191 1.3447 L .82649 1.29328 L .84635 1.24239 L .86114 1.195 L .87306 1.14511 L .87812 1.11738 L .88193 1.09097 L .88464 1.06582 L .88568 1.05309 L .88657 1.03903 L .88712 1.02707 L .88733 1.02034 L .88742 1.01714 L .88748 1.01417 L .88753 1.01132 L .88757 1.00874 L .88759 1.00586 L .88761 1.00279 L .88762 .99984 L .88761 .99663 L .88759 .99361 L .88756 .99082 L .88751 .98754 L .88745 .98442 L .8873 .97858 L .88709 .9723 L .8868 .96545 L Mistroke .88611 .95314 L .88529 .94174 L .88273 .91568 L .87898 .88803 L .87434 .86137 L .86822 .83275 L .85424 .7815 L .83678 .73173 L .81453 .68048 L .76482 .59209 L .69913 .50508 L .62453 .42963 L .5344 .35978 L .43855 .30375 L .33955 .2612 L .28636 .24383 L .22871 .22893 L .20261 .22347 L .17432 .21842 L .14973 .21476 L .12275 .2115 L .09782 .2092 L .0746 .20765 L .06173 .20704 L .05004 .20664 L .04356 .20649 L .04015 .20642 L .03654 .20636 L .03338 .20633 L .02991 .2063 L .02676 .20628 L .02381 .20628 L Mfstroke .02381 1.87111 m .02732 1.87111 L .03052 1.87109 L .03421 1.87106 L .0377 1.87101 L .0439 1.8709 L .05063 1.87073 L .05805 1.87049 L .0659 1.87017 L .07999 1.86944 L .09451 1.86846 L .1081 1.86734 L .13897 1.86405 L .17032 1.85965 L .19885 1.85471 L .26237 1.84038 L .31661 1.82438 L .3745 1.80322 L .48303 1.75094 L .57975 1.68787 L .67236 1.60832 L .75009 1.52085 L .81293 1.42809 L .83998 1.37831 L .86577 1.32188 L .88661 1.26602 L .90212 1.21401 L .91462 1.15926 L .91993 1.12882 L .92392 1.09984 L .92677 1.07224 L .92786 1.05827 L .92879 1.04283 L .92936 1.02971 L .92959 1.02232 L .92968 1.01881 L .92975 1.01555 L .9298 1.01243 L .92984 1.00959 L .92987 1.00643 L .92988 1.00307 L .92989 .99982 L .92988 .9963 L .92986 .99298 L .92983 .98992 L .92978 .98632 L .92972 .9829 L .92956 .97649 L .92934 .9696 L .92903 .96208 L Mistroke .92831 .94857 L .92745 .93606 L .92476 .90745 L .92083 .87711 L .91596 .84785 L .90955 .81644 L .89488 .7602 L .87657 .70557 L .85323 .64932 L .80108 .55232 L .73218 .45683 L .65392 .37402 L .55938 .29735 L .45884 .23586 L .355 .18916 L .29921 .1701 L .23874 .15375 L .21136 .14775 L .18168 .14221 L .15589 .13819 L .12759 .13462 L .10144 .13209 L .07709 .13039 L .06358 .12972 L .05132 .12929 L .04452 .12911 L .04095 .12904 L .03716 .12898 L .03385 .12894 L .03021 .12891 L .0269 .12889 L .02381 .12889 L Mfstroke .02381 1.95238 m .0275 1.95237 L .03086 1.95235 L .03474 1.95232 L .03841 1.95227 L .04493 1.95215 L .052 1.95196 L .0598 1.9517 L .06805 1.95135 L .08286 1.95055 L .09812 1.94948 L .11241 1.94825 L .14486 1.94466 L .17781 1.93985 L .20779 1.93444 L .27457 1.91878 L .33158 1.90128 L .39242 1.87815 L .5065 1.821 L .60816 1.75204 L .7055 1.66507 L .7872 1.56944 L .85326 1.46802 L .88169 1.41361 L .90879 1.3519 L .9307 1.29084 L .947 1.23397 L .96014 1.17412 L .96572 1.14084 L .96991 1.10916 L .97291 1.07898 L .97406 1.06371 L .97504 1.04683 L .97564 1.03248 L .97588 1.02441 L .97597 1.02057 L .97604 1.017 L .97609 1.01359 L .97613 1.01048 L .97616 1.00704 L .97618 1.00335 L .97619 .9998 L .97618 .99596 L .97616 .99233 L .97613 .98898 L .97607 .98505 L .97601 .98131 L .97584 .9743 L .97561 .96676 L .97529 .95854 L Mistroke .97453 .94377 L .97362 .9301 L .9708 .89882 L .96667 .86564 L .96155 .83365 L .95481 .79931 L .93939 .73782 L .92014 .6781 L .89561 .61661 L .8408 .51056 L .76837 .40615 L .68612 .31562 L .58675 .2318 L .48107 .16457 L .37192 .11352 L .31328 .09268 L .24972 .0748 L .22094 .06824 L .18975 .06219 L .16264 .05779 L .13289 .05389 L .10541 .05112 L .07981 .04927 L .06562 .04854 L .05273 .04806 L .04558 .04787 L .04182 .04779 L .03785 .04772 L .03436 .04768 L .03054 .04764 L .02706 .04762 L .02381 .04762 L Mfstroke [ .02 .02 ] 0 setdash .02381 1 m .02381 1.03284 L .02381 1.06872 L .02381 1.10254 L .02381 1.13523 L .02381 1.17026 L .02381 1.20428 L .02381 1.24081 L .02381 1.27643 L .02381 1.31116 L .02381 1.34867 L .02381 1.38539 L .02381 1.42131 L .02381 1.4603 L .02381 1.49859 L .02381 1.54022 L .02381 1.58127 L .02381 1.62171 L .02381 1.66585 L .02381 1.70948 L .02381 1.75716 L .02381 1.80447 L .02381 1.85134 L .02381 1.90271 L .02381 1.95113 L s .15203 1 m .15203 1.00098 L .15203 1.00187 L .15203 1.00289 L .15204 1.00386 L .15204 1.00559 L .15204 1.00746 L .15204 1.00952 L .15205 1.01171 L .15206 1.01563 L .15207 1.01968 L .15209 1.02348 L .15214 1.03212 L .15221 1.04094 L .15228 1.04902 L .1525 1.06722 L .15274 1.08307 L .15306 1.10041 L .15387 1.13457 L .15488 1.16769 L .15619 1.20319 L .15769 1.23776 L .15936 1.27143 L .16138 1.30772 L .16357 1.34321 L .16589 1.37789 L .16862 1.41548 L .17149 1.45236 L .17481 1.4924 L .17828 1.53185 L .18188 1.57067 L .18598 1.613 L .19021 1.65481 L .19455 1.69606 L .19946 1.74119 L .20448 1.78587 L .21015 1.8348 L .21594 1.8834 L .22167 1.93035 L s .27465 1 m .27465 1.00091 L .27465 1.00174 L .27466 1.0027 L .27466 1.00361 L .27466 1.00522 L .27467 1.00697 L .27467 1.0089 L .27468 1.01094 L .2747 1.0146 L .27473 1.01838 L .27476 1.02193 L .27486 1.03 L .27499 1.03824 L .27513 1.04578 L .27556 1.06278 L .27603 1.07759 L .27666 1.09378 L .27825 1.12568 L .28022 1.15661 L .28278 1.18977 L .28572 1.22206 L .28899 1.2535 L .29294 1.2874 L .29721 1.32054 L .30177 1.35294 L .3071 1.38804 L .31272 1.42249 L .31922 1.45988 L .32601 1.49672 L .33303 1.53298 L .34106 1.57251 L .34934 1.61156 L .35783 1.65009 L .36744 1.69224 L 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Note that you can write" }], "Text"], Cell[TextData[Cell[BoxData[ \(TraditionalForm\`\(\(\((x\/\(R\ cosh\ \[Mu]\))\)\^2 + \((z\/\(R\ sinh\ \ \[Mu]\))\)\^2 = \(sin\^2\ \[Nu] + cos\^2\ \[Nu] = 1\)\)\(,\)\)\)]]], "Text"], Cell[" which defines ellipses as the solid lines are. Similarly ", "Text"], Cell[TextData[Cell[BoxData[ \(TraditionalForm\`\(\(\((x\/\(R\ sin\ \[Nu]\))\)\^2 - \((z\/\(R\ cos\ \ \[Nu]\))\)\^2 = \(cosh\^2\ \[Mu] - \(sinh\^2\) \[Mu] = 1\)\)\(,\)\)\)]]], "Text"], Cell[" which defines hyperbolas, as the dotted lines are. ", "Text"], Cell[TextData[{ "The plot is only in the ", Cell[BoxData[ \(TraditionalForm\`x \[GreaterEqual] 0\)]], " region since ", Cell[BoxData[ \(TraditionalForm\`\[CurlyPhi] = 0\)]], ", ", Cell[BoxData[ \(TraditionalForm\`cosh\ \[Mu] > 0, \ \ and\ \ sin\ v \[GreaterEqual] 0\)]], "; the ", Cell[BoxData[ \(TraditionalForm\`x \[LessEqual] 0\)]], " region is covered by ", Cell[BoxData[ \(TraditionalForm\`\[CurlyPhi] = \(\(\[Pi]\)\(.