(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 12.2' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 129619, 3265] NotebookOptionsPosition[ 104984, 2810] NotebookOutlinePosition[ 110162, 2903] CellTagsIndexPosition[ 110082, 2898] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", "FirstSlide", CellTags-> "SlideShowHeader",ExpressionUUID->"ae861ed8-5fa1-4088-84fd-2651cc7b368a"], Cell["Introdu\[CCedilla]\[ATilde]o \[AGrave] F\[IAcute]sica Computacional I \ (2023.2)", "Title", CellChangeTimes->{ 3.559948400406288*^9, {3.699269571766136*^9, 3.699269573578824*^9}, { 3.7099080105038958`*^9, 3.7099080108133383`*^9}, {3.727189496068931*^9, 3.7271894962748613`*^9}, {3.83932299294512*^9, 3.839323000605739*^9}, { 3.8725882936406546`*^9, 3.872588293813486*^9}, {3.902925316275434*^9, 3.902925316485443*^9}},ExpressionUUID->"92508663-47f1-4deb-9615-\ d2c271e8e453"], Cell["Aula 4: solu\[CCedilla]\[ATilde]o de equa\[CCedilla]\[OTilde]es alg\ \[EAcute]bricas; derivadas; integrais", "Subtitle", CellChangeTimes->{{3.699269582406044*^9, 3.699269584285553*^9}, { 3.8393232649484053`*^9, 3.839323284264048*^9}, 3.840031268565436*^9, { 3.8407028843722677`*^9, 3.840702924713172*^9}, {3.840714600599773*^9, 3.8407146281949797`*^9}, {3.841157909295815*^9, 3.841157911477744*^9}},ExpressionUUID->"14f8f51b-58e9-4033-aea1-\ fd611ac95bb9"] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"ee3ea2a1-973b-4e9a-a704-0d8a3c30d1be"], Cell[CellGroupData[{ Cell["Solu\[CCedilla]\[ATilde]o de equa\[CCedilla]\[OTilde]es \ alg\[EAcute]bricas", "Section", CellChangeTimes->{ 3.699269664116198*^9, 3.70454748091461*^9, {3.704547516075143*^9, 3.7045475190812902`*^9}, {3.706294276733098*^9, 3.706294304716371*^9}, { 3.839323324298633*^9, 3.839323331059084*^9}, 3.840031288455207*^9, { 3.840714636688951*^9, 3.840714639748958*^9}, {3.841057130911109*^9, 3.84105713513932*^9}},ExpressionUUID->"7da6da63-6378-4f26-b58d-\ d42a7250afa4"], Cell["\<\ O Mathematica \[EAcute] capaz de fornecer solu\[CCedilla]\[OTilde]es anal\ \[IAcute]ticas (quando existem) ou num\[EAcute]ricas para equa\[CCedilla]\ \[OTilde]es alg\[EAcute]bricas. \ \>", "Text", CellChangeTimes->{{3.6992696952616243`*^9, 3.6992696957779827`*^9}, { 3.704547416597644*^9, 3.704547418388081*^9}, {3.8393233348613234`*^9, 3.839323366214238*^9}, {3.839323576559081*^9, 3.839323576563138*^9}, 3.840031301280365*^9, {3.840714647849332*^9, 3.840714700814382*^9}, { 3.902925408747019*^9, 3.9029254132790165`*^9}},ExpressionUUID->"3b730a40-c305-44bd-95b8-\ df5440115120"], Cell[TextData[{ "Antes de discutirmos as fun\[CCedilla]\[OTilde]es pertinentes, vale chamar \ a aten\[CCedilla]\[ATilde]o para como definimos uma ", StyleBox["equa\[CCedilla]\[ATilde]o", FontWeight->"Bold"], " no Mathematica. Em lugar da igualdade simples, que \[EAcute] reservada \ para ", StyleBox["atribui\[CCedilla]\[OTilde]es", FontWeight->"Bold"], ", utilizamos \[OpenCurlyDoubleQuote]", StyleBox["==", FontFamily->"Source Code Pro", FontWeight->"Regular"], "\[CloseCurlyDoubleQuote], como no exemplo abaixo:" }], "Text", CellChangeTimes->{{3.840715029587531*^9, 3.840715134031026*^9}},ExpressionUUID->"06361fc3-d0e5-4646-a4fa-\ 73129b0aa494"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"2", "*", "x"}], "+", "5"}], "\[Equal]", "3", " "}]], "Input", CellChangeTimes->{{3.840715137776206*^9, 3.840715146430374*^9}, 3.8725883264587917`*^9}, CellLabel->"",ExpressionUUID->"e05a18b9-8507-491c-b8bf-4e78d495e1a1"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"e87c0300-a443-4a7a-a867-2dd6904b1c93"], Cell[TextData[{ "A fun\[CCedilla]\[ATilde]o ", StyleBox[ButtonBox["Solve", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/Solve.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/Solve.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], " tenta obter solu\[CCedilla]\[OTilde]es anal\[IAcute]ticas de equa\ \[CCedilla]\[OTilde]es alg\[EAcute]bricas: " }], "Text", CellChangeTimes->{{3.840714805233673*^9, 3.840714848494001*^9}, { 3.840714990245832*^9, 3.8407150050952263`*^9}, {3.8407162178028297`*^9, 3.840716217806488*^9}},ExpressionUUID->"bc8716a3-ec22-4f95-8297-\ dc736158740b"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"x", "^", "2"}], "-", RowBox[{"3", "*", "x"}], "-", "10"}], "\[Equal]", "0"}], ",", "x"}], "]"}], " ", RowBox[{"(*", " ", RowBox[{ "Busca", " ", "a", " ", "solu\[CCedilla]\[ATilde]o", " ", "de", " ", "uma", " ", "equa\[CCedilla]\[ATilde]o", " ", "em", " ", RowBox[{"x", "."}]}], " ", "*)"}], " "}]], "Input", CellLabel->"",ExpressionUUID->"f6d29f23-317e-43ac-926b-ae4bed90b7e3"], Cell[TextData[{ "Note que ", StyleBox["Solve", FontFamily->"Source Code Pro", FontWeight->"Regular"], " fornece uma \[OpenCurlyDoubleQuote]lista aninhada\[CloseCurlyDoubleQuote] \ (lista de listas) de atribui\[CCedilla]\[OTilde]es tempor\[AAcute]rias (tamb\ \[EAcute]m chamadas de \[OpenCurlyDoubleQuote]regras de transforma\[CCedilla]\ \[ATilde]o\[CloseCurlyDoubleQuote]). Isso permite que realizemos \ c\[AAcute]lculos com as solu\[CCedilla]\[OTilde]es, utilizando o operador de \ substitui\[CCedilla]\[ATilde]o \[OpenCurlyDoubleQuote]", StyleBox["/.", FontFamily->"Source Code Pro", FontWeight->"Regular"], "\[CloseCurlyDoubleQuote]. Por exemplo, vamos verificar que de fato se trata \ de solu\[CCedilla]\[OTilde]es corretas:" }], "Text", CellGroupingRules->{"GroupTogetherGrouping", 10000.}, CellChangeTimes->{{3.840715987182619*^9, 3.8407160929320602`*^9}, 3.840716163797855*^9, 3.840716231972767*^9},ExpressionUUID->"e0ea5a5e-5f83-425f-bebc-\ b548d7385f5b"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{"x", "^", "2"}], "-", RowBox[{"3", "*", "x"}], "-", "10"}], "\[Equal]", "0"}], "/.", "%", " "}]], "Input", CellGroupingRules->{"GroupTogetherGrouping", 10000.