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This is of significance if you \ wish to know how a finite magnetic object might react and align itself with \ such a field. This Demonstration enables you to clearly see the \ non-uniformity and off-axis magnetic field components, thus allowing you to \ determine how such an object might react. The symmetry axis of the Helmholtz \ coil is aligned with the Earth's horizontal magnetic field. The coils are in \ the ", Cell[BoxData[ FormBox["x", TraditionalForm]], "InlineMath"], "-", Cell[BoxData[ FormBox["y", TraditionalForm]], "InlineMath"], " plane with one centered at ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", RowBox[{"x", ",", "y", ",", "z"}], ")"}], "=", RowBox[{"(", RowBox[{"0", ",", "0", ",", RowBox[{ RowBox[{"-", "R"}], "/", "2"}]}], ")"}]}], TraditionalForm]], "InlineMath"], " and the other separated by the radius ", Cell[BoxData[ FormBox["R", TraditionalForm]], "InlineMath"], " of the coils centered at ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"0", ",", "0", ",", RowBox[{"R", "/", "2"}]}], ")"}], TraditionalForm]], "InlineMath"], ". The number of turns for the coils ", Cell[BoxData[ FormBox["N", TraditionalForm]], "InlineMath"], " is initially set to 25 and the vacuum permeability ", Cell[BoxData[ FormBox[ SubscriptBox["\[Mu]", "0"], TraditionalForm]], "InlineMath"], " is included in the equations as well. You can vary the radius of the \ coils, the current supplied to them, the number of turns in the coils, and \ the Earth's field. All units are in the MKS system." }], "ManipulateCaption", CellChangeTimes->{ 3.529249259483221*^9, {3.529347825187688*^9, 3.5293478716813374`*^9}, { 3.5293479104242105`*^9, 3.52934800162433*^9}, {3.529356824046755*^9, 3.529356959379158*^9}, {3.5304654115277557`*^9, 3.530465567787768*^9}, { 3.5304657704411097`*^9, 3.530465783366272*^9}, {3.530465884643395*^9, 3.530466085118514*^9}, {3.530466145644917*^9, 3.530466243210525*^9}, { 3.5304662738163767`*^9, 3.530466338361918*^9}, {3.530466371615646*^9, 3.530466378311098*^9}, {3.5304664154236727`*^9, 3.5304664356856937`*^9}, { 3.530466493756443*^9, 3.5304667659654627`*^9}, {3.530466855574049*^9, 3.5304669225850353`*^9}, {3.530466958685066*^9, 3.530466978520074*^9}, { 3.530467009522079*^9, 3.530467017496902*^9}, {3.530487264069086*^9, 3.5304872803470173`*^9}, {3.5305590056678505`*^9, 3.530559008377006*^9}, { 3.5305590557207136`*^9, 3.5305590569767857`*^9}, {3.530982660976716*^9, 3.53098268926026*^9}, {3.530982797318375*^9, 3.5309828002805853`*^9}, { 3.531072024798503*^9, 3.5310720498750763`*^9}, {3.531075936327016*^9, 3.531075991079565*^9}}, CellID->1427452915] }, Open ]], Cell[CellGroupData[{ Cell["", "ThumbnailSection"], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`BE$$ = 0.000019399999999999997`, $CellContext`j$$ = 0.431, $CellContext`Nturns$$ = 25, $CellContext`R$$ = 0.5, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`j$$], 0.431, "current (A)"}, 0., 1., 0.05}, {{ Hold[$CellContext`BE$$], 0.000019399999999999997`, "horizontal Earth field (T)"}, 0, 0.000029999999999999997`, 1.*^-6}, {{ Hold[$CellContext`R$$], 0.5, "radius (m)"}, 0.01, 1, 0.1}, {{ Hold[$CellContext`Nturns$$], 25, "number of turns"}, 1, 50, 2}}, Typeset`size$$ = {550., {188., 192.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = False, $CellContext`j$16463$$ = 0, $CellContext`BE$16464$$ = 0, $CellContext`R$16465$$ = 0, $CellContext`Nturns$16466$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`BE$$ = 0.000019399999999999997`, $CellContext`j$$ = 0.431, $CellContext`Nturns$$ = 25, $CellContext`R$$ = 0.5}, "ControllerVariables" :> { Hold[$CellContext`j$$, $CellContext`j$16463$$, 0], Hold[$CellContext`BE$$, $CellContext`BE$16464$$, 0], Hold[$CellContext`R$$, $CellContext`R$16465$$, 0], Hold[$CellContext`Nturns$$, $CellContext`Nturns$16466$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> StreamPlot[{$CellContext`By[$CellContext`R$$, $CellContext`y, \ $CellContext`z + $CellContext`R$$/ 2, $CellContext`j$$, $CellContext`Nturns$$] + \ $CellContext`By[$CellContext`R$$, $CellContext`y, $CellContext`z - 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( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2)]) (($CellContext`R^2 + $CellContext`y^2 + $CellContext`z^2) EllipticE[ Pi/4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] + ($CellContext`R^2 + $CellContext`y^2 + $CellContext`z^2) EllipticE[(3 Pi)/ 4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] - ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2) EllipticF[ Pi/4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] - ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2) EllipticF[(3 Pi)/ 4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))]))/10^7, $CellContext`By[ Pattern[$CellContext`R, Blank[]], Pattern[$CellContext`y, Blank[]], Pattern[$CellContext`z, Blank[]], Pattern[$CellContext`i, Blank[]]] := (((((43 $CellContext`i) ( 1/(($CellContext`y Sqrt[$CellContext`R^2 + $CellContext`y^2 + $CellContext`z^2]) \ ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2)))) $CellContext`z) Sqrt[($CellContext`R^2 + $CellContext`y^2 + \ $CellContext`z^2)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2)]) (($CellContext`R^2 + $CellContext`y^2 + $CellContext`z^2) EllipticE[ Pi/4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] + ($CellContext`R^2 + $CellContext`y^2 + $CellContext`z^2) EllipticE[(3 Pi)/ 4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] - ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2) EllipticF[ Pi/4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] - ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2) EllipticF[(3 Pi)/ 4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))]))/ 10^7, $CellContext`y = -0.37570481168511055`, $CellContext`z = \ -0.41674912207414144`, $CellContext`Bz[ Pattern[$CellContext`R, Blank[]], Pattern[$CellContext`y, Blank[]], Pattern[$CellContext`z, Blank[]], Pattern[$CellContext`i, Blank[]], Pattern[N, Blank[]]] := (( N $CellContext`i) (((1/( Sqrt[$CellContext`R^2 + $CellContext`y^2 + $CellContext`z^2] \ ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))) Sqrt[($CellContext`R^2 + $CellContext`y^2 + \ $CellContext`z^2)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2)]) (($CellContext`R^2 - $CellContext`y^2 - $CellContext`z^2) EllipticE[ Pi/4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] + ($CellContext`R^2 - $CellContext`y^2 - $CellContext`z^2) EllipticE[(3 Pi)/ 4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] + ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2) EllipticF[ Pi/4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] + ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2) EllipticF[(3 Pi)/ 4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))])))/10^7, $CellContext`Bz[ Pattern[$CellContext`R, Blank[]], Pattern[$CellContext`y, Blank[]], Pattern[$CellContext`z, Blank[]], Pattern[$CellContext`i, Blank[]]] := (( 43 $CellContext`i) (((1/( Sqrt[$CellContext`R^2 + $CellContext`y^2 + $CellContext`z^2] \ ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))) Sqrt[($CellContext`R^2 + $CellContext`y^2 + \ $CellContext`z^2)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2)]) (($CellContext`R^2 - $CellContext`y^2 - $CellContext`z^2) EllipticE[ Pi/4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] + ($CellContext`R^2 - $CellContext`y^2 - $CellContext`z^2) EllipticE[(3 Pi)/ 4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] + ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2) EllipticF[ Pi/4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] + ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2) EllipticF[(3 Pi)/ 4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))])))/10^7}; Typeset`initDone$$ = True), SynchronousInitialization->False, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellID->452171779] }, Open ]], Cell[CellGroupData[{ Cell["", "SnapshotsSection"], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`BE$$ = 0.000019399999999999997`, $CellContext`j$$ = 0.431, $CellContext`Nturns$$ = 25, $CellContext`R$$ = 0.5, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`j$$], 0.431, "current (A)"}, 0., 1., 0.05}, {{ Hold[$CellContext`BE$$], 0.000019399999999999997`, "horizontal Earth field (T)"}, 0, 0.000029999999999999997`, 1.*^-6}, {{ Hold[$CellContext`R$$], 0.5, "radius (m)"}, 0.01, 1, 0.1}, {{ Hold[$CellContext`Nturns$$], 25, "number of turns"}, 1, 50, 2}}, Typeset`size$$ = {550., {188., 192.