\)\)\)]], " The disk in the ", Cell[BoxData[ \(TraditionalForm\`x - z\)]], " plane of radius ", Cell[BoxData[ \(TraditionalForm\`R\)]], " is special, since it is covered by a constant value 0 of the ", Cell[BoxData[ \(TraditionalForm\`\[Mu]\)]], " coordinate. Note that ", Cell[BoxData[ \(TraditionalForm\`\((\[Mu], v)\) = \((0, 0)\)\)]], " covers the top of this disk while", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{Cell[""], Cell[""], \((\[Mu], v)\)}], "=", \((0, \[Pi])\)}], TraditionalForm]]], " covers its bottom. Further, this coordinate system is like ordinary \ spherical polar coordinates in which the origin point has been smashed into a \ disk and we can distinguish one side of the disk from the other." }], "Text"], Cell[TextData[{ "The vectors ", Cell[BoxData[ \(TraditionalForm\`\((\[DifferentialD]r\&\[RightVector]\/\ \[DifferentialD]\ \[Mu], \[DifferentialD]r\&\[RightVector]\/\[DifferentialD]\ \ v, \[DifferentialD]r\&\[RightVector]\/\[DifferentialD]\ \[CurlyPhi])\)\)]], " can be written as ", Cell[BoxData[ \(TraditionalForm\`\((\(h\_\[Mu]\) e\&\[RightVector]\_\[Mu], \ \(h\_v\) e\&\[RightVector]\_v, \ \(h\_\[CurlyPhi]\) e\&\[RightVector]\_\[CurlyPhi])\)\)]], ", where the three vectors are unit length. Thus ", Cell[BoxData[ \(TraditionalForm\`h\_\[Mu] = \(\(\(|\)\(\[DifferentialD]r\&\ \[RightVector]\/\[DifferentialD]\ \[Mu]\)\(|\)\) = \@\((\[DifferentialD]r\&\ \[RightVector]\/\[DifferentialD]\ \[Mu])\)\^2\), \ h\_v = \(\(\(|\)\(\[DifferentialD]r\&\[RightVector]\/\[DifferentialD]v\)\ \(|\)\) = \@\((\[DifferentialD]r\&\[RightVector]\/\[DifferentialD]\ \ v)\)\^2\), \ h\_\[CurlyPhi] = \(\(\(|\)\(\[DifferentialD]r\&\[RightVector]\/\ \[DifferentialD]\ \[CurlyPhi]\)\(|\)\) = \(\(\@\((\[DifferentialD]r\&\ \[RightVector]\/\[DifferentialD]\ \[CurlyPhi])\)\^2\)\(.\)\)\)\)]], " Writing the three vectors as columns, the form ", Cell[BoxData[ \(TraditionalForm\`\((\[DifferentialD]r\&\[RightVector]\/\ \[DifferentialD]\ \[Mu], \[DifferentialD]r\&\[RightVector]\/\[DifferentialD]\ \ v, \[DifferentialD]r\&\[RightVector]\/\[DifferentialD]\ \[CurlyPhi])\)\)]], " is just the Jacobian matrix, \n", Cell[BoxData[ FormBox[ RowBox[{\(J\+\(~\+~\)\), "=", RowBox[{ RowBox[{"(", GridBox[{ {\(\[DifferentialD]x\/\[DifferentialD]\ \[Mu]\), \(\ \[DifferentialD]x\/\[DifferentialD]\ v\), \(\[DifferentialD]x\/\[DifferentialD]\ \ \[CurlyPhi]\)}, {\(\[DifferentialD]y\/\[DifferentialD]\ \[Mu]\), \(\ \[DifferentialD]y\/\[DifferentialD]\ v\), \(\[DifferentialD]y\/\[DifferentialD]\ \ \[CurlyPhi]\)}, {\(\[DifferentialD]z\/\[DifferentialD]\ \[Mu]\), \(\ \[DifferentialD]z\/\[DifferentialD]\ v\), \(\[DifferentialD]z\/\[DifferentialD]\ \ \[CurlyPhi]\)} }], ")"}], "=", RowBox[{\(O\+\(~\+~\)\ D\+\(~\+~\)\), "=", RowBox[{ RowBox[{"(", GridBox[{ {\(\(1\/h\_\[Mu]\) \[DifferentialD]x\/\[DifferentialD]\ \ \[Mu]\), \(\(1\/h\_v\) \[DifferentialD]x\/\[DifferentialD]\ v\), \(\(1\/h\_\ \[CurlyPhi]\) \[DifferentialD]x\/\[DifferentialD]\ \[CurlyPhi]\)}, {\(\(1\/h\_\[Mu]\) \[DifferentialD]y\/\[DifferentialD]\ \ \[Mu]\), \(\(1\/h\_v\) \[DifferentialD]y\/\[DifferentialD]\ v\), \(\(1\/h\_\ \[CurlyPhi]\) \[DifferentialD]y\/\[DifferentialD]\ \[CurlyPhi]\)}, {\(\(1\/h\_\[Mu]\) \[DifferentialD]z\/\[DifferentialD]\ \ \[Mu]\), \(\(1\/h\_v\) \[DifferentialD]z\/\[DifferentialD]\ v\), \(\(1\/h\_\ \[CurlyPhi]\) \[DifferentialD]z\/\[DifferentialD]\ \[CurlyPhi]\)} }], ")"}], " ", RowBox[{ RowBox[{"(", GridBox[{ {\(h\_\[Mu]\), "0", "0"}, {"0", \(h\_v\), "0"}, {"0", "0", \(h\_\[CurlyPhi]\)} }], ")"}], "."}]}]}]}]}], TraditionalForm]]] }], "Text"], Cell["For the coordinate system in hand,", "Text"], Cell[TextData[Cell[BoxData[{ FormBox[ RowBox[{\(\[DifferentialD]r\&\[RightVector]\/\[DifferentialD]\ \[Mu]\), "=", RowBox[{ RowBox[{ RowBox[{"(", GridBox[{ {GridBox[{ {\(R\ sinh\ \[Mu]\ \ sin\ v\ \ cos\ \[CurlyPhi]\)}, {\(R\ sinh\ \[Mu]\ \ \ sin\ v\ sin\ \[CurlyPhi]\)} }]}, {\(R\ cosh\ \[Mu]\ \ cos\ v\)} }], ")"}], " ", "\[Implies]", " ", \(h\_\[Mu]\)}], "=", \(R \@\(\(\ \)\(sinh\^2\ \[Mu]\ \ \ sin\^2\ v\ + \ \(\(cosh\^2\)\(\ \)\(\[Mu]\)\(\ \ \)\(cos\^2\)\(\ \)\(v\)\(\ \)\)\)\)\)}]}], TraditionalForm], "\[IndentingNewLine]", FormBox[\(\(=\)\(R \@\(\(\ \)\(sinh\^2\ \[Mu]\ \ \ sin\^2\ v\ + cosh\^2\ \ \[Mu]\ \ - cosh\^2\ \[Mu]\ \ sin\^2\ v\)\)\)\), TraditionalForm], "\[IndentingNewLine]", FormBox[\(\(=\)\(R \(\(\@\(\(\ \)\(cosh\^2\ \[Mu]\ \ - \ \ sin\^2\ v\)\)\ \)\(.\)\)\)\), TraditionalForm]}]]], "Text"], Cell[TextData[{ "Notice that ", Cell[BoxData[ FormBox[Cell[""], TraditionalForm]]], Cell[BoxData[ \(TraditionalForm\`cosh\^2\ \[Mu] \[GreaterEqual] 1\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\(\(sin\^2\ v\)\(\[LessEqual]\)\(1.\)\(\ \)\)\)]], "Thus ", Cell[BoxData[ \(TraditionalForm\`h\_\[Mu]\)]], "is real and positive except at the singular point ", Cell[BoxData[ \(TraditionalForm\`\[Mu] = \(v = 0. \)\)]] }], "Text"], Cell[TextData[{ " ", Cell[BoxData[{ FormBox[ RowBox[{\(\[DifferentialD]r\&\[RightVector]\/\[DifferentialD]\ v\), "=", RowBox[{ RowBox[{ RowBox[{"(", GridBox[{ {GridBox[{ {\(R\ cosh\ \[Mu]\ \ cos\ v\ \ cos\ \[CurlyPhi]\)}, {\(R\ cosh\ \[Mu]\ \ \ cos\ v\ sin\ \[CurlyPhi]\)} }]}, {\(\(-R\)\ sinh\ \[Mu]\ \ sin\ v\)} }], ")"}], " ", "\[Implies]", " ", \(h\_v\)}], "=", \(R \@\( cosh\^2\ \[Mu]\ \ cos\^2\ v\ + \ \(\(sinh\^2\)\(\ \ \)\(\[Mu]\)\(\ \ \ \)\(sin\^2\)\(\ \)\(v\)\(\ \)\)\)\)}]}], TraditionalForm], "\[IndentingNewLine]", FormBox[\(\(=\)\(R \@\(\(\ \)\(cosh\^2\ \[Mu]\ \ - \ \ \(\(sin\^2\)\(\ \ \)\(v\)\(\ \)\)\)\) = \(\(h\_\[Mu]\)\(.