}, CellChangeTimes->{{3.839947301413373*^9, 3.839947311139803*^9}, 3.84061327554305*^9, {3.840716111001123*^9, 3.84071616379874*^9}, 3.9029254229350185`*^9},ExpressionUUID->"aedb5a5b-0854-422d-9281-\ f487a24fc1c9"] }, Closed]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"32be91fb-777d-400b-86fa-9a049cede895"], Cell["\<\ Tanto as equa\[CCedilla]\[OTilde]es quanto as solu\[CCedilla]\[OTilde]es \ podem ser \[OpenCurlyDoubleQuote]valores\[CloseCurlyDoubleQuote] atribu\ \[IAcute]dos a vari\[AAcute]veis, o que evita ter que explicitamente repetir \ express\[OTilde]es ou fazer refer\[EHat]ncia a sa\[IAcute]das anteriores:\ \>", "Text", CellChangeTimes->{{3.840716256216364*^9, 3.8407162792437897`*^9}, { 3.84071639267841*^9, 3.8407164385558443`*^9}},ExpressionUUID->"0d223cbc-4c03-401e-905d-\ 973bce4d584c"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"eq1", "=", RowBox[{ RowBox[{ RowBox[{"x", "^", "2"}], "-", RowBox[{"3", "*", "x"}], "-", "10"}], "\[Equal]", "0"}]}], ";"}], " "}]], "Input", CellChangeTimes->{{3.840716285257362*^9, 3.84071631971697*^9}, 3.841216684087008*^9, 3.8725885241456585`*^9}, CellLabel->"",ExpressionUUID->"2869d96c-abc0-4783-9fa1-d4b178ad7213"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"sols1", "=", RowBox[{"Solve", "[", RowBox[{"eq1", ",", "x"}], "]"}], " "}]], "Input", CellChangeTimes->{{3.840716300085318*^9, 3.840716315897005*^9}, 3.841216685198595*^9, 3.872588537298909*^9}, CellLabel->"",ExpressionUUID->"53af0948-4cce-4355-bee2-c0eb400e404e"], Cell[BoxData[ RowBox[{"eq1", "/.", "sols1", " "}]], "Input", CellChangeTimes->{{3.8407163294275303`*^9, 3.840716338712697*^9}, 3.8412166860648212`*^9, 3.8725885399428873`*^9}, CellLabel->"",ExpressionUUID->"cd4d204c-cb86-4948-9431-32208be571ed"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"dbbc9628-10eb-4897-b200-5b6460fd6b1a"], Cell[TextData[{ "Embora funcione principalmente com equa\[CCedilla]\[OTilde]es polinomiais \ (at\[EAcute] quarto grau), ", StyleBox["Solve", FontFamily->"Source Code Pro", FontWeight->"Regular"], " tamb\[EAcute]m pode encontrar solu\[CCedilla]\[OTilde]es de alguns outros \ tipos de equa\[CCedilla]\[ATilde]o. Por exemplo, se quisermos expressar a \ velocidade de uma part\[IAcute]cula em termos de seu momento \ relativ\[IAcute]stico: " }], "Text", CellChangeTimes->{{3.840717295113817*^9, 3.84071729734042*^9}, { 3.840717343837861*^9, 3.840717386227171*^9}, {3.840717481133095*^9, 3.8407174814466467`*^9}, {3.8407177165619297`*^9, 3.840717758197268*^9}, { 3.840720388376389*^9, 3.840720390647537*^9}, {3.872588637425931*^9, 3.8725886374299316`*^9}, {3.872609909146102*^9, 3.8726099091531096`*^9}},ExpressionUUID->"651cfb64-d385-4f6a-b0f5-\ 32fba8fd40c5"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Solve", "[", RowBox[{ RowBox[{"p", "\[Equal]", RowBox[{"m", "*", RowBox[{"v", "/", RowBox[{"Sqrt", "[", RowBox[{"1", "-", RowBox[{ RowBox[{"v", "^", "2"}], "/", RowBox[{"c", "^", "2"}]}]}], "]"}]}]}]}], ",", "v"}], "]"}], " "}]], "Input", CellChangeTimes->{{3.84071749945927*^9, 