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = False, $CellContext`j$16533$$ = 0, $CellContext`BE$16534$$ = 0, $CellContext`R$16535$$ = 0, $CellContext`Nturns$16536$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`BE$$ = 0.000019399999999999997`, $CellContext`j$$ = 0.431, $CellContext`Nturns$$ = 25, $CellContext`R$$ = 0.5}, "ControllerVariables" :> { Hold[$CellContext`j$$, $CellContext`j$16533$$, 0], Hold[$CellContext`BE$$, $CellContext`BE$16534$$, 0], Hold[$CellContext`R$$, $CellContext`R$16535$$, 0], Hold[$CellContext`Nturns$$, $CellContext`Nturns$16536$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> StreamPlot[{$CellContext`By[$CellContext`R$$, $CellContext`y, \ $CellContext`z + $CellContext`R$$/ 2, $CellContext`j$$, $CellContext`Nturns$$] + \ $CellContext`By[$CellContext`R$$, $CellContext`y, $CellContext`z - \ $CellContext`R$$/ 2, $CellContext`j$$, $CellContext`Nturns$$], $CellContext`BE$$ - \ ($CellContext`Bz[$CellContext`R$$, $CellContext`y, $CellContext`z + \ $CellContext`R$$/ 2, $CellContext`j$$, $CellContext`Nturns$$] + \ $CellContext`Bz[$CellContext`R$$, $CellContext`y, $CellContext`z - \ $CellContext`R$$/ 2, $CellContext`j$$, $CellContext`Nturns$$])}, {$CellContext`y, \ -0.5, 0.5}, {$CellContext`z, -0.5, 0.5}, StreamPoints -> 150, FrameLabel -> {$CellContext`Y, $CellContext`Z}, LabelStyle -> Directive[Red, Medium], ImageSize -> {550, 380}], "Specifications" :> {{{$CellContext`j$$, 0.431, "current (A)"}, 0., 1., 0.05, Appearance -> "Labeled"}, {{$CellContext`BE$$, 0.000019399999999999997`, "horizontal Earth field (T)"}, 0, 0.000029999999999999997`, 1.*^-6, Appearance -> "Labeled"}, {{$CellContext`R$$, 0.5, "radius (m)"}, 0.01, 1, 0.1, Appearance -> "Labeled"}, {{$CellContext`Nturns$$, 25, "number of turns"}, 1, 50, 2, Appearance -> "Labeled"}}, "Options" :> { SynchronousUpdating -> False, TrackedSymbols -> True, SynchronousInitialization -> False}, "DefaultOptions" :> {ControllerLinking -> True}], ImageSizeCache->{593., {272., 278.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, Initialization:>({$CellContext`By[ Pattern[$CellContext`R, Blank[]], Pattern[$CellContext`y, Blank[]], Pattern[$CellContext`z, Blank[]], Pattern[$CellContext`i, Blank[]], Pattern[N, Blank[]]] := (((((N $CellContext`i) ( 1/(($CellContext`y Sqrt[$CellContext`R^2 + $CellContext`y^2 + $CellContext`z^2]) \ ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2)))) $CellContext`z) Sqrt[($CellContext`R^2 + $CellContext`y^2 + \ $CellContext`z^2)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2)]) (($CellContext`R^2 + $CellContext`y^2 + $CellContext`z^2) EllipticE[ Pi/4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] + ($CellContext`R^2 + $CellContext`y^2 + $CellContext`z^2) EllipticE[(3 Pi)/ 4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] - ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2) EllipticF[ Pi/4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] - ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2) EllipticF[(3 Pi)/ 4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))]))/10^7, $CellContext`By[ Pattern[$CellContext`R, Blank[]], Pattern[$CellContext`y, Blank[]], Pattern[$CellContext`z, Blank[]], Pattern[$CellContext`i, Blank[]]] := (((((43 $CellContext`i) ( 1/(($CellContext`y Sqrt[$CellContext`R^2 + $CellContext`y^2 + $CellContext`z^2]) \ ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2)))) $CellContext`z) Sqrt[($CellContext`R^2 + $CellContext`y^2 + \ $CellContext`z^2)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2)]) (($CellContext`R^2 + $CellContext`y^2 + $CellContext`z^2) EllipticE[ Pi/4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] + ($CellContext`R^2 + $CellContext`y^2 + $CellContext`z^2) EllipticE[(3 Pi)/ 4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] - ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2) EllipticF[ Pi/4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] - ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2) EllipticF[(3 Pi)/ 4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))]))/ 10^7, $CellContext`y = -0.37570481168511055`, $CellContext`z = \ -0.41674912207414144`, $CellContext`Bz[ Pattern[$CellContext`R, Blank[]], Pattern[$CellContext`y, Blank[]], Pattern[$CellContext`z, Blank[]], Pattern[$CellContext`i, Blank[]], Pattern[N, Blank[]]] := (( N $CellContext`i) (((1/( Sqrt[$CellContext`R^2 + $CellContext`y^2 + $CellContext`z^2] \ ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))) Sqrt[($CellContext`R^2 + $CellContext`y^2 + \ $CellContext`z^2)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2)]) (($CellContext`R^2 - $CellContext`y^2 - $CellContext`z^2) EllipticE[ Pi/4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] + ($CellContext`R^2 - $CellContext`y^2 - $CellContext`z^2) EllipticE[(3 Pi)/ 4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] + ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2) EllipticF[ Pi/4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] + ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2) EllipticF[(3 Pi)/ 4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))])))/10^7, $CellContext`Bz[ Pattern[$CellContext`R, Blank[]], Pattern[$CellContext`y, Blank[]], Pattern[$CellContext`z, Blank[]], Pattern[$CellContext`i, Blank[]]] := (( 43 $CellContext`i) (((1/( Sqrt[$CellContext`R^2 + $CellContext`y^2 + $CellContext`z^2] \ ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))) Sqrt[($CellContext`R^2 + $CellContext`y^2 + \ $CellContext`z^2)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2)]) (($CellContext`R^2 - $CellContext`y^2 - $CellContext`z^2) EllipticE[ Pi/4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] + ($CellContext`R^2 - $CellContext`y^2 - $CellContext`z^2) EllipticE[(3 Pi)/ 4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] + ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2) EllipticF[ Pi/4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] + ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2) EllipticF[(3 Pi)/ 4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))])))/10^7}; Typeset`initDone$$ = True), SynchronousInitialization->False, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellID->518384485], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`BE$$ = 0.000024, $CellContext`j$$ = 0.7000000000000001, $CellContext`Nturns$$ = 25, $CellContext`R$$ = 0.31000000000000005`, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`j$$], 0.7000000000000001, "current (A)"}, 0., 1., 0.05}, {{ Hold[$CellContext`BE$$], 0.000024, "horizontal Earth field (T)"}, 0, 0.000029999999999999997`, 1.*^-6}, {{ Hold[$CellContext`R$$], 0.31000000000000005`, "radius (m)"}, 0.01, 1, 0.1}, {{ Hold[$CellContext`Nturns$$], 25, "number of turns"}, 1, 50, 2}}, Typeset`size$$ = {550., {188., 192.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = False, $CellContext`j$16603$$ = 0, $CellContext`BE$16604$$ = 0, $CellContext`R$16605$$ = 0, $CellContext`Nturns$16606$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`BE$$ = 0.000024, $CellContext`j$$ = 0.7000000000000001, $CellContext`Nturns$$ = 25, $CellContext`R$$ = 0.31000000000000005`}, "ControllerVariables" :> { Hold[$CellContext`j$$, $CellContext`j$16603$$, 0], Hold[$CellContext`BE$$, $CellContext`BE$16604$$, 0], Hold[$CellContext`R$$, $CellContext`R$16605$$, 0], Hold[$CellContext`Nturns$$, $CellContext`Nturns$16606$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> StreamPlot[{$CellContext`By[$CellContext`R$$, $CellContext`y, \ $CellContext`z + $CellContext`R$$/ 2, $CellContext`j$$, $CellContext`Nturns$$] + \ $CellContext`By[$CellContext`R$$, $CellContext`y, $CellContext`z - \ $CellContext`R$$/ 2, $CellContext`j$$, $CellContext`Nturns$$], $CellContext`BE$$ - \ ($CellContext`Bz[$CellContext`R$$, $CellContext`y, $CellContext`z + \ $CellContext`R$$/ 2, $CellContext`j$$, $CellContext`Nturns$$] + \ $CellContext`Bz[$CellContext`R$$, $CellContext`y, $CellContext`z - \ $CellContext`R$$/ 2, $CellContext`j$$, $CellContext`Nturns$$])}, {$CellContext`y, \ -0.5, 0.5}, {$CellContext`z, -0.5, 0.5}, StreamPoints -> 150, FrameLabel -> {$CellContext`Y, $CellContext`Z}, LabelStyle -> Directive[Red, Medium], ImageSize -> {550, 380}], "Specifications" :> {{{$CellContext`j$$, 0.7000000000000001, "current (A)"}, 0., 1., 0.05, Appearance -> "Labeled"}, {{$CellContext`BE$$, 0.000024, "horizontal Earth field (T)"}, 0, 0.000029999999999999997`, 1.*^-6, Appearance -> "Labeled"}, {{$CellContext`R$$, 0.31000000000000005`, "radius (m)"}, 0.01, 1, 0.1, Appearance -> "Labeled"}, {{$CellContext`Nturns$$, 25, "number of turns"}, 1, 50, 2, Appearance -> "Labeled"}}, "Options" :> { SynchronousUpdating -> False, TrackedSymbols -> True, SynchronousInitialization -> False}, "DefaultOptions" :> {ControllerLinking -> True}], ImageSizeCache->{593., {272., 278.