\)\)\)\), TraditionalForm]}]] }], "Text"], Cell[TextData[{ "Finally,\n", Cell[BoxData[ FormBox[ RowBox[{\(\[DifferentialD]r\&\[RightVector]\/\[DifferentialD]\ \ \[CurlyPhi]\), "=", RowBox[{ RowBox[{ RowBox[{"(", GridBox[{ {GridBox[{ {\(\(-R\)\ cosh\ \[Mu]\ sin\ v\ \ sin\ \ \[CurlyPhi]\)}, {\(R\ cosh\ \[Mu]\ \ sin\ v\ cos\ \[CurlyPhi]\)} }]}, {"0"} }], ")"}], " ", "\[Implies]", " ", \(h\_\[CurlyPhi]\)}], "=", \(\(R\)\(\ \)\(cosh\)\(\ \)\(\[Mu]\)\(\ \)\(sin\)\(\ \)\(v\)\ \(\ \)\)}]}], TraditionalForm]]], " (note that I have used ", Cell[BoxData[ \(TraditionalForm\`sin\ v \[GreaterEqual] 0\)]], " and ", Cell[BoxData[ \(TraditionalForm\`cosh\ \[Mu] > 0\)]], ")." }], "Text"], Cell[TextData[{ "Now write the matrix ", Cell[BoxData[ \(TraditionalForm\`O\+\(~\+~\)\)]], " and show that it is orthogonal with a unit determinate which is just ", Cell[BoxData[ \(TraditionalForm\`e\&\[RightVector]\_\[Mu]\[CenterDot]\(\((e\&\ \[RightVector]\_v\[Cross]e\&\[RightVector]\_\[CurlyPhi])\)\(.\)\(\ \)\)\)]], " This will show that the system is a right-handed orthogonal curvilinear \ coordinate system (with the order of the unit vectors as specified)." }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{"Om", "=", RowBox[{"(", GridBox[{ {\(\(Sinh[\[Mu]] Sin[v] Cos[\[CurlyPhi]]\)\/\@\(Cosh[\[Mu]]\^2 - Sin[v]\^2\)\), \ \(\(Cosh[\[Mu]] Cos[v] Cos[\[CurlyPhi]]\)\/\@\(Cosh[\[Mu]]\^2 - Sin[v]\^2\)\), \ \(-\(\(Cosh[\[Mu]] Sin[v] Sin[\[CurlyPhi]]\)\/\(Cosh[\[Mu]] Sin[v]\)\)\)}, {\(\(Sinh[\[Mu]] Sin[v] Sin[\[CurlyPhi]]\)\/\@\(Cosh[\[Mu]]\^2 - Sin[v]\^2\)\), \ \(\(Cosh[\[Mu]] Cos[v] Sin[\[CurlyPhi]]\)\/\@\(Cosh[\[Mu]]\^2 - Sin[v]\^2\)\), \ \(\(Cosh[\[Mu]] Sin[v] Cos[\[CurlyPhi]]\)\/\(Cosh[\[Mu]] Sin[v]\)\)}, {\(\(Cosh[\[Mu]] Cos[v]\)\/\@\(Cosh[\[Mu]]\^2 - Sin[v]\^2\)\), \ \(-\(\(Sinh[\[Mu]] Sin[v]\)\/\@\(Cosh[\[Mu]]\^2 - Sin[v]\^2\)\)\), "0"} }], ")"}]}], ";"}]], "Input", CellLabel->"In[29]:="], Cell["Check the orthogonality and the determinate of this matrix", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[Transpose[Om] . Om]\)], "Input", CellLabel->"In[30]:="], Cell[BoxData[ \({{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}\)], "Output", CellLabel->"Out[30]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[Det[Om]]\)], "Input", CellLabel->"In[31]:="], Cell[BoxData[ \(1\)], "Output", CellLabel->"Out[31]="] }, Open ]], Cell[TextData[{ "So we have a nice coordinate system, i.e., the unit vectors are mutually \ orthogonal. This formally shows that the lines of fixed ", Cell[BoxData[ \(TraditionalForm\`\[Mu]\)]], " and ", Cell[BoxData[ \(TraditionalForm\`v\)]], " intersect orthogonally in the diagram above. Further from the equation \ for the ", "Jacob", "ian of the transformation from cartesian to oblate spheroidal coordinates \ we have \n", Cell[BoxData[ \(TraditionalForm\`J = \(\(\(|\)\(det\ J\+\(~\+~\)\)\(|\)\) = \(\(\(|\)\ \(det\ \((O\+\(~\+~\)\ D\+\(~\+~\))\)\)\(|\)\) = \(\(\(|\)\(det\ \ O\+\(~\+~\)\)\(|\)\(\ \)\(|\)\(det\ D\+\(~\+~\)\)\(|\)\) = \(h\_\[Mu]\ h\_v\ \ h\_\[CurlyPhi] = R\^3\ \((\ cosh\^2\ \[Mu]\ \ - \ \ sin\^2\ v\ )\)\ cosh\ \[Mu]\ sin\ \ \(\(v\)\(.\)\)\)\)\)\)\)]] }], "Text"], Cell[TextData[{ "The gradient is \n\n", Cell[BoxData[ \(TraditionalForm\`\[Del]\&\[RightVector] \[CapitalPhi] = \(1\/\(R \@\(\ \(\ \)\(cosh\^2\ \[Mu]\ \ - \ \ sin\^2\ v\)\)\)\) \(\[PartialD]\ \ \[CapitalPhi]\/\[PartialD]\ \[Mu]\) e\&\[RightVector]\_\[Mu] + \(1\/\(R \@\(\(\ \)\(cosh\^2\ \[Mu]\ \ \ - \ \ sin\^2\ v\)\)\)\) \(\[PartialD]\ \[CapitalPhi]\/\[PartialD]\ v\) e\&\[RightVector]\_v + \(1\/\(R\ cosh\ \[Mu]\ sin\ v\)\) \(\ \[PartialD]\ \[CapitalPhi]\/\[PartialD]\ \[CurlyPhi]\) e\&\[RightVector]\_\[CurlyPhi]\)]], "." }], "Text"], Cell[TextData[{ "The Lagrangian density is the coefficient of ", Cell[BoxData[ \(TraditionalForm\`\[DifferentialD]\[Mu]\ \[DifferentialD]v\ \ \[DifferentialD]\[CurlyPhi]\)]], " in the action integral and so is" }], "Text"], Cell[TextData[{ " ", Cell[BoxData[{ FormBox[ RowBox[{ FormBox[\(\[ScriptCapitalL] = \((\(\[CurlyEpsilon]\_0\/2\) \ \((\((\(1\/\(R \@\(\(\ \)\(cosh\^2\ \[Mu]\ \ - \ \ sin\^2\ v\)\)\)\) \ \[PartialD]\ \[CapitalPhi]\/\[PartialD]\ \[Mu])\)\^2 + \((\(1\/\(R \@\(\(\ \)\ \(cosh\^2\ \[Mu]\ \ - \ \ sin\^2\ v\)\)\)\) \[PartialD]\ \[CapitalPhi]\/\ \[PartialD]\ v)\)\^2 + \((\(1\/\(R\ cosh\ \[Mu]\ sin\ v\)\) \[PartialD]\ \ \[CapitalPhi]\/\[PartialD]\ \[CurlyPhi])\)\^2)\)\[IndentingNewLine] - \[Rho]\ \ \[CapitalPhi])\)\ R\^3\ \((\ cosh\^2\ \[Mu]\ \ - \ \ sin\^2\ v\ )\)\ cosh\ \[Mu]\ sin\ \ v\), "TraditionalForm"], "\[IndentingNewLine]"}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{ FormBox[\(\(=\)\(\((\(\[CurlyEpsilon]\_0\/2\) R\ \((\(1\/\(cosh\^2\ \[Mu]\ \ - \ \ sin\^2\ v\)\) \((\ \ \((\[PartialD]\ \[CapitalPhi]\/\[PartialD]\ \[Mu])\)\^2 + \((\[PartialD]\ \ \[CapitalPhi]\/\[PartialD]\ v)\)\^2\ )\) + \(1\/\(cosh\^2\ \[Mu]\ sin\^2\ v\)\ \) \((\[PartialD]\ \[CapitalPhi]\/\[PartialD]\ \[CurlyPhi])\)\^2\ )\)\ \[IndentingNewLine] - R\^3\ \[Rho]\ \[CapitalPhi])\)\ \ \((\ cosh\^2\ \[Mu]\ \ - \ \ sin\^2\ v\ )\)\ cosh\ \[Mu]\ sin\ \ v\)\), "TraditionalForm"], "\[IndentingNewLine]"}], TraditionalForm], "\[IndentingNewLine]", FormBox[ FormBox[\(\(=\)\(R\ \(\[CurlyEpsilon]\_0\/2\) \((cosh\ \[Mu]\ sin\ v\ \ \((\ \((\[PartialD]\ \[CapitalPhi]\/\[PartialD]\ \[Mu])\)\^2 + \((\[PartialD]\ \ \[CapitalPhi]\/\[PartialD]\ v)\)\^2\ )\) + \(\(cosh\^2\ \[Mu]\ \ - \ \ sin\ \^2\ v\)\/\(cosh\ \[Mu]\ sin\ v\)\) \((\[PartialD]\ \ \[CapitalPhi]\/\[PartialD]\ \[CurlyPhi])\)\^2\ )\)\)\), "TraditionalForm"], TraditionalForm], "\[IndentingNewLine]", FormBox[ FormBox[\(\(-R\^3\)\ \[Rho]\ \[CapitalPhi]\ \ \((\ cosh\^2\ \[Mu]\ \ - \ \ sin\^2\ v\ )\)\ cosh\ \[Mu]\ sin\ \ \(\(v\)\(.