(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 9.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 157, 7] NotebookDataLength[ 3331319, 55154] NotebookOptionsPosition[ 3329220, 55079] NotebookOutlinePosition[ 3329575, 55095] CellTagsIndexPosition[ 3329532, 55092] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Exemplo 3.5 do Griffiths", "Section", CellChangeTimes->{{3.585906150953682*^9, 3.585906178879013*^9}}], Cell[CellGroupData[{ Cell[TextData[{ "\nEm sala de aula resolvemos o problema (Exemplo 3.5 do livro do Griffiths) \ de uma \[OpenCurlyDoubleQuote]caixa\[CloseCurlyDoubleQuote] infinita na dire\ \[CCedilla]\[ATilde]o ", StyleBox["x", FontSlant->"Italic"], ", e de lados ", StyleBox["a", FontSlant->"Italic"], " (na dire\[CCedilla]\[ATilde]o ", StyleBox["y", FontSlant->"Italic"], ") e ", StyleBox["b", FontSlant->"Italic"], " na dire\[CCedilla]\[ATilde]o ", StyleBox["z", FontSlant->"Italic"], ".\n\nAs condi\[CCedilla]\[OTilde]es de contorno s\[ATilde]o tais que na \ tampa \[OpenCurlyDoubleQuote]da esquerda\[CloseCurlyDoubleQuote], em ", StyleBox["x", FontSlant->"Italic"], "=0, o potencial \[EAcute] mantido constante (", Cell[BoxData[ FormBox[ SubscriptBox["\[CurlyPhi]", "0"], TraditionalForm]], FormatType->"TraditionalForm"], "), e nas outras paredes ele \[EAcute] nulo.\n\nAp\[OAcute]s eliminarmos as \ solu\[CCedilla]\[OTilde]es que crescem exponencialmente na dire\[CCedilla]\ \[ATilde]o ", StyleBox["x,", FontSlant->"Italic"], " e limitarmos as solu\[CCedilla]\[OTilde]es nas dire\[CCedilla]\[OTilde]es ", StyleBox["y", FontSlant->"Italic"], " e ", StyleBox["z", FontSlant->"Italic"], " a senos (j\[AAcute] que os cossenos n\[ATilde]o se anulam quando ", StyleBox["y", FontSlant->"Italic"], "=0 ou ", StyleBox["z", FontSlant->"Italic"], "=0), conclu\[IAcute]mos que a solu\[CCedilla]\[ATilde]o para esse problema \ deveria ser escrita como:\n\n", StyleBox["\t", FontSize->24], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[CurlyPhi]", "(", RowBox[{"x", ",", "y", ",", "z"}], ")"}], " ", "=", " ", RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"n", "=", "1"}], "\[Infinity]"], RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"m", "=", "1"}], "\[Infinity]"], RowBox[{ SubscriptBox["C", "nm"], SuperscriptBox["\[ExponentialE]", RowBox[{ RowBox[{"-", SubscriptBox["\[Alpha]", "nm"]}], "x"}]], " ", "sen", " ", FractionBox[ RowBox[{"n", " ", "\[Pi]", " ", "y"}], "a"], " ", "sen", " ", FractionBox[ RowBox[{"m", " ", "\[Pi]", " ", "z"}], "b"]}]}]}]}], TraditionalForm]], FormatType->"TraditionalForm", FontSize->24], "\n\nPara encontrar quem s\[ATilde]o as constante ", Cell[BoxData[ SubscriptBox[ StyleBox["C", FontSlant->"Italic"], RowBox[{" ", RowBox[{ StyleBox["n", FontSlant->"Italic"], StyleBox[" ", FontSlant->"Italic"], StyleBox["m", FontSlant->"Italic"], " "}]}]]]], " temos que impor a condi\[CCedilla]\[ATilde]o de que ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[CurlyPhi]", "(", RowBox[{ RowBox[{"x", "=", "0"}], ",", "y", ",", "z"}], ")"}], " ", "=", " ", RowBox[{ SubscriptBox["\[CurlyPhi]", "0"], " ", RowBox[{"(", RowBox[{ RowBox[{"0", "<", "y", "<", "a"}], ",", " ", RowBox[{"0", "<", "z", "<", "b"}]}], ")"}]}]}], TraditionalForm]], FormatType->"TraditionalForm"], " . Isso nos permite calcular esse coeficientes, que s\[ATilde]o dados por:\n\ \n", StyleBox["\t", FontSize->24], Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["C", RowBox[{"n", " ", "m", " "}]], "=", " ", RowBox[{ SubscriptBox["\[CurlyPhi]", "0"], FractionBox["16", SuperscriptBox["\[Pi]", "2"]], " ", FractionBox["1", RowBox[{"n", " ", "m"}]]}]}], TraditionalForm]], FormatType->"TraditionalForm", FontSize->24], " \n\t\nonde ", StyleBox["n", FontSlant->"Italic"], " e ", StyleBox["m", FontSlant->"Italic"], " s\[ATilde]o n\[UAcute]meros \[IAcute]mpares (ou seja, ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["C", RowBox[{"n", " ", "m"}]], "=", "0"}], TraditionalForm]], FormatType->"TraditionalForm"], " se ", StyleBox["n", FontSlant->"Italic"], " or par ", StyleBox["ou", FontWeight->"Bold"], " se ", StyleBox["m", FontSlant->"Italic"], " for par).\n\nPor simplicidade, vou escolher ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["\[CurlyPhi]", "0"], "=", "1"}], TraditionalForm]], FormatType->"TraditionalForm"], ", e tamb\[EAcute]m ", StyleBox["a", FontSlant->"Italic"], "=", StyleBox["b", FontSlant->"Italic"], "=1 .\n\nAqui est\[AAcute] a s\[EAcute]rie de Fourier de uma fun\[CCedilla]\ \[ATilde]o que \[EAcute] constante em {y,z}, no intervalo entre 0{{3.585906019062043*^9, 3.585906146175959*^9}, { 3.585906185199936*^9, 3.585906368126606*^9}, {3.5859064000173187`*^9, 3.585906717560812*^9}, {3.585906748407505*^9, 3.585907056376382*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"fun5", "[", RowBox[{"y_", ",", "z_"}], "]"}], ":=", RowBox[{ RowBox[{"16", "/", RowBox[{"\[Pi]", "^", "2"}]}], "*", RowBox[{"Sum", "[", RowBox[{ RowBox[{ RowBox[{"1", "/", RowBox[{"(", RowBox[{"m", "*", "n"}], ")"}]}], "*", RowBox[{"Sin", "[", RowBox[{"n", "*", "\[Pi]", "*", "y"}], "]"}], "*", RowBox[{"Sin", "[", RowBox[{"m", "*", "\[Pi]", "*", "z"}], "]"}]}], ",", RowBox[{"{", RowBox[{"m", ",", "1", ",", "5", ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"n", ",", "1", ",", "5", ",", "2"}], 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