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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[ 30264, 712]*) (*NotebookOutlinePosition[ 59977, 1783]*) (* CellTagsIndexPosition[ 59906, 1777]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[StyleBox["Classical electrodynamics"]], "Subtitle", CellMargins->{{Inherited, 71}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times New Roman"], Cell[TextData[StyleBox["doug "]], "Subsubtitle", CellMargins->{{Inherited, 71}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times New Roman"], Cell[TextData[StyleBox[ "Introduction\nThe Maxwell equations\nThe 4-potential A\nThe Lorentz force\n\ Conservation laws\nImplications"]], "Text", CellMargins->{{Inherited, 71}, {Inherited, Inherited}}, FontFamily->"Times New Roman"], Cell[CellGroupData[{ Cell[TextData[StyleBox[" Introduction"]], "Subsection", CellMargins->{{0, 71}, {Inherited, Inherited}}, FontFamily->"Times New Roman"], Cell[TextData[StyleBox[ "Maxwell speculated that someday quaternions would be useful in the analysis \ of electromagnetism. Hopefully after a 130 year wait, in this notebook we \ can begin that process. This approach relies on a judicious use of \ commutators and anticommutators."]], "Text", CellMargins->{{Inherited, 71}, {Inherited, Inherited}}, FontFamily->"Times New Roman"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[" The Maxwell equations"]], "Subsection", CellMargins->{{0, 71}, {Inherited, Inherited}}, FontFamily->"Times New Roman"], Cell[TextData[StyleBox[ "The Maxwell equations are formed from a combinations of commutators and \ anticommutators of the differential operator and the electric and magnetic \ fields E and B respectively (for isolated charges in a vacuum."]], "Text", CellMargins->{{Inherited, 71}, {Inherited, Inherited}}, FontFamily->"Times New Roman"], Cell[BoxData[{ \({0, \ \(0\& \[RightVector] \)} == \((evenop[{q[dt, \(\[EmptyDownTriangle]\& \[RightVector] \)]}, q[0, \(B\& \[RightVector] \)[t]]]\ \n \t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ + \ oddop[{q[dt, \(\[EmptyDownTriangle]\& \[RightVector] \)]}, q[0, \(E\& \[RightVector] \)[t]]])\).{1, 0}\), \({0, \(0\& \[RightVector] \)} == { \(-\((\ \(\[EmptyDownTriangle]\& \[RightVector] \)\ \[CenterDot]\ \(B\& \[RightVector] \)[t]\ \ )\)\), \(\ \(\[EmptyDownTriangle]\& \[RightVector] \)\ X\ \(E\& \[RightVector] \)[t]\ \) + \[PartialD]\(B\& \[RightVector] \)[t]\/\[PartialD]t}\), \(4\ \[Pi]\ {\[Rho], \ \(J\& \[RightVector] \)}\ == \((oddop[{q[dt, \(\[EmptyDownTriangle]\& \[RightVector] \)]}, q[0, \(B\& \[RightVector] \)[t]]]\ \n \t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ - \ evenop[{q[dt, \(\[EmptyDownTriangle]\& \[RightVector] \)]}, q[0, \(E\& \[RightVector] \)[t]]])\).{1, 0}\), \({4\ \[Pi]\ \[Rho], 4\ \[Pi]\ \(J\& \[RightVector] \)} == {\(\ \(\[EmptyDownTriangle]\& \[RightVector] \)\ \[CenterDot]\ \(E\& \[RightVector] \)[t]\ \ \), \(\ \(\[EmptyDownTriangle]\& \[RightVector] \)\ X\ \(B\& \[RightVector] \)[t]\ \) - \[PartialD]\(E\& \[RightVector] \)[t]\/\[PartialD]t}\)}], "Input", CellMargins->{{9, 71}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ StyleBox["{", FontWeight->"Bold"], \(\((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector])\), \((0, B\&\[RightVector])\)\), StyleBox["}", FontWeight->"Bold"]}], " ", "+", " ", RowBox[{ StyleBox["[", FontWeight->"Bold"], \(\((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector])\), \((0, E\&\[RightVector])\)\), StyleBox["]", FontWeight->"Bold"]}]}], "\n", "=", RowBox[{ \((\(-\[EmptyDownTriangle]\&\[RightVector]\)\[CenterDot]B \&\[RightVector]\ \ , \ \[EmptyDownTriangle]\&\[RightVector]\ X\ E\&\[RightVector]\ + \[PartialD]B\&\[RightVector]\/\[PartialD]t)\), "=", " ", RowBox[{ RowBox[{ RowBox[{\((0, \ 0\&\[RightVector])\), "\n", "\t", "\n", StyleBox["[", FontWeight->"Bold"], \(\((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector])\), \((0, B\&\[RightVector])\)\), StyleBox["]", FontWeight->"Bold"]}], "-", " ", RowBox[{ StyleBox["{", FontWeight->"Bold"], \(\((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector])\), \((0, E\&\[RightVector])\)\), StyleBox["}", FontWeight->"Bold"]}]}], "\n", "=", \(\((\ \[EmptyDownTriangle]\&\[RightVector]\[CenterDot]E \&\[RightVector]\ \ , \ \[EmptyDownTriangle]\&\[RightVector]\ X\ B\&\[RightVector]\ - \[PartialD]E\&\[RightVector]\/\[PartialD]t)\) = 4\ \[Pi]\ \((\[Rho], J\&\[RightVector])\)\)}]}]}], "\n", "\t\t\t\t"}], RowBox[{ RowBox[{"where", " ", StyleBox["[", FontWeight->"Bold"], \(A, \ B\), StyleBox["]", FontWeight->"Bold"]}], " ", "=", " ", RowBox[{ RowBox[{\(\(AB\ - \ BA\)\/2\), " ", "and", " ", RowBox[{ StyleBox["{", FontWeight->"Bold"], \(A, \ B\), StyleBox["}", FontWeight->"Bold"]}]}], " ", "=", " ", \(\(AB\ + \ BA\)\/2\)}]}]}], "Input", CellMargins->{{9, 71}, {Inherited, Inherited}}, FontSize->16], Cell[TextData[StyleBox[ "The first quaternion equation embodies the homogeneous Maxwell equations. \ The scalar term says that there are no magnetic monopoles. The vector term \ is Faraday's law."]], "Text", CellMargins->{{Inherited, 71}, {Inherited, Inherited}}, FontFamily->"Times New Roman"], Cell[TextData[StyleBox[ "The second quaternion equation is the source term. The scalar equation is \ Gauss' law. The vector term is Ampere's law, with Maxwell's correction."]], "Text", CellMargins->{{Inherited, 71}, {Inherited, Inherited}}, FontFamily->"Times New Roman"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[" The 4-potential A"]], "Subsection", CellMargins->{{0, 71}, {Inherited, Inherited}}, FontFamily->"Times New Roman"], Cell[TextData[StyleBox[ "The electric and magnetic fields are often viewed as arising from the same \ 4-potential A. These can also be expressed easily using quaternions."]], "Text", CellMargins->{{Inherited, 71}, {Inherited, Inherited}}, FontFamily->"Times New Roman"], Cell[BoxData[{ \(\(E\ == \ vector[{q[dt, \(-\[EmptyDownTriangle]\&\[RightVector]\)]}, q[\[Phi][t], \(-A\&\[RightVector][t]\)]]]\)\n \ \ \ \ \ \ \ \ \ \ \ .{1, 0}\), \(E == {0, \(-\((\ \[EmptyDownTriangle]\&\[RightVector]\ \[Phi][t]\ )\)\) - \[PartialD]A\&\[RightVector][t]\/\[PartialD]t}\), \(B\ == \ oddop[{q[dt, \[EmptyDownTriangle]\&\[RightVector]]}, q[\[Phi][t], A\&\[RightVector][t]]].