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, Initialization:>({$CellContext`By[ Pattern[$CellContext`R, Blank[]], Pattern[$CellContext`y, Blank[]], Pattern[$CellContext`z, Blank[]], Pattern[$CellContext`i, Blank[]], Pattern[N, Blank[]]] := (((((N $CellContext`i) ( 1/(($CellContext`y Sqrt[$CellContext`R^2 + $CellContext`y^2 + $CellContext`z^2]) \ ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2)))) $CellContext`z) Sqrt[($CellContext`R^2 + $CellContext`y^2 + \ $CellContext`z^2)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2)]) (($CellContext`R^2 + $CellContext`y^2 + $CellContext`z^2) EllipticE[ Pi/4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] + ($CellContext`R^2 + $CellContext`y^2 + $CellContext`z^2) EllipticE[(3 Pi)/ 4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] - ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2) EllipticF[ Pi/4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] - ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2) EllipticF[(3 Pi)/ 4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))]))/10^7, $CellContext`By[ Pattern[$CellContext`R, Blank[]], Pattern[$CellContext`y, Blank[]], Pattern[$CellContext`z, Blank[]], Pattern[$CellContext`i, Blank[]]] := (((((43 $CellContext`i) ( 1/(($CellContext`y Sqrt[$CellContext`R^2 + $CellContext`y^2 + $CellContext`z^2]) \ ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2)))) $CellContext`z) Sqrt[($CellContext`R^2 + $CellContext`y^2 + \ $CellContext`z^2)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2)]) (($CellContext`R^2 + $CellContext`y^2 + $CellContext`z^2) EllipticE[ Pi/4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] + ($CellContext`R^2 + $CellContext`y^2 + $CellContext`z^2) EllipticE[(3 Pi)/ 4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] - ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2) EllipticF[ Pi/4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] - ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2) EllipticF[(3 Pi)/ 4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))]))/ 10^7, $CellContext`y = -0.37570481168511055`, $CellContext`z = \ -0.41674912207414144`, $CellContext`Bz[ Pattern[$CellContext`R, Blank[]], Pattern[$CellContext`y, Blank[]], Pattern[$CellContext`z, Blank[]], Pattern[$CellContext`i, Blank[]], Pattern[N, Blank[]]] := (( N $CellContext`i) (((1/( Sqrt[$CellContext`R^2 + $CellContext`y^2 + $CellContext`z^2] \ ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))) Sqrt[($CellContext`R^2 + $CellContext`y^2 + \ $CellContext`z^2)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2)]) (($CellContext`R^2 - $CellContext`y^2 - $CellContext`z^2) EllipticE[ Pi/4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] + ($CellContext`R^2 - $CellContext`y^2 - $CellContext`z^2) EllipticE[(3 Pi)/ 4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] + ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2) EllipticF[ Pi/4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] + ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2) EllipticF[(3 Pi)/ 4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))])))/10^7, $CellContext`Bz[ Pattern[$CellContext`R, Blank[]], Pattern[$CellContext`y, Blank[]], Pattern[$CellContext`z, Blank[]], Pattern[$CellContext`i, Blank[]]] := (( 43 $CellContext`i) (((1/( Sqrt[$CellContext`R^2 + $CellContext`y^2 + $CellContext`z^2] \ ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))) Sqrt[($CellContext`R^2 + $CellContext`y^2 + \ $CellContext`z^2)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2)]) (($CellContext`R^2 - $CellContext`y^2 - $CellContext`z^2) EllipticE[ Pi/4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] + ($CellContext`R^2 - $CellContext`y^2 - $CellContext`z^2) EllipticE[(3 Pi)/ 4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] + ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2) EllipticF[ Pi/4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] + ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2) EllipticF[(3 Pi)/ 4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))])))/10^7}; Typeset`initDone$$ = True), SynchronousInitialization->False, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellID->351776995], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`BE$$ = 9.*^-6, $CellContext`j$$ = 0.5, $CellContext`Nturns$$ = 25, $CellContext`R$$ = 0.26, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`j$$], 0.25, "current (A)"}, 0., 1., 0.05}, {{ Hold[$CellContext`BE$$], 0., "horizontal Earth field (T)"}, 0, 0.000029999999999999997`, 1.