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Transforming to Cartesian coordinates in the vertical plane \ bisecting the ring, the following approximation is found to be \ indistinguishable graphically from the exact solution: ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["A", "\[Phi]"], "(", RowBox[{"x", ",", "z"}], ")"}], "=", RowBox[{ FractionBox[ SubscriptBox["\[Mu]", "0"], RowBox[{"4", "\[Pi]"}]], FractionBox[ RowBox[{"\[Pi]", " ", SuperscriptBox["a", RowBox[{"2", " "}]], "I", " ", "x"}], SuperscriptBox[ RowBox[{"(", RowBox[{ SuperscriptBox["a", "2"], "+", SuperscriptBox["x", "2"], "+", SuperscriptBox["z", "2"]}], ")"}], RowBox[{"3", "/", "2"}]]], RowBox[{"(", RowBox[{"1", "+", FractionBox[ RowBox[{"15", SuperscriptBox["a", "2"], SuperscriptBox["x", "2"]}], RowBox[{"8", SuperscriptBox[ RowBox[{"(", RowBox[{ SuperscriptBox["a", "2"], "+", SuperscriptBox["x", "2"], "+", SuperscriptBox["z", "2"]}], ")"}], "2"]}]]}], ")"}]}]}], TraditionalForm]], "InlineMath",ExpressionUUID-> "9b4c3408-3d4a-4e6a-a6b5-59ce8b052532"], ". 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The magnetic lines of force are represented by contours ", Cell[BoxData[ FormBox[ RowBox[{"\[CapitalPsi]", "(", RowBox[{"x", ",", "z"}], ")"}], TraditionalForm]], "InlineMath", ExpressionUUID->"84e59201-e1fd-46e6-8c1d-7e2725fbaa0a"], " orthogonal to those of the potential ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["\[CapitalPhi]", "m"], "(", RowBox[{"x", ",", "z"}], ")"}], TraditionalForm]], "InlineMath", ExpressionUUID->"f55255e5-6ea9-41d4-a944-b91bf0cbf417"], ", thus ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"\[Del]", RowBox[{"\[CapitalPsi]", "(", RowBox[{"x", ",", "z"}], ")"}]}], "\[CenterDot]", RowBox[{"\[Del]", RowBox[{ SubscriptBox["\[CapitalPhi]", "m"], "(", RowBox[{"x", ",", "z"}], ")"}]}]}], "=", "0"}], TraditionalForm]], "InlineMath",ExpressionUUID->"162cca6f-fbb4-4cd8-b0d4-597a6cc772b9"], ". This equation can be satisfied by ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[CapitalPsi]", "(", RowBox[{"x", ",", "z"}], ")"}], "=", RowBox[{ RowBox[{ SubscriptBox["A", "\[Phi]"], "(", RowBox[{"x", ",", "z"}], ")"}], "."}]}], TraditionalForm]], "InlineMath",ExpressionUUID->"d594b2b0-ef13-43cb-99e1-a35c49b3a1bc"] }], "DetailNotes", CellChangeTimes->{ 3.35696210375764*^9, {3.47328560867675*^9, 3.473285724978177*^9}, { 3.473285830615075*^9, 3.473285913594384*^9}, {3.473285943652048*^9, 3.473285999342504*^9}, {3.473286514915247*^9, 3.473286525246027*^9}, { 3.4732865598331337`*^9, 3.473286774278658*^9}, {3.4732870387479*^9, 3.473287242360038*^9}, {3.473287294319437*^9, 3.473287297092342*^9}, { 3.473287357196142*^9, 3.473287665947299*^9}, {3.4732877158138113`*^9, 3.473287727456603*^9}, {3.4732878066294622`*^9, 3.473287807473081*^9}, { 3.473287842459509*^9, 3.473287868233622*^9}, {3.473287927376733*^9, 3.473287980440406*^9}, {3.473288013733542*^9, 3.4732880673090277`*^9}, { 3.4732881051601477`*^9, 3.473288162535335*^9}, {3.473288192937314*^9, 3.473288344028216*^9}, {3.4732921895102262`*^9, 3.4732923255618763`*^9}, 3.473292658373323*^9, {3.473292808899003*^9, 3.4732928110265303`*^9}, { 3.47329294814373*^9, 3.473293009725862*^9}, 3.4732930470009937`*^9, { 3.4736317711066995`*^9, 3.4736318282004495`*^9}, {3.499438999481839*^9, 3.499439012674038*^9}, {3.499439340799464*^9, 3.4994393590985727`*^9}, { 3.499441983480875*^9, 3.499442014855875*^9}}, CellID->983735429,ExpressionUUID->"4ba55f89-1aa8-429c-9bd0-cffd483f6f25"], Cell[TextData[{ "Reference: J. 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