{1, 0}\), \(B == {0, \(\ \[EmptyDownTriangle]\&\[RightVector]\ X\ A\&\[RightVector][t]\ \)} \)}], "Input", CellMargins->{{9, 71}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[{ RowBox[{"E", " ", "=", RowBox[{ RowBox[{"vector", RowBox[{ StyleBox["{", FontWeight->"Bold"], \(\((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector])\), \((\[Phi], \(-A\&\[RightVector]\))\)\), StyleBox["}", FontWeight->"Bold"]}]}], "=", \((0, \(-\[EmptyDownTriangle]\&\[RightVector]\)\ \[Phi] - \[PartialD]A\&\[RightVector]\/\[PartialD]t)\)}]}], RowBox[{"B", " ", "=", " ", RowBox[{ RowBox[{ StyleBox["[", FontWeight->"Bold"], \(\((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector])\), \((\[Phi], A\&\[RightVector])\)\), StyleBox["]", FontWeight->"Bold"]}], "=", \((0, \ \[EmptyDownTriangle]\&\[RightVector]\ X\ A\&\[RightVector]\ ) \)}]}]}], "Input", CellMargins->{{9, 71}, {Inherited, Inherited}}, FontSize->16], Cell[TextData[StyleBox[ "The electric field E is the vector part of the anticommutator of the \ conjugates of the differential operator and the 4-potential. The magnetic \ field B involves the commutator."]], "Text", CellMargins->{{Inherited, 71}, {Inherited, Inherited}}, FontFamily->"Times New Roman"], Cell[TextData[StyleBox[ "These forms can be directly placed into the Maxwell equations."]], "Text", CellMargins->{{Inherited, 71}, {Inherited, Inherited}}, FontFamily->"Times New Roman"], Cell[BoxData[{ \({0, \ \(0\& \[RightVector] \)} == \n \((evenop[{q[dt, \(\[EmptyDownTriangle]\& \[RightVector] \)]}, oddop[{q[dt, \(\[EmptyDownTriangle]\& \[RightVector] \)]}, q[\[Phi][t], \(A\& \[RightVector] \)[t]]]]\ \n + \ oddop[{q[dt, \(\[EmptyDownTriangle]\& \[RightVector] \)]}, \n \t\t\t\tvector[ evenop[{ q[dt, \(-\(\[EmptyDownTriangle]\& \[RightVector] \)\)]}, q[\[Phi][t], \(-\(A\& \[RightVector] \)[t]\)]]]])\)\n \ \ \ \ \ \ \ \ \ \ .{1, 0}\), RowBox[{\({0, \(0\& \[RightVector] \)}\), "==", RowBox[{"{", RowBox[{ \(-\((\ \(\[EmptyDownTriangle]\& \[RightVector] \)\ \[CenterDot]\ \((\ \(\[EmptyDownTriangle]\& \[RightVector] \)\ X\ \(A\& \[RightVector] \)[t]\ )\)\ \ )\)\), ",", " ", RowBox[{ \(-\(\[EmptyDownTriangle]\& \[RightVector] \)\), " ", "X", " ", SuperscriptBox[\((\ \(\[EmptyDownTriangle]\& \[RightVector] \)\ \[Phi][t])\), TagBox["", Derivative], MultilineFunction->None]}]}], "}"}]}], \({0, \ \(0\& \[RightVector] \)} == \n\t \((oddop[{q[dt, \(\[EmptyDownTriangle]\& \[RightVector] \)]}, oddop[{q[dt, \(\[EmptyDownTriangle]\& \[RightVector] \)]}, q[\[Phi][t], \(A\& \[RightVector] \)[t]]]]\ \n\t\t - \ evenop[{q[dt, \(\[EmptyDownTriangle]\& \[RightVector] \)]}, \n \t\t\t\tvector[ evenop[{ q[dt, \(-\(\[EmptyDownTriangle]\& \[RightVector] \)\)]}, q[\[Phi][t], \(-\(A\& \[RightVector] \)[t]\)]]]])\)\n \ \ \ \ \ \ \ \ \ .{1, 0}\), \({0, \(0\& \[RightVector] \)} == { \(\[EmptyDownTriangle]\& \[RightVector] \)\ \[CenterDot]\ \((\(-\((\ \(\[EmptyDownTriangle]\& \[RightVector] \)\ \[Phi][t]\ ) \)\) - \[PartialD]\(A\& \[RightVector] \)[t]\/\[PartialD]t) \), \n\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \(\[EmptyDownTriangle]\& \[RightVector] \)\ X\ \((\ \(\[EmptyDownTriangle]\& \[RightVector] \)\ X\ \(A\& \[RightVector] \)[t]\ )\)\ + \[PartialD]\^2\( A\& \[RightVector] \)[t]\/\[PartialD]t\^2 + \(\[PartialD]\(\[EmptyDownTriangle]\& \[RightVector] \)\[InvisibleComma] \[Phi][t]\)\/\[PartialD]t}\)}], "Input", CellMargins->{{9, 71}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ StyleBox["{", FontWeight->"Bold"], RowBox[{ \((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector]) \), ",", RowBox[{ StyleBox["[", FontWeight->"Bold"], \(\((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector])\), \((\[Phi], A\&\[RightVector])\)\), StyleBox["]", FontWeight->"Bold"]}]}], StyleBox["}", FontWeight->"Bold"]}], " ", "\n", "+", RowBox[{ StyleBox["[", FontWeight->"Bold"], RowBox[{ \((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector]) \), ",", RowBox[{"vector", RowBox[{ StyleBox["{", FontWeight->"Bold"], \(\((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector])\), \((\[Phi], \(-A\&\[RightVector]\))\)\), StyleBox["}", FontWeight->"Bold"]}]}]}], StyleBox["]", FontWeight->"Bold"]}]}], "\n", "=", RowBox[{ RowBox[{"(", RowBox[{ \(\(-\[EmptyDownTriangle]\&\[RightVector]\)\[CenterDot]\ \[EmptyDownTriangle]\&\[RightVector]\ X\ A\&\[RightVector]\), " ", ",", RowBox[{ \(-\[EmptyDownTriangle]\&\[RightVector]\), " ", "X", " ", \(\[EmptyDownTriangle]\&\[RightVector]\), " ", SuperscriptBox["\[Phi]", TagBox["", Derivative], MultilineFunction->None]}]}], ")"}], "=", RowBox[{ \((\(-\[EmptyDownTriangle]\&\[RightVector]\)\[CenterDot]B \&\[RightVector], \[EmptyDownTriangle]\&\[RightVector]\ X\ E\&\[RightVector] + \[PartialD]B\&\[RightVector]\/\[PartialD]t)\), "=", RowBox[{ RowBox[{ RowBox[{\((0, \ 0\&\[RightVector])\), "\n", "\t", "\n", StyleBox["[", FontWeight->"Bold"], RowBox[{ \((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector])\), ",", RowBox[{ StyleBox["[", FontWeight->"Bold"], \(\((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector])\), \((\[Phi], A\&\[RightVector])\)\), StyleBox["]", FontWeight->"Bold"]}]}], StyleBox["]", FontWeight->"Bold"]}], "\n", "\t", "-", " ", RowBox[{ StyleBox["{", FontWeight->"Bold"], RowBox[{ \((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector])\), ",", RowBox[{"vector", RowBox[{ StyleBox["{", FontWeight->"Bold"], \(\((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector])\), \((\[Phi], \(-A\&\[RightVector]\))\)\), StyleBox["}", FontWeight->"Bold"]}]}]}], StyleBox["}", FontWeight->"Bold"]}]}], "\n", "=", \(\((\(-\[EmptyDownTriangle]\&\[RightVector]\)\[CenterDot]\ \[EmptyDownTriangle]\&\[RightVector]\ \[Phi]\ - \[EmptyDownTriangle]\&\[RightVector]\[CenterDot]\[PartialD]A \&\[RightVector]\/\[PartialD]t, \ \[EmptyDownTriangle]\&\[RightVector]\ X\ \ \[EmptyDownTriangle]\&\[RightVector]\ X\ A\&\[RightVector]\ \ + \[PartialD]\^2 A\&\[RightVector]\/\[PartialD]t\^2 + \(\[PartialD]\[EmptyDownTriangle]\&\[RightVector]\ \[InvisibleComma] \[Phi]\)\/\[PartialD]t)\)\n = \(\((\ \[EmptyDownTriangle]\&\[RightVector]\ \[CenterDot]\ E\&\[RightVector]\ \ , \ \[EmptyDownTriangle]\&\[RightVector]\ X\ B\&\[RightVector]\ - \[PartialD]E\&\[RightVector]\/\[PartialD]t)\) = 4\ \[Pi]\ \((\[Rho], J\&\[RightVector])\)\)\)}]}]}]}]], "Input", CellMargins->{{9, 71}, {Inherited, Inherited}}, FontSize->16], Cell["\<\ The homogeneous terms are formed from the sum of both orders of the \ commutator and anticommutator. The source terms arise from the difference of \ two commutators and two anticommutators.\ \>", "Text", CellMargins->{{Inherited, 71}, {Inherited, Inherited}}, FontFamily->"Times New Roman"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[" The Lorentz force"]], "Subsection", CellMargins->{{0, 71}, {Inherited, Inherited}}, FontFamily->"Times New Roman"], Cell[TextData[StyleBox[ "The Lorentz force is generated similarly to the source term of the Maxwell \ equations, but there a small game required to get the signs correct for the \ 4-force."]], "Text", CellMargins->{{Inherited, 71}, {Inherited, Inherited}}, FontFamily->"Times New Roman"], Cell[BoxData[{ \(\((oddop[{q[\[Gamma], \ \[Gamma]\ \[Beta]]}, q[0, \(B\& \[RightVector] \)[t]]]\ \n\t - \ evenop[{q[\(-\[Gamma]\), \ \[Gamma]\ \[Beta]]}, q[0, \(E\& \[RightVector] \)[t]]])\).