*^-6}, {{ Hold[$CellContext`R$$], 0.26, "radius (m)"}, 0.01, 1, 0.1}, {{ Hold[$CellContext`Nturns$$], 25, "number of turns"}, 1, 50, 2}}, Typeset`size$$ = {550., {188., 192.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = False, $CellContext`j$16673$$ = 0, $CellContext`BE$16674$$ = 0, $CellContext`R$16675$$ = 0, $CellContext`Nturns$16676$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`BE$$ = 0., $CellContext`j$$ = 0.25, $CellContext`Nturns$$ = 25, $CellContext`R$$ = 0.26}, "ControllerVariables" :> { Hold[$CellContext`j$$, $CellContext`j$16673$$, 0], Hold[$CellContext`BE$$, $CellContext`BE$16674$$, 0], Hold[$CellContext`R$$, $CellContext`R$16675$$, 0], Hold[$CellContext`Nturns$$, $CellContext`Nturns$16676$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> StreamPlot[{$CellContext`By[$CellContext`R$$, $CellContext`y, \ $CellContext`z + $CellContext`R$$/ 2, $CellContext`j$$, $CellContext`Nturns$$] + \ $CellContext`By[$CellContext`R$$, $CellContext`y, $CellContext`z - \ $CellContext`R$$/ 2, $CellContext`j$$, $CellContext`Nturns$$], $CellContext`BE$$ - \ ($CellContext`Bz[$CellContext`R$$, $CellContext`y, $CellContext`z + \ $CellContext`R$$/ 2, $CellContext`j$$, $CellContext`Nturns$$] + \ $CellContext`Bz[$CellContext`R$$, $CellContext`y, $CellContext`z - \ $CellContext`R$$/ 2, $CellContext`j$$, $CellContext`Nturns$$])}, {$CellContext`y, \ -0.5, 0.5}, {$CellContext`z, -0.5, 0.5}, StreamPoints -> 150, FrameLabel -> {$CellContext`Y, $CellContext`Z}, LabelStyle -> Directive[Red, Medium], ImageSize -> {550, 380}], "Specifications" :> {{{$CellContext`j$$, 0.25, "current (A)"}, 0., 1., 0.05, Appearance -> "Labeled"}, {{$CellContext`BE$$, 0., "horizontal Earth field (T)"}, 0, 0.000029999999999999997`, 1.*^-6, Appearance -> "Labeled"}, {{$CellContext`R$$, 0.26, "radius (m)"}, 0.01, 1, 0.1, Appearance -> "Labeled"}, {{$CellContext`Nturns$$, 25, "number of turns"}, 1, 50, 2, Appearance -> "Labeled"}}, "Options" :> { SynchronousUpdating -> False, TrackedSymbols -> True, SynchronousInitialization -> False}, "DefaultOptions" :> {ControllerLinking -> True}], ImageSizeCache->{593., {272., 278.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, Initialization:>({$CellContext`By[ Pattern[$CellContext`R, Blank[]], Pattern[$CellContext`y, Blank[]], Pattern[$CellContext`z, Blank[]], Pattern[$CellContext`i, Blank[]], Pattern[N, Blank[]]] := (((((N $CellContext`i) ( 1/(($CellContext`y Sqrt[$CellContext`R^2 + $CellContext`y^2 + $CellContext`z^2]) \ ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2)))) $CellContext`z) Sqrt[($CellContext`R^2 + $CellContext`y^2 + \ $CellContext`z^2)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2)]) (($CellContext`R^2 + $CellContext`y^2 + $CellContext`z^2) EllipticE[ Pi/4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] + ($CellContext`R^2 + $CellContext`y^2 + $CellContext`z^2) EllipticE[(3 Pi)/ 4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] - ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2) EllipticF[ Pi/4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] - ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2) EllipticF[(3 Pi)/ 4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))]))/10^7, $CellContext`By[ Pattern[$CellContext`R, Blank[]], Pattern[$CellContext`y, Blank[]], Pattern[$CellContext`z, Blank[]], Pattern[$CellContext`i, Blank[]]] := (((((43 $CellContext`i) ( 1/(($CellContext`y Sqrt[$CellContext`R^2 + $CellContext`y^2 + $CellContext`z^2]) \ ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2)))) $CellContext`z) Sqrt[($CellContext`R^2 + $CellContext`y^2 + \ $CellContext`z^2)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2)]) (($CellContext`R^2 + $CellContext`y^2 + $CellContext`z^2) EllipticE[ Pi/4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] + ($CellContext`R^2 + $CellContext`y^2 + $CellContext`z^2) EllipticE[(3 Pi)/ 4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] - ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2) EllipticF[ Pi/4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] - ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2) EllipticF[(3 Pi)/ 4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))]))/ 10^7, $CellContext`y = -0.37570481168511055`, $CellContext`z = \ -0.41674912207414144`, $CellContext`Bz[ Pattern[$CellContext`R, Blank[]], Pattern[$CellContext`y, Blank[]], Pattern[$CellContext`z, Blank[]], Pattern[$CellContext`i, Blank[]], Pattern[N, Blank[]]] := (( N $CellContext`i) (((1/( Sqrt[$CellContext`R^2 + $CellContext`y^2 + $CellContext`z^2] \ ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))) Sqrt[($CellContext`R^2 + $CellContext`y^2 + \ $CellContext`z^2)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2)]) (($CellContext`R^2 - $CellContext`y^2 - $CellContext`z^2) EllipticE[ Pi/4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] + ($CellContext`R^2 - $CellContext`y^2 - $CellContext`z^2) EllipticE[(3 Pi)/ 4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] + ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2) EllipticF[ Pi/4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] + ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2) EllipticF[(3 Pi)/ 4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))])))/10^7, $CellContext`Bz[ Pattern[$CellContext`R, Blank[]], Pattern[$CellContext`y, Blank[]], Pattern[$CellContext`z, Blank[]], Pattern[$CellContext`i, Blank[]]] := (( 43 $CellContext`i) (((1/( Sqrt[$CellContext`R^2 + $CellContext`y^2 + $CellContext`z^2] \ ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))) Sqrt[($CellContext`R^2 + $CellContext`y^2 + \ $CellContext`z^2)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2)]) (($CellContext`R^2 - $CellContext`y^2 - $CellContext`z^2) EllipticE[ Pi/4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] + ($CellContext`R^2 - $CellContext`y^2 - $CellContext`z^2) EllipticE[(3 Pi)/ 4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] + ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2) EllipticF[ Pi/4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))] + ($CellContext`R^2 + ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2) EllipticF[(3 Pi)/ 4, -(((4 $CellContext`R) $CellContext`y)/($CellContext`R^2 - ( 2 $CellContext`R) $CellContext`y + $CellContext`y^2 + \ $CellContext`z^2))])))/10^7}; Typeset`initDone$$ = True), SynchronousInitialization->False, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellID->33096210] }, Open ]], Cell[CellGroupData[{ Cell["", "DetailsSection"], Cell[TextData[{ "This present Demonstration gives a detailed understanding of the \ interactions between a magnetic torsion pendulum and the net horizontal \ magnetic field it feels while hanging at the nominal center of a Helmholtz \ coil pair used to cancel the horizontal component of the Earth's magnetic \ field. A Helmholtz coil, known for its ability to provide a relatively \ uniform magnetic field between its coils, is traditionally used to cancel out \ the Earth's magnetic field. With a Helmholtz coil suitably aligned so that \ the longitudinal axis of the coils points along the magnetic north-south \ direction, you can cancel the horizontal component of Earth's field (", Cell[BoxData[ FormBox[ SubscriptBox["B", "h"], TraditionalForm]], "InlineMath"], ") when sufficient current is supplied through its wires. At this critical \ current, even though you have succeeded in cancelling ", Cell[BoxData[ FormBox[ SubscriptBox["B", "h"], TraditionalForm]], "InlineMath"], " at the exact center of the coils, there remain small but non-zero, \ off-axis horizontal field components that can interact with a finite, \ imperfectly centered magnetic dipole. The above Demonstration gives this \ detailed rendering of the interplay between the Earth's field with the \ Helmholtz coil. The Demonstration thus allows you to see the net magnetic \ field vector plot anywhere between the two coils. Since the net field does \ not flip everywhere at once, you can see how the net field varies in space, \ in order to see how a finite magnetic object might react in such a field. " }], "DetailNotes", CellChangeTimes->{{3.530461208633819*^9, 3.5304614184285917`*^9}, { 3.5304614707552767`*^9, 3.53046154171346*^9}, {3.530461593839923*^9, 3.530461713309021*^9}, {3.530461774628624*^9, 3.530461776987412*^9}, { 3.5304618226582317`*^9, 3.530462017925906*^9}, {3.530462115316634*^9, 3.530462183433051*^9}, {3.530462330205982*^9, 3.530462407011231*^9}, { 3.530463144933462*^9, 3.530463197263815*^9}, {3.530463261314314*^9, 3.53046357075662*^9}, {3.530463607277089*^9, 3.530463693904387*^9}, { 3.530463775910655*^9, 3.5304637766128817`*^9}, {3.530463813389226*^9, 3.530464141843748*^9}, {3.530464176818756*^9, 3.5304641872568827`*^9}, { 3.530464310332341*^9, 3.5304643395825357`*^9}, {3.530464421276219*^9, 3.530464437395996*^9}, {3.5309827476787786`*^9, 3.5309827650310793`*^9}, { 3.5309828750151854`*^9, 3.530982879019875*^9}, {3.531072013078034*^9, 3.5310720251935186`*^9}, {3.5310720561303263`*^9, 3.531072061233031*^9}, { 3.531076093719542*^9, 3.531076153905246*^9}, 3.531076868972342*^9}, CellID->1005000733], Cell["\<\ For a full derivation of the equations used to produce the vector plot, see \ [1].