3.8407175187226753`*^9}, 3.840717871926985*^9, 3.8412166884061537`*^9, 3.8725887317502003`*^9, 3.9029254352240205`*^9},ExpressionUUID->"53fafa51-3e7c-4654-8f30-\ dd9bb83c94ea"], Cell[CellGroupData[{ Cell[TextData[{ "Como apenas a solu\[CCedilla]\[ATilde]o com sinal positivo interessa (por \ qu\[EHat]?), podemos selecion\[AAcute]-la com o aux\[IAcute]lio da fun\ \[CCedilla]\[ATilde]o ", StyleBox[ButtonBox["Last", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/Last.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/Last.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], ", que recupera o \[UAcute]ltimo elemento de uma lista:" }], "Text", CellGroupingRules->{"GroupTogetherGrouping", 10000.}, CellChangeTimes->{{3.840717772181724*^9, 3.840717824122362*^9}, 3.840717871929182*^9, {3.840718783758773*^9, 3.840718783764414*^9}},ExpressionUUID->"f561e8f7-847f-4811-8640-\ 790492594788"], Cell[BoxData[ RowBox[{ RowBox[{"Last", "[", "%", "]"}], " "}]], "Input", CellGroupingRules->{"GroupTogetherGrouping", 10000.}, CellChangeTimes->{{3.840717827189011*^9, 3.8407178293553667`*^9}, 3.840717871930538*^9, 3.841216895811414*^9, 3.872594315002001*^9}, CellLabel->"",ExpressionUUID->"f1991d11-bf06-4284-8514-901f6afbb72c"] }, Open ]] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"f4abf8ae-e5a5-4155-bfef-c2f83419c5de"], Cell["\<\ Mais um exemplo, agora com uma equa\[CCedilla]\[ATilde]o envolvendo fun\ \[CCedilla]\[OTilde]es trigonom\[EAcute]tricas:\ \>", "Text", CellChangeTimes->{{3.840717900027441*^9, 3.8407179161894913`*^9}},ExpressionUUID->"c0e271b6-f898-4758-97f5-\ de10218251d3"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"eq2", "=", RowBox[{ RowBox[{ RowBox[{"9", "*", RowBox[{ RowBox[{"Cos", "[", "x", "]"}], "^", "2"}]}], "+", RowBox[{"8", "*", RowBox[{"Cos", "[", "x", "]"}]}]}], "\[Equal]", "1"}]}], ";"}], " "}]], "Input", CellChangeTimes->{{3.8407179863975554`*^9, 3.8407180068247623`*^9}, 3.8412166901043797`*^9, 3.8725887772737255`*^9, 3.8725943909268446`*^9}, CellLabel->"",ExpressionUUID->"e9519620-0bf7-4372-bddd-1de49e270bad"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"sols2", "=", RowBox[{"Solve", "[", RowBox[{"eq2", ",", "x"}], "]"}], " "}]], "Input", CellChangeTimes->{{3.8407179260173187`*^9, 3.840717942956999*^9}, { 3.840717992268073*^9, 3.8407180012268267`*^9}, 3.840719503264101*^9, 3.841216691267064*^9, 3.872588779277785*^9}, CellLabel->"",ExpressionUUID->"f894fce0-7b93-42e8-a649-7a3833cd58cc"], Cell[CellGroupData[{ Cell["\<\ Note a presen\[CCedilla]a de express\[OTilde]es condicionais nas solu\ \[CCedilla]\[OTilde]es. Discutiremos condicionais em aulas futuras. Em todo \ caso, podemos verificar que as solu\[CCedilla]\[OTilde]es s\[ATilde]o \ corretas dentro das condi\[CCedilla]\[OTilde]es indicadas.\ \>", "ItemParagraph", CellGroupingRules->{"GroupTogetherGrouping", 10000.