\)\)\), "TraditionalForm"], TraditionalForm]}]] }], "Text"], Cell["This yields for the equation of electrostatics", "Text"], Cell[TextData[{ Cell[BoxData[{ \(TraditionalForm\`R\ \(\[CurlyEpsilon]\_0\) \(\(\(\[PartialD]\)\(\ \ \)\)\/\[PartialD]\ \[Mu]\) \((cosh\ \[Mu]\ sin\ v\ \ \[PartialD]\ \ \[CapitalPhi]\/\[PartialD]\ \[Mu])\) + R\ \(\[CurlyEpsilon]\_0\) \(\(\(\[PartialD]\)\(\ \)\)\/\[PartialD]\ v\) \((cosh\ \[Mu]\ sin\ v\ \ \[PartialD]\ \[CapitalPhi]\/\ \[PartialD]\ v)\) + R\ \(\[CurlyEpsilon]\_0\) \(\(\(\[PartialD]\)\(\ \)\)\/\[PartialD]\ \ \[CurlyPhi]\) \((\(\(cosh\^2\ \[Mu]\ \ - \ \ sin\^2\ v\)\/\(cosh\ \[Mu]\ sin\ \ v\)\) \[PartialD]\ \[CapitalPhi]\/\[PartialD]\ \[CurlyPhi])\)\), "\[IndentingNewLine]", \(TraditionalForm\`\(+\ R\^3\)\ \[Rho]\ \((\ cosh\^2\ \[Mu]\ \ - \ \ sin\^2\ v\ )\)\ cosh\ \[Mu]\ sin\ v\ = \ \ 0. \)}]], "\nFinally then for the Laplacian and divergence:" }], "Text"], Cell[TextData[Cell[BoxData[ \(TraditionalForm\`\[Del]\^2 \[CapitalPhi] = \(1\/\(\(R\^2\)\(\ \)\((\ cosh\^2\ \[Mu]\ \ - \ \ sin\^2\ v\ )\)\(\ \)\(cosh\)\(\ \)\ \(\[Mu]\)\(\ \)\)\) \(\(\(\[PartialD]\)\(\ \)\)\/\[PartialD]\ \[Mu]\) \((cosh\ \ \[Mu]\ \ \[PartialD]\ \[CapitalPhi]\/\[PartialD]\ \[Mu])\) + \(1\/\(R\^2\ \ \((\ cosh\^2\ \[Mu]\ \ - \ \ sin\^2\ v\ )\)\ \ sin\ v\)\) \ \(\(\(\[PartialD]\)\(\ \)\)\/\[PartialD]\ v\) \((sin\ v\ \ \[PartialD]\ \[CapitalPhi]\/\[PartialD]\ \ v)\) + \(1\/\(R\^2\ cosh\^2\ \[Mu]\ sin\^2\ v\)\) \[PartialD]\^2\ \ \[CapitalPhi]\/\[PartialD]\ \[CurlyPhi]\^2\)]]], "Text"], Cell["and, remembering", "Text"], Cell[TextData[Cell[BoxData[ \(TraditionalForm\`\(\(\[Del]\&\[RightVector] \[CapitalPhi] = \(1\/\(R \@\ \(\(\ \)\(cosh\^2\ \[Mu]\ \ - \ \ sin\^2\ v\)\)\)\) \(\[PartialD]\ \ \[CapitalPhi]\/\[PartialD]\ \[Mu]\) e\&\[RightVector]\_\[Mu] + \(1\/\(R \@\(\(\ \)\(cosh\^2\ \[Mu]\ \ \ - \ \ sin\^2\ v\)\)\)\) \(\[PartialD]\ \[CapitalPhi]\/\[PartialD]\ v\) e\&\[RightVector]\_v + \(1\/\(R\ cosh\ \[Mu]\ sin\ v\)\) \(\ \[PartialD]\ \[CapitalPhi]\/\[PartialD]\ \[CurlyPhi]\) e\&\[RightVector]\_\[CurlyPhi]\)\(,\)\)\)]]], "Text"], Cell["we get", "Text"], Cell[TextData[Cell[BoxData[ \(TraditionalForm\`\[Del]\&\[RightVector]\(\(\[CenterDot]\)\(E\&\ \[RightVector]\)\) = \(1\/\(\(R\)\(\ \)\((\ cosh\^2\ \[Mu]\ \ - \ \ sin\^2\ v\ )\)\(\ \)\(cosh\)\(\ \)\ \(\[Mu]\)\(\ \)\)\) \(\(\(\[PartialD]\)\(\ \)\)\/\[PartialD]\ \[Mu]\) \((cosh\ \ \[Mu]\ \ \(\@\(\(\ \)\(cosh\^2\ \[Mu]\ \ - \ \ sin\^2\ v\)\)\) E\_\[Mu])\) + \(1\/\(\(R\)\(\ \)\((\ cosh\^2\ \[Mu]\ \ - \ \ sin\^2\ v\ )\)\(\ \)\(sin\)\(\ \ \)\(v\)\(\ \)\)\) \(\(\(\[PartialD]\)\(\ \)\)\/\[PartialD]\ v\) \((sin\ v\ \ \(\@\(\(\ \)\(cosh\^2\ \[Mu]\ \ - \ \ \ sin\^2\ v\)\)\) E\_v)\) + \(1\/\(R\ \ \ cosh\ \[Mu]\ \ sin\ v\)\) \[PartialD]\ \ E\_\(\(\ \)\(\[CurlyPhi]\)\)\/\[PartialD]\ \[CurlyPhi]\)]]], "Text"], Cell[TextData[{ "The expression for the Laplacian looks pretty formidable, but in the case \ that ", Cell[BoxData[ \(TraditionalForm\`\[Rho] = 0\)]], " and there is circular symmetry around the ", Cell[BoxData[ \(TraditionalForm\`z\)]], " axis (so no ", Cell[BoxData[ \(TraditionalForm\`\[CurlyPhi]\)]], " dependence) the equation for the potential simplifies considerable. In \ this case," }], "Text"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`\(1\/\(\(\ \)\(\(cosh\)\(\ \)\(\[Mu]\)\(\ \)\)\)\) \ \(\(\(\[PartialD]\)\(\ \)\)\/\[PartialD]\ \[Mu]\) \((cosh\ \[Mu]\ \ \ \[PartialD]\ \[CapitalPhi]\/\[PartialD]\ \[Mu])\) + \(1\/\(\(\ \)\(sin\ v\ \)\)\) \(\(\(\[PartialD]\)\(\ \)\)\/\[PartialD]\ v\) \((sin\ v\ \ \[PartialD]\ \[CapitalPhi]\/\[PartialD]\ \ v)\) = 0\)]], "," }], "Text"], Cell["which is much nicer.", "Text"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ 11. Approximate Electrostatic Solutions by the Relaxation Method\ \>", "Section"], Cell[TextData[{ "The principle of minimizing (or, more generally, finding a stationary \ point for) the action to find the electrostatic potential in a specified 3D \ volume ", Cell[BoxData[ \(TraditionalForm\`V\)]], " with specified boundary potentials on ", Cell[BoxData[ \(TraditionalForm\`\[PartialD]V\)]], " can be used in practice to find an approximate solution to the problem. \ The idea is to represent the potential in some way with a function that \ contains adjustable parameters, for any values of which the boundary \ conditions are satisfied. Then calculate the action as a function of the \ parameters, and finally choose that set of parameters that makes the action \ stationary; this will be the best possible approximation of the given form to \ the actual potential. " }], "Text"], Cell[TextData[{ "The extreme case of this strategy is to parametrize the potential with \ itself, i.e., with its values on the vertices of a rectangular grid, defined \ by a (small) regular spacing ", Cell[BoxData[ \(TraditionalForm\`\[CapitalDelta]\)]], " in ", Cell[BoxData[ \(TraditionalForm\`x, y, \ and\ \(\(z\)\(.\)\)\)]], " Deform the boundary slightly, as necessary, to follow the 3D grid. The \ vertices can be parametrized by integers ", Cell[BoxData[ \(TraditionalForm\`\((i, j, k)\)\)]], " so that the parameters are just ", Cell[BoxData[ \(TraditionalForm\`\[CapitalPhi]\_\(i, j, k\)\)]], " when ", Cell[BoxData[ \(TraditionalForm\`\((i, j, k)\)\)]], " does ", StyleBox["not", FontWeight->"Bold"], " identify a point on the boundary. Then approximate ", Cell[BoxData[ \(TraditionalForm\`\((\[PartialD]\[CapitalPhi]\/\[PartialD]x)\)\_\(i . \ j . k\) \[TildeEqual] \(\[CapitalPhi]\_\(\(i + 1\)\(,\)\(j\)\(,\)\(k\)\(\ \ \)\) - \ \[CapitalPhi]\_\(i, j, k\)\)\/\[CapitalDelta]\)]], " and similarly for the other two dimensions. Also take ", Cell[BoxData[ \(TraditionalForm\`\(\(\[Rho]\_\(i, j, k\)\)\(\ \)\)\)]], "as the given charge density in ", Cell[BoxData[ \(TraditionalForm\`V\)]], " at the vertex ", Cell[BoxData[ \(TraditionalForm\`\((i, j, k)\)\)]], ". Finally, approximate the action integral by a sum and find the equations \ in the values of the potential that make it stationary. These equations can \ be solved iteratively by starting with, for example, zero potential on all \ the grid points inside ", Cell[BoxData[ \(TraditionalForm\`V\)]], " and the appropriate specified value on grid points on ", Cell[BoxData[ \(TraditionalForm\`\[PartialD]V\)]], ". This allows the calculation of a next approximation to the field. Then \ use it in the same equations, etc., and continue iterating until exhaustion \ calls a halt to the process (or more formally, the potential values become \ stable in value. In fact this procedure leads to a matrix equation and the \ problem becomes that of inverting a large sparse matrix. The solution by \ iteration