\ \>", "DetailNotes", CellChangeTimes->{{3.5274394267670794`*^9, 3.5274394567401457`*^9}, { 3.527439641105802*^9, 3.5274396574569263`*^9}, {3.5304901612257943`*^9, 3.5304901779427505`*^9}, {3.531076303474988*^9, 3.5310763108496437`*^9}, 3.531076766639579*^9}, CellID->1191252205], Cell["\<\ For a more detailed discussion of a magnetic torsion pendulum interacting \ with the Helmholtz coil and Earth's field, see [2].\ \>", "DetailNotes", CellChangeTimes->{ 3.35696210375764*^9, {3.5304898660029087`*^9, 3.5304899004658794`*^9}, { 3.5304901895234127`*^9, 3.530490196969839*^9}, {3.5304907421810226`*^9, 3.5304907421830235`*^9}, {3.531076512269932*^9, 3.5310765301087713`*^9}}, CellID->157011213], Cell["References", "DetailNotes", CellChangeTimes->{{3.531076327615447*^9, 3.531076331745482*^9}}, CellID->860260813], Cell[TextData[{ "[1] C. Goolsby. \"Finding the Magnetic Field from a Helmholtz Coil.\" ", StyleBox["Scribd", FontSlant->"Italic"], ". (Nov 16, 2011) ", ButtonBox["www.scribd.com/fullscreen/72977243?access_key=key-\ 2ly21ms0kg761vpsit6", BaseStyle->"Hyperlink", ButtonData->{ URL["http://www.scribd.com/fullscreen/72977243?access_key=key-\ 2ly21ms0kg761vpsit6"], None}, ButtonNote-> "http://www.scribd.com/fullscreen/72977243?access_key=key-\ 2ly21ms0kg761vpsit6"], "." }], "DetailNotes", CellChangeTimes->{{3.53107633344064*^9, 3.531076359856174*^9}, { 3.531076405576359*^9, 3.531076464130838*^9}}, CellID->1543572487], Cell[TextData[{ "[2] C. Goolsby. \"Rotational Behavior of a Magnetic Torsion Pendulum in a \ Helmholtz Coil.\" ", StyleBox["Scribd", FontSlant->"Italic"], ". (Nov 16, 2011) ", ButtonBox["www.scribd.com/fullscreen/72977908?access_key=key-\ mqw1eunbsgg3citsfpc", BaseStyle->"Hyperlink", ButtonData->{ URL["http://www.scribd.com/fullscreen/72977908?access_key=key-\ mqw1eunbsgg3citsfpc"], None}, ButtonNote-> "http://www.scribd.com/fullscreen/72977908?access_key=key-\ mqw1eunbsgg3citsfpc"], "." }], "DetailNotes", CellChangeTimes->{{3.531076467180065*^9, 3.531076470021332*^9}, { 3.5310765365642014`*^9, 3.531076602092349*^9}}, CellID->1220333249], Cell[TextData[{ "[3] Wikipedia. \"Helmholtz Coil.\" (Oct 24, 2011) ", ButtonBox["en.wikipedia.org/wiki/Helmholtz_coil", BaseStyle->"Hyperlink", ButtonData->{ URL["http://en.wikipedia.org/wiki/Helmholtz_coil"], None}, ButtonNote->"http://en.wikipedia.org/wiki/Helmholtz_coil"], "." }], "DetailNotes", CellChangeTimes->{{3.531076608328102*^9, 3.53107660957023*^9}, { 3.531076690409175*^9, 3.5310767540595512`*^9}}, CellID->1672087093], Cell[TextData[{ "[4] Wikipedia. \"Earth's Magnetic Field.\" (Nov 23, 2011) ", ButtonBox["en.wikipedia.org/wiki/Earth's_magnetic _field", BaseStyle->"Hyperlink", ButtonData->{ URL["http://en.wikipedia.org/wiki/Earth's_magnetic_field"], None}, ButtonNote->"http://en.wikipedia.org/wiki/Earth's_magnetic_field"], "." }], "DetailNotes", CellChangeTimes->{{3.531076758464259*^9, 3.531076809571756*^9}}, CellID->796767409] }, Open ]], Cell[CellGroupData[{ Cell["", "ControlSuggestionsSection"], Cell[BoxData[ TooltipBox[ RowBox[{ CheckboxBox[True], Cell[" Resize Images"]}], "\"Click inside an image to reveal its orange resize frame.\\nDrag any of \ the orange resize handles to resize the image.\"", TooltipDelay->0.35]], "ControlSuggestions", CellChangeTimes->{3.5293483757457385`*^9}, FontFamily->"Verdana", CellTags->"ResizeImages"], Cell[BoxData[ TooltipBox[ RowBox[{ CheckboxBox[False], Cell[" Rotate and Zoom in 3D"]}], RowBox[{ "\"Drag a 3D graphic to rotate it. 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Goolsby", BaseStyle->"Hyperlink", ButtonData->{ URL["http://demonstrations.wolfram.com/author.html?author=Cody+A.+\ Goolsby"], None}, ButtonNote-> "http://demonstrations.wolfram.com/author.html?author=Cody+A.+Goolsby"], " and ", ButtonBox["Paul A. 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