}, CellChangeTimes->{{3.840718490456517*^9, 3.840718561600259*^9}, 3.8407195032654257`*^9},ExpressionUUID->"4e1b715d-fa20-48c0-92ed-\ 32dd06ab64aa"], Cell[BoxData[ RowBox[{ RowBox[{"Simplify", "[", RowBox[{"eq2", "/.", "sols2"}], "]"}], " "}]], "Input", CellGroupingRules->{"GroupTogetherGrouping", 10000.}, CellChangeTimes->{{3.840718013979251*^9, 3.840718019080924*^9}, { 3.8407180623784533`*^9, 3.8407180669396143`*^9}, 3.840719503266327*^9, 3.841216736539151*^9, 3.872594398277274*^9}, CellLabel->"",ExpressionUUID->"36a7559f-6ecf-4169-9bd8-a38bc4aeba9b"] }, Open ]] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"efec382b-9290-417c-adf8-5eb1685c8d98"], Cell[TextData[{ "A fun\[CCedilla]\[ATilde]o ", StyleBox[ButtonBox["Solve", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/Solve.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/Solve.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], " tamb\[EAcute]m sabe lidar com sistemas de equa\[CCedilla]\[OTilde]es:" }], "Text", CellChangeTimes->{{3.840718729492465*^9, 3.8407187526139708`*^9}, { 3.8725888090178347`*^9, 3.872588809020834*^9}},ExpressionUUID->"03b85a2d-9264-41c5-8e5c-\ 02b63143665d"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Solve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"x", "+", "y"}], "\[Equal]", "z"}], ",", RowBox[{ RowBox[{"10", "-", RowBox[{"6", "*", "x"}], "-", RowBox[{"2", "*", "z"}]}], "\[Equal]", "0"}], ",", RowBox[{ RowBox[{ RowBox[{"6", "*", "x"}], "-", "24", "-", RowBox[{"4", "*", "y"}]}], "\[Equal]", "0"}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", ",", "y", ",", "z"}], "}"}]}], "]"}], " "}]], "Input", 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No \ exemplo, a varredura em ", Cell[BoxData[ FormBox["x", TraditionalForm]],ExpressionUUID-> "c4737cd9-f325-4417-948e-3c9b2acf6bbe"], " \[EAcute] feita primeiro, e a varredura em ", Cell[BoxData[ FormBox["y", TraditionalForm]],ExpressionUUID-> "8d8efa67-5d5e-455e-b630-ff72b0cf0fc4"], " \[EAcute] feita por \[UAcute]ltimo. Como o limite superior da integral em ", Cell[BoxData[ FormBox["y", TraditionalForm]],ExpressionUUID-> "6edbcccd-318a-42ea-8306-9a12778acf70"], " depende de ", Cell[BoxData[ FormBox["x", TraditionalForm]],ExpressionUUID-> "e354b7c6-14a1-42c2-a4b4-b78046503de3"], ", inverter os argumentos causaria problemas. 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V\[AAcute]rios deles envolvem \ aproximar a integra\[CCedilla]\[ATilde]o por uma soma sobre intervalos de \ largura adapt\[AAcute]vel, verificando a converg\[EHat]ncia \[AGrave] medida \ que a discretiza\[CCedilla]\[ATilde]o \[EAcute] refinada. Mais sobre esses m\ \[EAcute]todos na disciplina de C\[AAcute]lculo Num\[EAcute]rico.