can be shown to converge to the solution of the linear equations. \ This may seem pretty crude, but in fact it is quite practical and, with some \ embellishments, yields quite accurate solutions." }], "Text"], Cell[CellGroupData[{ Cell["Solution", "Subsection"], Cell[TextData[{ "Begin by setting up a simple lattice of points in the volume ", Cell[BoxData[ \(TraditionalForm\`V\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\(r\& \[RightVector] \)\_\(i, j, k\)\)]], " where ", Cell[BoxData[ \(TraditionalForm\`\((i, j, k)\)\)]], " are integers and \n", Cell[BoxData[{ \(TraditionalForm\`\(r\& \[RightVector] \)\_\(i + 1, j, k\) - \(r\& \ \[RightVector] \)\_\(i, j, k\) = \[CapitalDelta]\ \(e\& \[RightVector] \ \)\_x\), "\[IndentingNewLine]", \(TraditionalForm\`\(r\& \[RightVector] \)\_\(i, j + 1, k\) - \(r\& \ \[RightVector] \)\_\(i, j, k\) = \[CapitalDelta]\ \(e\& \[RightVector] \ \)\_y\), "\[IndentingNewLine]", \(TraditionalForm\`\(r\& \[RightVector] \)\_\(i, j, k + 1\) - \(r\& \ \[RightVector] \)\_\(i, j, k\) = \[CapitalDelta]\ \(\(\(e\& \[RightVector] \)\ \_z\)\(.\)\)\)}]] }], "Text"], Cell[TextData[{ "Let ", Cell[BoxData[ FormBox[ RowBox[{\(\[CapitalPhi]\_\(i, j, k\)\), "=", RowBox[{\(\[CapitalPhi](\(r\& \[RightVector] \)\_\(i, j, k\))\), Cell[""]}]}], TraditionalForm]]], "and ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\_\(i, j, k\) = \[Rho](\(r\& \[RightVector] \)\ \_\(i, j, k\))\)]], ". " }], "Text"], Cell[TextData[{ "In general, the lattice points will not fall on the surface ", Cell[BoxData[ \(TraditionalForm\`\[PartialD]V\)]], ", but in general, the any point on the surface will be on order of \ \[CapitalDelta] from a lattice point. So deform ", Cell[BoxData[ \(TraditionalForm\`\[PartialD]V\)]], " slightly (assume \[CapitalDelta] is small) so that the deformed boundary \ ", Cell[BoxData[ \(TraditionalForm\`\[PartialD]V\^\(\(\[Prime]\)\(\ \)\)\)]], "passes through lattice points. At each of these assign the value of the \ potential to that on a nearby place in ", Cell[BoxData[ \(TraditionalForm\`\(\(\[PartialD]V\)\(.\)\)\)]], " " }], "Text"], Cell[TextData[{ "Next, approximate ", Cell[BoxData[ \(TraditionalForm\`\[PartialD]\(\[CapitalPhi](\(r\& \[RightVector] \)\_\ \(i, j, k\))\)\/\[PartialD]\ x = \(\[CapitalPhi]\_\(\(i + \ 1\)\(,\)\(j\)\(,\)\(k\)\(\ \)\) - \ \[CapitalPhi]\_\(i, j, k\)\)\/\(\(\ \[CapitalDelta]\)\(\ \)\)\)]], "and similarly for the other dimensions and finally approximate the action \ integral by a sum. Thus we have the approximation" }], "Text"], Cell[TextData[{ " ", Cell[BoxData[ \(TraditionalForm\`\(\(\[ScriptCapitalA]\)\(\[TildeEqual]\)\)\)]], Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(i, j, k\)\((\(\[CurlyEpsilon]\_0\/2\) \ \((\((\(\[CapitalPhi]\_\(\(i + 1\)\(,\)\(j\)\(,\)\(k\)\(\ \)\) - \ \ \[CapitalPhi]\_\(i, j, k\)\)\/\(\(\[CapitalDelta]\)\(\ \)\))\)\^2 + \((\(\ \[CapitalPhi]\_\(\(i\)\(,\)\(j + 1\)\(,\)\(k\)\(\ \)\) - \ \ \[CapitalPhi]\_\(i, j, k\)\)\/\(\(\[CapitalDelta]\)\(\ \)\))\)\^2 + \((\(\ \[CapitalPhi]\_\(i, j, \(\(k\)\(+\)\(1\)\(\ \)\)\) - \ \[CapitalPhi]\_\(i, j, \ k\)\)\/\(\(\[CapitalDelta]\)\(\ \)\))\)\^2)\) - \(\[Rho]\_\(i, j, k\)\) \[CapitalPhi]\_\(i, j, k\))\) \ \[CapitalDelta]\^3\)]], "." }], "Text"], Cell[TextData[{ "The values of the potential at all of the interior points of ", Cell[BoxData[ \(TraditionalForm\`V\)]], " are to be thought of as parameters to be chosen to make the action \ stationary under variation of these parameters, i.e., the derivative of the \ action with respect to each of these parameters must vanish. Thus if ", Cell[BoxData[ \(TraditionalForm\`\((\[ScriptL], m, n)\)\)]], " defines a point in the interior we get the equation" }], "Text"], Cell[TextData[Cell[BoxData[{ FormBox[ RowBox[{ FractionBox[ RowBox[{" ", FormBox[\(\[CurlyEpsilon]\_0\), "TraditionalForm"]}], \(\[CapitalDelta]\^2\)], \(\(\ \[CapitalDelta]\^3\)(\[CapitalPhi]\_\(\(\[ScriptL]\)\(,\)\(m\)\(,\)\(n\)\(\ \ \)\) - \ \[CapitalPhi]\_\(\[ScriptL] - 1, m, n\) - \[CapitalPhi]\_\(\(\ \[ScriptL] + 1\)\(,\)\(m\)\(,\)\(n\)\(\ \)\) + \ \[CapitalPhi]\_\(\[ScriptL], \ m, n\)\[IndentingNewLine] + \[CapitalPhi]\_\(\(\[ScriptL]\)\(,\)\(m\)\(,\)\(n\ \)\(\ \)\) - \ \[CapitalPhi]\_\(\[ScriptL], m - 1, n\) - \[CapitalPhi]\_\(\(\ \[ScriptL]\)\(,\)\(m + 1\)\(,\)\(n\)\(\ \)\) + \ \[CapitalPhi]\_\(\[ScriptL], \ m, n\)\[IndentingNewLine] + \[CapitalPhi]\_\(\(\[ScriptL]\)\(,\)\(m\)\(,\)\(n\ \)\(\ \)\) - \ \[CapitalPhi]\_\(\[ScriptL], m, n - 1\) - \[CapitalPhi]\_\(\ \[ScriptL], m, \(\(n\)\(+\)\(1\)\(\ \)\)\) + \ \[CapitalPhi]\_\(\[ScriptL], \ m, n\))\)}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{ " ", \(\(-\[Rho]\_\(\(\[ScriptL]\)\(,\)\(m\)\(,\)\(n\)\(\ \ \)\)\) \[CapitalDelta]\^3 = 0. \)}], TraditionalForm]}]]], "Text"], Cell["Rearranging, we get a form easy to visualize:", "Text"], Cell[TextData[{ " ", Cell[BoxData[ \(TraditionalForm\`\[CapitalPhi]\_\(\(\[ScriptL]\)\(,\)\(m\)\(,\)\(n\)\(\ \ \)\) = \(1\/6\) \((\ \[CapitalPhi]\_\(\[ScriptL] - 1, m, n\) + \ \[CapitalPhi]\_\(\(\[ScriptL] + 1\)\(,\)\(m\)\(,\)\(n\)\(\ \)\)\ \[IndentingNewLine] + \ \[CapitalPhi]\_\(\[ScriptL], m - 1, n\) + \ \[CapitalPhi]\_\(\(\[ScriptL]\)\(,\)\(m + 1\)\(,\)\(n\)\(\ \)\)\ \[IndentingNewLine] + \ \[CapitalPhi]\_\(\[ScriptL], m, n - 1\) + \ \[CapitalPhi]\_\(\[ScriptL], m, \(\(n\)\(+\)\(1\)\(\ \)\)\))\) + \(\ \[CapitalDelta]\^2\/\(6 \[CurlyEpsilon]\_0\)\) \[Rho]\_\(\(\[ScriptL]\)\(,\)\ \(m\)\(,\)\(n\)\(\ \)\)\)]] }], "Text"], Cell[TextData[{ "which just says the potential at a lattice point is the average of the \ potentials on its six nearest lattice points, plus a contribution from any \ charge density there. If there is no dependence along one of the axes, say \ the ", Cell[BoxData[ \(TraditionalForm\`z\)]], " axis because of translational symmetry along it, we have a two \ dimensional problem and it is easy to see that the relaxation equation \ becomes" }], "Text"], Cell[BoxData[ \(TraditionalForm\`\[CapitalPhi]\_\(\(\[ScriptL]\)\(,\)\(m\)\(,\)\(n\)\(\ \ \)\) = \(1\/6\) \((\ \[CapitalPhi]\_\(\[ScriptL] - 