{1, 0}\), \({\(\ \((\[Beta]\ \[Gamma])\)\ \[CenterDot]\ \(E\& \[RightVector] \)[t]\ \ \), \(\ \((\[Beta]\ \[Gamma])\)\ X\ \(B\& \[RightVector] \)[t]\ \) + \(\ \(E\& \[RightVector] \)[t]\ \[Gamma]\ \)}\)}], "Input", CellMargins->{{18, 71}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ StyleBox["[", FontWeight->"Bold"], \(\((\[Gamma], \ \[Gamma]\ \[Beta])\), \((0, B\&\[RightVector])\)\), StyleBox["]", FontWeight->"Bold"]}], " ", "-", RowBox[{ StyleBox["{", FontWeight->"Bold"], \(\((\(-\[Gamma]\), \ \[Gamma]\ \[Beta])\), \((0, E\&\[RightVector])\)\), StyleBox["}", FontWeight->"Bold"]}]}], "\n", "\n", "=", \((\[Beta]\ \[Gamma]\[CenterDot]E\&\[RightVector]\ \ , \[Gamma] E\&\[RightVector]\ \ + \ \[Gamma]\[Beta]\ \ X\ B\&\[RightVector])\)}]], "Input", CellMargins->{{18, 71}, {Inherited, Inherited}}, FontSize->16], Cell["\<\ This is the covariant form of the Lorentz force. The additional minus sign \ required may be a convention handed down through the ages.\ \>", "Text", CellMargins->{{Inherited, 71}, {Inherited, Inherited}}, FontFamily->"Times New Roman"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[" Conservation laws"]], "Subsection", CellMargins->{{0, 71}, {Inherited, Inherited}}, FontFamily->"Times New Roman"], Cell[TextData[StyleBox[ "The continuity equation--conservation of charge--is formed by applying the \ conjugate of the differential operator to the source terms of the Maxwell \ equations."]], "Text", CellMargins->{{Inherited, 71}, {Inherited, Inherited}}, FontFamily->"Times New Roman"], Cell[BoxData[{ \(scalar[ op[{q[dt, \(-\(\[EmptyDownTriangle]\& \[RightVector] \)\)]}, 4\ \[Pi]\ q[\[Rho][t], \ \(J\& \[RightVector] \)]]].{1, 0}\n == scalar[op[{q[dt, \(-\(\[EmptyDownTriangle]\& \[RightVector] \)\)]}, \n \t\t\toddop[{q[dt, \(\[EmptyDownTriangle]\& \[RightVector] \)]}, q[0, \(B\& \[RightVector] \)[t]]]\ \n\t\t\t - \ evenop[{q[dt, \(\[EmptyDownTriangle]\& \[RightVector] \)]}, q[0, \(E\& \[RightVector] \)[t]]]]].{1, 0}\), \({4\ \((\ \(\[EmptyDownTriangle]\& \[RightVector] \)\ \[CenterDot]\ \((\[Pi]\ \(J\& \[RightVector] \))\)\ \ )\) + 4\ \[Pi]\ \[PartialD]\[Rho][t]\/\[PartialD]t, 0}\n == {\(\ \(\[EmptyDownTriangle]\& \[RightVector] \)\ \[CenterDot]\ \((\(\ \(\[EmptyDownTriangle]\& \[RightVector] \)\ X\ \(B\& \[RightVector] \)[t]\ \) - \[PartialD]\(E\& \[RightVector] \)[t]\/\[PartialD]t)\)\ \ \) + \[PartialD]\(E\& \[RightVector] \)[t]\/\[PartialD]t\ \(\[EmptyDownTriangle]\& \[RightVector] \)\[CenterDot]\ \(E\& \[RightVector] \)[t], 0}\)}], "Input", CellMargins->{{Inherited, 71}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ \(scalar \((\((\[PartialD]\/\[PartialD]t, \(-\[EmptyDownTriangle]\&\[RightVector]\))\), 4\ \[Pi]\ \((\[Rho], \ J\&\[RightVector])\))\)\n = \(scalar \(( \((\[PartialD]\/\[PartialD]t, \(-\[EmptyDownTriangle]\&\[RightVector]\))\) \((\ \[EmptyDownTriangle]\&\[RightVector]\[CenterDot]E \&\[RightVector], \ \[EmptyDownTriangle]\&\[RightVector]\ X\ B\&\[RightVector]\ - \[PartialD]E\&\[RightVector]\/\[PartialD]t)\))\)\n = \(4\ \[Pi] \((E\&\[RightVector]\[CenterDot]\ J\&\[RightVector]\ + \[PartialD]\[Rho]\/\[PartialD]t, 0)\)\n = \((\[EmptyDownTriangle]\&\[RightVector]\[CenterDot]\ \[EmptyDownTriangle]\&\[RightVector]\ X\ B\&\[RightVector]\ - \[EmptyDownTriangle]\&\[RightVector]\[CenterDot]\[PartialD]E \&\[RightVector]\/\[PartialD]t\ \ + \[PartialD]E\&\[RightVector]\/\[PartialD]t\ \[EmptyDownTriangle]\&\[RightVector]\[CenterDot]\ E\&\[RightVector], 0)\)\)\)\)], "Input", CellMargins->{{Inherited, 71}, {Inherited, Inherited}}, FontSize->16], Cell[TextData[StyleBox[ "The right hand side is zero, so the divergence of the current density plus \ the rate of change of the charge density must equal zero. That means that \ charge is conserved."]], "Text", CellMargins->{{Inherited, 71}, {Inherited, Inherited}}, FontFamily->"Times New Roman"], Cell[TextData[StyleBox[ "Poynting's theorem for energy conservation is formed in a very similar way, \ except that the conjugate of electric field is used instead of the conjgate \ of the differential operator."]], "Text", CellMargins->{{Inherited, 71}, {Inherited, Inherited}}, FontFamily->"Times New Roman"], Cell[BoxData[{ \(scalar[ op[{q[0, \(-\(E\& \[RightVector] \)\)]}, 4\ \[Pi]\ q[\[Rho][t], \ \(J\& \[RightVector] \)]]].{1, 0}\n == scalar[op[{q[0, \(-\(E\& \[RightVector] \)\)]}, \n\t\t\t oddop[{q[dt, \(\[EmptyDownTriangle]\& \[RightVector] \)]}, q[0, \(B\& \[RightVector] \)[t]]]\ \n\t\t\t - \ evenop[{q[dt, \(\[EmptyDownTriangle]\& \[RightVector] \)]}, q[0, \(E\& \[RightVector] \)[t]]]]].{1, 0}\), \({4\ \((\ \(E\& \[RightVector] \)\ \[CenterDot]\ \((\[Pi]\ \(J\& \[RightVector] \))\)\ \ )\), 0} == {\(\ \(E\& \[RightVector] \)\ \[CenterDot]\ \((\(\ \(\[EmptyDownTriangle]\& \[RightVector] \)\ X\ \(B\& \[RightVector] \)[t]\ \) - \[PartialD]\(E\& \[RightVector] \)[t]\/\[PartialD]t)\)\ \ \), 0} \)}], "Input", CellMargins->{{9, 71}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ \(scalar \((\((0, \(-E\&\[RightVector]\))\), 4\ \[Pi]\ \((\[Rho], \ J\&\[RightVector])\))\)\n = \(scalar \(( \((0, \(-E\&\[RightVector]\))\) \((\ \[EmptyDownTriangle]\&\[RightVector]\[CenterDot]E \&\[RightVector]\ \ , \ \[EmptyDownTriangle]\&\[RightVector]\ X\ B\&\[RightVector]\ - \[PartialD]E\&\[RightVector]\/\[PartialD]t)\))\)\n = \(4\ \[Pi] \((E\&\[RightVector]\[CenterDot]J\&\[RightVector], 0)\)\n = \((\ E \&\[RightVector]\[CenterDot]\[EmptyDownTriangle]\&\ \[RightVector]\ X\ B\&\[RightVector]\ - E\&\[RightVector]\[CenterDot]\[PartialD]E \&\[RightVector]\/\[PartialD]t\ , 0)\)\)\)\)], "Input", CellMargins->{{9, 71}, {Inherited, Inherited}}, FontSize->16], Cell[TextData[StyleBox[ "Additional vector identities are required before the final form is \ reached."]], "Text", CellMargins->{{Inherited, 71}, {Inherited, Inherited}}, FontFamily->"Times New Roman"], Cell[BoxData[{ \(E\&\[RightVector]\ \[CenterDot]\ \((\ \[EmptyDownTriangle]\&\[RightVector]\ X\ B\&\[RightVector][t]) \)\ == \ B\&\[RightVector]\ \[CenterDot]\ \((\ \[EmptyDownTriangle]\&\[RightVector]\ X\ E\&\[RightVector][t]) \)\ + \ \[EmptyDownTriangle]\&\[RightVector]\[CenterDot]\ \((B\&\[RightVector][t]\ X\ E\&\[RightVector][t])\)\ \), \(\[EmptyDownTriangle]\&\[RightVector]\ X\ E\&\[RightVector][t]\ == \ \(-\(\[PartialD]B\&\[RightVector][t]\/\[PartialD]t\)\)\), \(E\&\[RightVector]\ \[CenterDot]\[PartialD]E\&\[RightVector][t]\/\[PartialD]t\ == \ \[PartialD]\(E\&\[RightVector]\^2\)[t]\/\(2\ \[PartialD]t\)\ \), \(B\&\[RightVector]\ \[CenterDot]\[PartialD]B\&\[RightVector][t]\/\[PartialD]t\ == \ \[PartialD]\(B\&\[RightVector]\^2\)[t]\/\(2\ \[PartialD]t\)\ \)}], "Input", CellMargins->{{Inherited, 71}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[{ \(E\&\[RightVector]\ \[CenterDot]\ \((\ \[EmptyDownTriangle]\&\[RightVector]\ X\ B\&\[RightVector])\)\ = \ B\&\[RightVector]\[CenterDot]\((\ \[EmptyDownTriangle]\&\[RightVector]\ X\ E\&\[RightVector])\)\ + \ \[EmptyDownTriangle]\&\[RightVector]\[CenterDot]\(( B\&\[RightVector]\ X\ E\&\[RightVector])\)\ \), \(\[EmptyDownTriangle]\&\[RightVector]\ X\ E\&\[RightVector]\ = \ \(-\(\[PartialD]B\&\[RightVector]\/\[PartialD]t\)\)\), \(E\&\[RightVector]\ \[CenterDot]\[PartialD]E\&\[RightVector]\/\[PartialD]t\ = \ \(\[PartialD]E\&\[RightVector]\^2\/\(2\ \[PartialD]t\)\ \ \ \ \ \ \ \ B\&\[RightVector]\ \[CenterDot]\[PartialD]B\&\[RightVector]\/\[PartialD]t\ = \ \[PartialD]B\&\[RightVector]\^2\/\(2\ \[PartialD]t\)\)\ \)}], "Input",\ CellMargins->{{Inherited, 71}, {Inherited, Inherited}}, FontSize->16], Cell[TextData[StyleBox[ "Use these equations to simplify to the following."]], "Text", CellMargins->{{Inherited, 71}, {Inherited, Inherited}}, FontFamily->"Times New Roman"], Cell[BoxData[ \({4\ \((\ E\&\[RightVector]\ \[CenterDot]\ \((\[Pi]\ J\&\[RightVector])\)\ \ )\), 0}\n == {\[EmptyDownTriangle]\&\[RightVector]\[CenterDot]\ \((B\&\[RightVector][t]\ X\ E\&\[RightVector][t])\) - 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In this notebook, these basic \ elements have been written as quaternion equations, exploiting the actions of \ commutators and anticommutators. There is an interesting link between the E \ field and a differential operator for gerenrating conservation laws. More \ importantly, the means to generate these equations using quaternion operators \ has been displayed. 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Styles", "Section"], Cell["\<\ The cells below define styles that define standard ButtonFunctions, for use \ in palette buttons.\ \>", "Text"], Cell[StyleData["Paste"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, After]}]&)}], Cell[StyleData["Evaluate"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], SelectionEvaluate[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["EvaluateCell"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionMove[ FrontEnd`InputNotebook[ ], All, Cell, 1], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["CopyEvaluate"], 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The \ \"Hyperlink\" style is for links within the same Notebook, or between \ Notebooks.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Hyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonNote->ButtonData}], Cell[StyleData["Hyperlink", "Presentation"]], Cell[StyleData["Hyperlink", "Condensed"]], Cell[StyleData["Hyperlink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell["\<\ The following styles are for linking automatically to the on-line help \ system.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["MainBookLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "MainBook", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["MainBookLink", "Presentation"]], Cell[StyleData["MainBookLink", "Condensed"]], Cell[StyleData["MainBookLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["AddOnsLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "AddOns", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["AddOnsLink", "Presentation"]], Cell[StyleData["AddOnsLink", "Condensed"]], Cell[StyleData["AddOnLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["RefGuideLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "RefGuideLink", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["RefGuideLink", "Presentation"]], Cell[StyleData["RefGuideLink", "Condensed"]], Cell[StyleData["RefGuideLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["GettingStartedLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "GettingStarted", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["GettingStartedLink", "Presentation"]], Cell[StyleData["GettingStartedLink", "Condensed"]], Cell[StyleData["GettingStartedLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["OtherInformationLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "OtherInformation", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["OtherInformationLink", "Presentation"]], Cell[StyleData["OtherInformationLink", "Condensed"]], Cell[StyleData["OtherInformationLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Placeholder Styles", "Section"], Cell["\<\ The cells below define styles useful for making placeholder objects in \ palette templates.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Placeholder"], Editable->False, Selectable->False, StyleBoxAutoDelete->True, Placeholder->True, StyleMenuListing->None], Cell[StyleData["Placeholder", "Presentation"]], Cell[StyleData["Placeholder", "Condensed"]], Cell[StyleData["Placeholder", "Printout"]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["SelectionPlaceholder"], Editable->False, Selectable->False, StyleBoxAutoDelete->True, Placeholder->Primary, StyleMenuListing->None, DrawHighlighted->True], Cell[StyleData["SelectionPlaceholder", "Presentation"]], Cell[StyleData["SelectionPlaceholder", "Condensed"]], Cell[StyleData["SelectionPlaceholder", "Printout"]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["FormatType Styles", "Section"], Cell["\<\ The cells below define styles that are mixed in with the styles of most \ cells. If a cell's FormatType matches the name of one of the styles defined \ below, then that style is applied between the cell's style and its own \ options.\ \>", "Text"], Cell[StyleData["CellExpression"], PageWidth->Infinity, CellMargins->{{6, Inherited}, {Inherited, Inherited}}, ShowCellLabel->False, ShowSpecialCharacters->False, AllowInlineCells->False, AutoItalicWords->{}, StyleMenuListing->None, FontFamily->"Courier", Background->GrayLevel[1]], Cell[StyleData["InputForm"], AllowInlineCells->False, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["OutputForm"], PageWidth->Infinity, TextAlignment->Left, LineSpacing->{1, -5}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["StandardForm"], LineSpacing->{1.25, 0}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["TraditionalForm"], LineSpacing->{1.25, 0}, SingleLetterItalics->True, TraditionalFunctionNotation->True, DelimiterMatching->None, StyleMenuListing->None], Cell["\<\ The style defined below is mixed in to any cell that is in an inline cell \ within another.\ \>", "Text"], Cell[StyleData["InlineCell"], TextAlignment->Left, ScriptLevel->1, StyleMenuListing->None], Cell[StyleData["InlineCellEditing"], StyleMenuListing->None, Background->RGBColor[1, 0.749996, 0.8]] }, Closed]] }, Open ]] }] ] (*********************************************************************** Cached data follows. 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