\ \>", "ItemParagraph", CellChangeTimes->{{3.84113648219901*^9, 3.841136595429772*^9}, { 3.84113671281429*^9, 3.841136715187635*^9}, 3.841167096909854*^9},ExpressionUUID->"0775d9f7-fabf-48b5-a29c-\ 5d515f42502c"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"614e9c64-7bcd-43a7-8246-543404508669"], Cell[TextData[{ "Integrais m\[UAcute]ltiplas tamb\[EAcute]m podem ser calculadas, como no \ exemplo abaixo, relevante para o c\[AAcute]lculo do potencial \ el\[EAcute]trico, em um ponto qualquer, produzido por um disco uniformemente \ carregado de raio", Cell[BoxData[ FormBox[ RowBox[{" ", "a"}], TraditionalForm]],ExpressionUUID-> "6db511dd-02b4-47ac-9643-b0289eef6026"], ", centrado na origem e perpendicular ao eixo ", Cell[BoxData[ FormBox["z", TraditionalForm]],ExpressionUUID-> "04b0f5cc-1fc8-46d9-9306-fcd3246a54d8"], ". Se ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", RowBox[{"x", ",", "y", ",", "z"}], ")"}], " "}], TraditionalForm]], ExpressionUUID->"3745c1c1-1292-4a19-9c6e-0296d708b564"], " s\[ATilde]o as coordenadas do ponto, a integral relevante, em coordenadas \ cil\[IAcute]ndricas, \[EAcute] " }], "Text", CellChangeTimes->{{3.841155034832841*^9, 3.841155235009308*^9}, { 3.841155293427744*^9, 3.8411552954650173`*^9}, {3.8411554383965197`*^9, 3.841155438474719*^9}, {3.8411571152309723`*^9, 3.841157120056892*^9}},ExpressionUUID->"af78f9c8-cc88-4480-8c29-\ 88f54460dbf6"], Cell[TextData[Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["\[Integral]", "0", "a"], RowBox[{ StyleBox["dr", FontSlant->"Italic"], RowBox[{ SubsuperscriptBox["\[Integral]", "0", RowBox[{"2", "\[Pi]"}]], RowBox[{ StyleBox["d\[Phi]", FontSlant->"Italic"], RowBox[{ FractionBox["r", SqrtBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"x", "-", RowBox[{"r", " ", "cos", " ", "\[Phi]"}]}], ")"}], "2"], "+", SuperscriptBox[ RowBox[{"(", RowBox[{"y", "-", RowBox[{"r", " ", "sin", " ", "\[Phi]"}]}], ")"}], "2"], "+", SuperscriptBox["z", "2"]}]]], "."}]}]}]}]}], TraditionalForm]],ExpressionUUID->"4ff0a94c-86da-4996-bac9-c6d090e795a0"]], \ "Text", CellChangeTimes->{{3.841155372632465*^9, 3.8411557795560007`*^9}, { 3.841155815680887*^9, 3.841155815680979*^9}, {3.841156158855932*^9, 3.8411561590109653`*^9}},ExpressionUUID->"7d71f862-8fa6-4df5-866c-\ 5388d7b2ebbe"], Cell[TextData[{ "(O potencial el\[EAcute]trico corresponde \[AGrave] integral multiplicada \ por ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[Sigma]", "/", "4"}], SubscriptBox["\[Pi]\[Epsilon]", "0"]}], TraditionalForm]],ExpressionUUID-> "37f56885-98d7-473a-9358-63db2409d654"], ".) Se quisermos manter o valor de ", Cell[BoxData[ FormBox["a", TraditionalForm]],ExpressionUUID-> "d00cdec8-4784-4047-a2a0-663984c7f09c"], " como vari\[AAcute]vel, podemos definir uma fun\[CCedilla]\[ATilde]o ", Cell[BoxData[ FormBox[ RowBox[{"V", "(", RowBox[{"a", ",", "x", ",", "y", ",", "z"}], ")"}], TraditionalForm]], ExpressionUUID->"ba7ca620-883d-4e66-8552-9b37e3d92c87"], " que forne\[CCedilla]a o valor da integral:" }], "Text", CellChangeTimes->{{3.8411558191024227`*^9, 3.8411558404288*^9}, { 