1, m, n\) + \ \[CapitalPhi]\_\(\(\[ScriptL] + 1\)\(,\)\(m\)\(,\)\(n\)\(\ \)\)\ \[IndentingNewLine] + \ \[CapitalPhi]\_\(\[ScriptL], m - 1, n\) + \ \[CapitalPhi]\_\(\(\[ScriptL]\)\(,\)\(m + 1\)\(,\)\(n\)\(\ \)\)\ \[IndentingNewLine] + \ \[CapitalPhi]\_\(\[ScriptL], m, n - 1\) + \ \[CapitalPhi]\_\(\[ScriptL], m, \(\(n\)\(+\)\(1\)\(\ \)\)\))\) + \(\ \[CapitalDelta]\^2\/\(6 \[CurlyEpsilon]\_0\)\) \[Rho]\_\(\(\[ScriptL]\)\(,\)\ \(m\)\(,\)\(n\)\(\ \)\)\ \ \((3 D, \ \[Rho]\ is\ charge\ per\ unit\ volume)\)\)], "Text"], Cell[TextData[{ " ", Cell[BoxData[ \(TraditionalForm\`\[CapitalPhi]\_\(\(\[ScriptL]\)\(,\)\(m\)\(,\)\(n\)\(\ \ \)\) = \(1\/4\) \((\ \ \[CapitalPhi]\_\(m - 1, n\) + \[CapitalPhi]\_\(\(m + \ 1\)\(,\)\(n\)\(\ \)\)\[IndentingNewLine] + \ \[CapitalPhi]\_\(m, n - 1\) + \ \[CapitalPhi]\_\(m, \(\(n\)\(+\)\(1\)\(\ \)\)\))\) + \ \(\[CapitalDelta]\^2\/\(4 \[CurlyEpsilon]\_0\)\) \ \[Lambda]\_\(\(m\)\(,\)\(n\)\(\ \)\)\ \ \ \ \ \ \ \ \ \ \((2 D, \ \[Lambda]\ \ is\ charge\ per\ \ unit\ area\ per\ unit\ \ length\ in\ the\ z\ direction)\)\)]], " " }], "Text"], Cell[CellGroupData[{ Cell["Example 1 in 2D", "Subsubsection"], Cell[TextData[{ "Suppose the 2D volume and boundaries are as indicated, with no charge in \ the volume:\n", Cell[BoxData[ FormBox[ RowBox[{"\[CapitalPhi]", "~", RowBox[{"(", GridBox[{ {"0", "0", "0", "0", "0", "0"}, {"1", "\[Placeholder]", "\[Placeholder]", "\[Placeholder]", "\[Placeholder]", "1"}, {"2", "\[Placeholder]", "\[Placeholder]", "\[Placeholder]", "\[Placeholder]", "2"}, {"3", "\[Placeholder]", "\[Placeholder]", "\[Placeholder]", "\[Placeholder]", "3"}, {"4", "\[Placeholder]", "\[Placeholder]", "\[Placeholder]", "\[Placeholder]", "4"}, {"5", "5", "5", "5", "5", "5"} }], ")"}]}], TraditionalForm]]], " " }], "Text"], Cell[TextData[{ "Take the ", Cell[BoxData[ \(TraditionalForm\`\(\(0\^th\)\(\ \)\)\)]], "iteration to be zero potential on each of the interior points and show the \ first few iterations. Here is a little function to do the arithmetic." }], "Text"], Cell[BoxData[{ \(relax[times_] := Module[{\[CapitalPhi] = {{0, 0, 0, 0, 0, 0}, {1, 0, 0, 0, 0, 1}, {2, 0, 0, 0, 0, 2}, {3, 0, 0, 0, 0, 3}, {4, 0, 0, 0, 0, 4}, {5, 5, 5, 5, 5, 5}}}, \[IndentingNewLine]Do[\[CapitalPhi]old = \[CapitalPhi]; For[n = 2, n < 6, \(n++\), For[m = 2, m < 6, \(m++\), \[CapitalPhi]\[LeftDoubleBracket]n, m\[RightDoubleBracket] = \(1\/4. \) \((\[CapitalPhi]old\ \[LeftDoubleBracket]n - 1, m\[RightDoubleBracket] + \[CapitalPhi]old\ \[LeftDoubleBracket]n + 1, m\[RightDoubleBracket] + \[CapitalPhi]old\ \[LeftDoubleBracket]n, m - 1\[RightDoubleBracket] + \[CapitalPhi]old\ \[LeftDoubleBracket]n, m + 1\[RightDoubleBracket])\)]], {times}]; Print[MatrixForm@\[CapitalPhi]]]\), "\[IndentingNewLine]", \(relax[100]\)}], "Input", CellLabel->"In[85]:="], Cell["\<\ The following shows the first few iterations, showing how the boundary \ propagate into the interior as the iterations proceed. Finally, since this \ volume has just constant electric fields on the \"side\" walls and \ equipotentials on the bottom and top walls, the result should be a uniform \ electric field inside the volume \[Dash] as many iterations show is the case. \ The uniform potential gradient on the side walls can be created \ experimentally by running a uniform current through a uniform resistor making \ up the side walls. This idea is used in high voltage devices when a constant \ field is needed in some portion of space in an apparatus.\ \>", "Text"], Cell[TextData[{ " ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", GridBox[{ {"0", "0", "0", "0", "0", "0"}, {"1", "0", "0", "0", "0", "1"}, {"2", "0", "0", "0", "0", "2"}, {"3", "0", "0", "0", "0", "3"}, {"4", "0", "0", "0", "0", "4"}, {"5", "5", "5", "5", "5", "5"} }], ")"}], \( \[Rule] \+1\), RowBox[{ RowBox[{"(", GridBox[{ {"0", "0", "0", "0", "0", "0"}, {"1", ".25", "0", "0", ".25", "1"}, {"2", ".5", "0", "0", ".5", "2"}, {"3", ".75", "0", "0", ".75", "3"}, {"4", "2.25", "1.25", "1.25", "2.25", "4"}, {"5", "5", "5", "5", "5", "5"} }], ")"}], \( \[Rule] \+2\), "\[IndentingNewLine]", RowBox[{ RowBox[{"(", GridBox[{ {"0", "0", "0", "0", "0", "0"}, {"1", "0.375`", "0.0625`", "0.0625`", "0.375`", "1"}, {"2", ".75", "0.125`", "0.125`", ".75", "2"}, {"3", "1.4375`", ".5", ".5", "1.4375`", "3"}, {"4", "2.75`", "2.125`", "2.125`", "2.75`", "4"}, {"5", "5", "5", "5", "5", "5"} }], ")"}], \( \[Rule] \+3\), "\[IndentingNewLine]", RowBox[{ RowBox[{"(", GridBox[{ {"0", "0", "0", "0", "0", "0"}, {"1", "0.453125`", "0.140625`", "0.140625`", "0.453125`", "1"}, {"2", "0.984375`", "0.359375`", "0.359375`", "0.984375`", "2"}, {"3", "1.75", "1.046875`", "1.046875`", "1.75", "3"}, {"4", "3.140625`", "2.59375`", "2.59375`", "3.140625`", "4"}, {"5", "5", "5", "5", "5", "5"} }], ")"}], "\[Rule]", RowBox[{"\[Ellipsis]", \( \[Rule] \+30\), RowBox[{ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "0", "0", "0", "0", "0"}, {"1", "0.9977318615854806`", "0.9963300718297847`", "0.9963300718297847`", "0.9977318615854806`", "1"}, {"2", "1.996330045360005`", "1.994061893710596`", "1.9940618937105958`", "1.9963300453600052`", "2"}, {"3", "2.9963300188902258`", "2.994061840771036`", "2.994061840771036`", "2.9963300188902258`", "3"}, {"4", "3.997731808645921`", "3.9963299924204456`", "3.9963299924204456`", "3.9977318086459213`", "4"}, {"5", "5", "5", "5", "5", "5"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]], "\[Rule]", RowBox[{"\[Ellipsis]", \( \[Rule] \+100\), TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "0", "0", "0", "0", "0"}, {"1", "0.999999999182082`", "0.999999998676581`", "0.9999999986765808`", "0.999999999182082`", "1"}, {"2", "1.999999998676581`", "1.999999997858663`", "1.9999999978586631`", "1.9999999986765808`", "2"}, {"3", "2.9999999986765813`", "2.9999999978586636`", "2.999999997858663`", "2.9999999986765813`", "3"}, {"4", "3.999999999182082`", "3.999999998676581`", "3.9999999986765813`", "3.999999999182082`", "4"}, {"5", "5", "5", "5", "5", "5"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]}]}]}]}]}]}]}], TraditionalForm]]] }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Example 2 in 2D", "Subsubsection"], Cell[TextData[{ "Again restrict to 2D, but this time put a line charge of ", Cell[BoxData[ \(TraditionalForm\`\[CurlyEpsilon]\_0\)]], " (taken as a pure number) Coulombs per meter (a lattice-sized 2D point \ charge} at the origin with a grounded circular boundary around it. Let the \ lattice spacing be ", Cell[BoxData[ \(TraditionalForm\`\[CapitalDelta]\)]], " meters. Then the approximation for the charge density is that it have a \ value of ", Cell[BoxData[ \(TraditionalForm\`\[CurlyEpsilon]\_0\/\[CapitalDelta]\^2\)]], "in a single lattice cell. Thus the lattice size drops out of the problem \ and need not be specified." }], "Text"], Cell[BoxData[{ \(relaxch[times_] := Module[{mxx = 11}, mid = IntegerPart[mxx/2. ] + 1; \[IndentingNewLine]\[CapitalPhi] = \(\((\(0 &\) /@ Range[mxx])\) &\) /@ Range[mxx]; \[IndentingNewLine]\[Rho] = \[CapitalPhi]; \[Rho]\ \[LeftDoubleBracket]mid, mid\[RightDoubleBracket] = 1; \[IndentingNewLine]Do[\[CapitalPhi]old = \[CapitalPhi]; For[n = 2, n < mxx, \(n++\), For[m = 2, m < mxx, \(m++\), If[\((n - mid)\)\^2 + \((m - mid)\)\^2 < \((mid - 1)\)\^2, \ \[CapitalPhi]\[LeftDoubleBracket]n, m\[RightDoubleBracket] = \(1\/4. \) \((\[CapitalPhi]old\ \[LeftDoubleBracket]n - 1, m\[RightDoubleBracket] + \[CapitalPhi]old\ \[LeftDoubleBracket]n + 1, m\[RightDoubleBracket] + \[CapitalPhi]old\ \[LeftDoubleBracket]n, m - 1\[RightDoubleBracket] + \[CapitalPhi]old\ \[LeftDoubleBracket]n, m + 1\[RightDoubleBracket] + \[Rho]\[LeftDoubleBracket] n, m\[RightDoubleBracket])\)]]], {times}]; 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")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]}]}]}]}]}]}]], "Text"], Cell["\<\ You can clearly see the effect of the charge spreading out to the walls as \ the iterations grow. Further, there is clear circular symmetry. Finally since \ there is little change between 100 iterations and 500, stop at 500.\ \>", "Text"], Cell[TextData[{ "It is interesting to solve the analytic problem of the potential around a \ line charge centered in a grounded conducting cylinder of radius ", Cell[BoxData[ \(TraditionalForm\`R\)]], " and compare with this numerical solution. " }], "Text"], Cell[TextData[{ "Getting the free space 2D Green function is usually done with Gauss's law, \ but let's use the field equation for 2D electrostatics as n example of \ solving electrostatics problems from the differential equations. This problem \ is just the case of cylindrical geometry given above but with no ", Cell[BoxData[ \(TraditionalForm\`z\)]], " or \[CurlyPhi] dependence. Calling the potential ", Cell[BoxData[ \(TraditionalForm\`\(\(\[CapitalPhi]\_2 = \ \(\[CapitalPhi]\_2\)(\[Rho])\)\(,\)\)\)]], " we have " }], "Text"], Cell[TextData[{ " ", Cell[BoxData[ \(TraditionalForm\`\(1\/\[Rho]\) \(\[PartialD]\/\[PartialD]\[Rho]\) \((\ \[Rho] \[PartialD]\ \[CapitalPhi]\_2\/\[PartialD]\[Rho])\) = \ \(-\(\[Rho]\_charge\/\[CurlyEpsilon]\_0\)\)\)]] }], "Text"], Cell[TextData[{ "where I will choose ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\_charge\)]], " so that a spatial integral over a unit length in ", Cell[BoxData[ \(TraditionalForm\`z\)]], " and any area in the ", Cell[BoxData[ \(TraditionalForm\`x - y\)]], " plane containing the origin yields ", Cell[BoxData[ \(TraditionalForm\`\[CurlyEpsilon]\_0\)]], ". The expression surely must contain a factor \[Delta](\[Rho]) to express \ the concentration of charge along the ", Cell[BoxData[ \(TraditionalForm\`z\)]], " axis. But is that all? Let ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\_charge = \(\[CurlyEpsilon]\_0\) A\ \(\[Delta](\[Rho])\)\)]], " where the prefactor ", Cell[BoxData[ \(TraditionalForm\`A\)]], " is to be chosen so that" }], "Text"], Cell[TextData[{ " ", Cell[BoxData[ \(TraditionalForm\`\[CurlyEpsilon]\_0 = \(\[Integral]\_\(z = 0\)\%1\(\ \[Integral]\_\(\[CurlyPhi] = 0\)\%\(2 \[Pi]\)\ \(\[Integral]\_\(\[Rho] = 0\)\ \%\(\[Rho] > 0\)\ \(\[Rho]\_charge\) \[Rho] \[DifferentialD]\[Rho]\ \ \[DifferentialD]\[CurlyPhi]\ \[DifferentialD]z\)\) = \(\[Integral]\_\(z = 0\)\ \%1\(\[Integral]\_\(\[CurlyPhi] = 0\)\%\(2 \[Pi]\)\ \(\[Integral]\_\(\[Rho] \ = 0\)\%\(\[Rho] > 0\)\ \(\[CurlyEpsilon]\_0\) A\ \(\[Delta](\[Rho])\) \[Rho] \[DifferentialD]\[Rho]\ \ \[DifferentialD]\[CurlyPhi]\ \[DifferentialD]z\)\) = 2 \( \[Pi]\[CurlyEpsilon]\_0\) \(\[Integral]\_\(\[Rho] = 0\)\%\(\ \[Rho] > 0\)\ A\ \(\[Delta](\[Rho])\) \[Rho] \ \[DifferentialD]\[Rho]\)\)\)\)]], " and so we must choose ", Cell[BoxData[ \(TraditionalForm\`A = 1\/\(2 \[Pi]\ \[Rho]\)\)]], " and interpret ", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_\(\[Rho] = 0\)\%\(\[Rho] > 0\)\ \ \(\ \[Delta](\[Rho])\) \[DifferentialD]\[Rho]\)]], " as ", Cell[BoxData[ \(TraditionalForm\`\(\(1.\)\(\ \)\)\)]], "Thus the equation to be solved is" }], "Text"], Cell[TextData[{ " ", Cell[BoxData[ \(TraditionalForm\`\(1\/\[Rho]\) \(\[PartialD]\/\[PartialD]\[Rho]\) \((\ \[Rho] \[PartialD]\ \[CapitalPhi]\_2\/\[PartialD]\[Rho])\) = \(\(-\(1\/\(2 \ \[Pi]\ \[Rho]\)\)\) \(\[Delta](\[Rho])\)\ \ \ \[Implies] \ \ \ \(\[PartialD]\ \/\[PartialD]\[Rho]\) \((\[Rho] \[PartialD]\ \[CapitalPhi]\_2\/\[PartialD]\ \[Rho])\) = \(\(-\(1\/\(\(2\) \(\[Pi]\)\(\ \)\)\)\) \(\[Delta](\[Rho])\)\(\ \ \ \ \ \ \ \ \ \ \ \ \)\)\)\)]], "(not worrying about multiplying by zero when \[Rho]=0!)." }], "Text"], Cell[TextData[{ "Clearly if we choose ", Cell[BoxData[ \(TraditionalForm\`\[CapitalPhi]\_2 = K\ log\ R\/\[Rho]\)]], ", where ", Cell[BoxData[ \(TraditionalForm\`K\)]], " is any constant, we solve Poisson's equation and the boundary condition \ for ", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\[Rho] \[NotEqual] 0\)\)\)]], ". Next we integrate over \[Rho] from 0 to any positive value to get" }], "Text"], Cell[BoxData[ FormBox[ RowBox[{" ", Cell[TextData[Cell[BoxData[ \(TraditionalForm\`\[Integral]\_\(\(\ \)\(0\)\)\%\(\[Rho] > 0\)\(\ \[PartialD]\/\[PartialD]\[Rho]\) \((\[Rho] \[PartialD]\ \[CapitalPhi]\_2\/\ \[PartialD]\[Rho])\) \[DifferentialD]\[Rho] = \(\(\(\[Rho] \[PartialD]\ \ \[CapitalPhi]\_2\/\[PartialD]\[Rho]\)\( | \+\(\[Rho] = 0\)\%\(\[Rho] > 0\)\)\) = \(\(-\(1\/\(\(2\) \(\[Pi]\)\(\ \)\)\)\) \(\ \[Integral]\_\(\(\ \)\(0\)\)\%\(\[Rho] > 0\)\(\[Delta](\[Rho])\)\ \ \[DifferentialD]\[Rho]\) = \(-\(\(1\/\(\(2\) \(\[Pi]\)\(\ \ \)\)\)\(.