3.841156163159802*^9, 3.841156226452874*^9}, {3.84115698195581*^9, 3.841157038628912*^9}, {3.841157138440261*^9, 3.841157140984133*^9}},ExpressionUUID->"71c51c06-1138-44fa-acb7-\ 25cb0fadb4f4"], Cell[BoxData[ RowBox[{ RowBox[{"V", "[", RowBox[{"a_", ",", "x_", ",", "y_", ",", "z_"}], "]"}], ":=", RowBox[{"NIntegrate", " ", "[", RowBox[{ RowBox[{"r", "/", RowBox[{"Sqrt", "[", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"x", "-", RowBox[{"r", "*", RowBox[{"Cos", "[", "\[Phi]", "]"}]}]}], ")"}], "^", "2"}], "+", RowBox[{ RowBox[{"(", RowBox[{"y", "-", RowBox[{"r", "*", RowBox[{"Sin", "[", "\[Phi]", "]"}]}]}], ")"}], "^", "2"}], "+", RowBox[{"z", "^", "2"}]}], "]"}]}], ",", RowBox[{"{", RowBox[{"r", ",", "0", ",", "a"}], "}"}], ",", RowBox[{"{", RowBox[{"\[Phi]", ",", "0", ",", RowBox[{"2", "*", "\[Pi]"}]}], "}"}]}], "]"}], " "}]], "Input", CellChangeTimes->{{3.8411470012166033`*^9, 3.841147005704012*^9}, { 3.841156092960197*^9, 3.84115612228367*^9}, 3.841216858697858*^9, 3.872611512653496*^9},ExpressionUUID->"9e69cdd9-c492-47cd-98bf-\ 3bbc182e214b"] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"da14d007-df89-4bea-bf86-9b7920014b9a"], Cell["\<\ Em seguida, podemos utilizar a fun\[CCedilla]\[ATilde]o para calcular o \ potencial nos pontos que desejarmos. Por exemplo:\ \>", "Text", CellChangeTimes->{{3.841156245848516*^9, 3.841156298171711*^9}, { 3.841157152220684*^9, 3.841157155832122*^9}},ExpressionUUID->"9937b1d5-2413-425d-b31c-\ cffd16f2ea64"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"V", "[", RowBox[{"1", ",", "1", ",", "2", ",", "5"}], "]"}], " "}]], "Input", CellChangeTimes->{{3.841156129863968*^9, 3.841156133426775*^9}, 3.841216861520163*^9, 3.8725941961703854`*^9, 3.872611507008329*^9},ExpressionUUID->"b8c17879-63a5-42a1-b89b-\ c23bfa12e807"], Cell[TextData[{ "Podemos inclusive fazer gr\[AAcute]ficos do potencial como fun\[CCedilla]\ \[ATilde]o de uma das coordenadas. Vamos calcular o potencial ao longo do \ eixo perpendicular ao disco (nesse caso h\[AAcute] uma solu\[CCedilla]\ \[ATilde]o anal\[IAcute]tica fechada, que voc\[EHat] pode obter pela fun\ \[CCedilla]\[ATilde]o ", StyleBox[ButtonBox["Integrate", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/Integrate.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/Integrate.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], "):" }], "Text", CellChangeTimes->{{3.841156320211883*^9, 3.8411563695759907`*^9}, { 3.8411575224831142`*^9, 3.84115754976449*^9}, {3.8726114943349714`*^9, 3.8726114943449736`*^9}},ExpressionUUID->"4b0fc16a-4f75-4af9-9490-\ 08b39dd48b48"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Plot", "[", RowBox[{ RowBox[{"V", "[", RowBox[{"1", ",", "0", ",", "0", ",", "z"}], "]"}], ",", RowBox[{"{", RowBox[{"z", ",", "0", ",", "10"}], "}"}]}], "]"}], " "}]], "Input", CellChangeTimes->{{3.841156371949*^9, 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