\)\)\)\)\)\)]]]]}], TraditionalForm]], "Text"], Cell[TextData[{ "It is not obvious how to handle the evaluation in the integrated part at \ ", Cell[BoxData[ \(TraditionalForm\`\[Rho] = 0\)]], ". One way to handle the value of ", Cell[BoxData[ \(TraditionalForm\`\[Rho] \[PartialD]\ \[CapitalPhi]\_2\/\[PartialD]\ \[Rho]\)]], " at ", Cell[BoxData[ \(TraditionalForm\`\[Rho] = 0\)]], " is to calculate it for a finite diameter cylinder of radius \[Epsilon] \ filled with a constant charge density ", Cell[BoxData[ FormBox[ FractionBox[ FormBox[\(\[CurlyEpsilon]\_0\), "TraditionalForm"], \(\[Pi]\ e\^2\)], TraditionalForm]]], " per unit length centered in a grounded cylinder (just replace the delta \ function with a simple cylinder of charge) and then take the limit as \ \[Epsilon] goes to ", Cell[BoxData[ \(TraditionalForm\`\(0\^+\)\)]], ", i.e., ", Cell[BoxData[ \(TraditionalForm\`lim\+\(\[Epsilon] \[Rule] 0, \ \ \[Epsilon] > 0\) . \ \)]], " It is easy to solve the problem ", Cell[BoxData[ FormBox[ RowBox[{\(\(1\/\[Rho]\) \(\[DifferentialD]\/\[DifferentialD]\[Rho]\) \ \((\[Rho]\ \[DifferentialD]\[CapitalPhi]\_rod\/\[DifferentialD]\[Rho])\)\), "=", RowBox[{"(", GridBox[{ {\(-\(1\/\(\[Pi]\ \[Epsilon]\^2\)\)\), \(0 \[LessEqual] \ \[Rho] < \[Epsilon]\)}, {"0", \(\[Epsilon] \[LessEqual] \[Rho] \[LessEqual] R\)} }], ")"}]}], TraditionalForm]]], " giving ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\ \[DifferentialD]\[CapitalPhi]\_rod\/\ \[DifferentialD]\[Rho]\)]], "= ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", GridBox[{ {\(\(-\(\[Rho]\^2\/\(2\ \[Pi]\ \[Epsilon]\^2\)\)\) + K\_1\), \(0 \[LessEqual] \[Rho] < \[Epsilon]\)}, {\(-K\_2\), \(\[Epsilon] \[LessEqual] \[Rho] \[LessEqual] R\)} }], ")"}], " ", "where", " ", \(K\_1\), " ", "and", " ", \(K\_2\), " ", "are", " ", \(\(constants\)\(.\)\)}], TraditionalForm]]], " This function must be continuous since otherwise its derivative at the \ discontinuity would generate a delta function and Poisson's equation would be \ violated. Thus we must have ", Cell[BoxData[ \(TraditionalForm\`\(-1\)\/\(2 \[Pi]\) + K\_1 = \(-K\_2\)\)]], " or ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\ \[DifferentialD]\[CapitalPhi]\_rod\/\ \[DifferentialD]\[Rho]\)]], "=", Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ {\(\(-\(\[Rho]\^2\/\(2\ \[Pi]\ \[Epsilon]\^2\)\)\) - K\_2 + 1\/\(2 \[Pi]\)\), \(0 \[LessEqual] \[Rho] < \[Epsilon]\)}, {\(-K\_2\), \(\[Epsilon] \[LessEqual] \[Rho] \[LessEqual] R\)} }], ")"}], TraditionalForm]]], ". Integrating again gives ", Cell[BoxData[ FormBox[ RowBox[{\(\[CapitalPhi]\_rod\), "=", RowBox[{"(", GridBox[{ {\(\(-\(\[Rho]\^2\/\(4 \[Pi]\ \[Epsilon]\^2\)\)\) + \ \((\(-K\_2\) + 1\/\(2 \[Pi]\))\) log\ \[Rho]\ + \ K\_3\), \(0 \[LessEqual] \[Rho] < \[Epsilon]\)}, {\(K\_2\ log\ R\/\[Rho]\), \(\[Epsilon] \[LessEqual] \[Rho] \ \[LessEqual] R\)} }], ")"}], " "}], TraditionalForm]]], " where ", Cell[BoxData[ \(TraditionalForm\`K\_2\)]], " and ", Cell[BoxData[ \(TraditionalForm\`K\_3\)]], " are constants and the boundary condition at ", Cell[BoxData[ \(TraditionalForm\`\(\(\[Rho]\)\(=\)\(R\)\(\ \)\)\)]], " has been satisfied. " }], "Text"], Cell[TextData[{ "As above impose the condition that ", Cell[BoxData[ \(TraditionalForm\`\[CapitalPhi]\_rod\)]], " must be continuous (so it does not have a delta function in its \ derivative which would violate Poisson's equation) and that it be regular \ everywhere. This then gives the conditions ", Cell[BoxData[ \(TraditionalForm\`K\_2 = 1\/\(2 \(\[AliasDelimiter]\[Pi]\)\)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`K\_3 = K\_2\ log\ R\/\[Epsilon] + \(\(1\/\(4 \[Pi]\)\)\(.\)\(\ \)\)\)]], "We get\n ", Cell[BoxData[ \(TraditionalForm\`\(\(\[CapitalPhi]\_rod\)\(=\)\)\)]], Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ {\(\(-\(\(\[Rho]\^2 - \[Epsilon]\^2\)\/\(2\ \[Pi]\ \ \[Epsilon]\^2\)\)\) + 1\/\(2 \[Pi]\)\ log\ R\/\[Epsilon]\), \(0 \[LessEqual] \ \[Rho] < \[Epsilon]\)}, {\(1\/\(2 \[Pi]\)\ \ log\ R\/\[Rho]\), \(\[Epsilon] \ \[LessEqual] \[Rho] \[LessEqual] R\)} }], ")"}], TraditionalForm]]], " .\nFrom this get ", Cell[BoxData[ FormBox[ RowBox[{\(\[Rho] \[PartialD]\ \[CapitalPhi]\_2\/\[PartialD]\[Rho]\), "=", RowBox[{"(", GridBox[{ {\(-\(\[Rho]\^2\/\(\[Pi]\ \[Epsilon]\^2\)\)\), \(0 \ \[LessEqual] \[Rho] < \[Epsilon]\)}, {\(\(-\)\(1\/\(2 \[Pi]\)\)\(\ \)\), \(\[Epsilon] \ \[LessEqual] \[Rho] \[LessEqual] R\)} }], ")"}], " "}], TraditionalForm]]], ", \nso we get the result that ", Cell[BoxData[ \(TraditionalForm\`lim\+\(\[Epsilon] \[Rule] 0, \ \[Epsilon] > \ 0\)\[Rho] \(\[PartialD]\ \[CapitalPhi]\_2\/\[PartialD]\[Rho]\) \((\[Rho] = 0)\) = 0. \)]], " Finally then we get from \n", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_\(\(\ \)\(0\)\)\%\(\[Rho] > 0\)\(\ \[PartialD]\/\[PartialD]\[Rho]\) \((\[Rho] \[PartialD]\ \[CapitalPhi]\_2\/\ \[PartialD]\[Rho])\) \[DifferentialD]\[Rho] = \(\(\(\[Rho] \[PartialD]\ \ \[CapitalPhi]\_2\/\[PartialD]\[Rho]\)\( | \+\(\[Rho] = 0\)\%\(\[Rho] > 0\)\)\) = \(\[Rho] \(\[PartialD]\ \[CapitalPhi]\_2\/\ \[PartialD]\[Rho]\) \((\[Rho] > 0)\) = \(K = \(-\(1\/\(\(2\) \(\[Pi]\)\(\ \ \)\)\)\)\)\)\)\)]], "\nand \n", Cell[BoxData[ \(TraditionalForm\`\[CapitalPhi]\_2 = 1\/\(2 \[Pi]\)\ log\ R\/\[Rho]\)]], ".\nwhich is the usual result obtained from applying Gauss's law in three \ dimensions. " }], "Text"], Cell[TextData[{ "For our present purposes all we need is that ", Cell[BoxData[ \(TraditionalForm\`\[CapitalPhi]\_2 = \(-\(1\/\(2 \[Pi]\)\)\) log\ \[Rho] + cnst\)]], ". I will choose the constant so that the potential is zero at ", Cell[BoxData[ \(TraditionalForm\`\[Rho] = 5\)]], ", approximating the square geometry above. 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