(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 92605, 2244] NotebookOptionsPosition[ 88462, 2117] NotebookOutlinePosition[ 88901, 2136] CellTagsIndexPosition[ 88858, 2133] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["The electrostatic potential between two plates", "Title", CellChangeTimes->{{3.3996673838626633`*^9, 3.3996673940552425`*^9}}], Cell[TextData[{ "Given two infinite parallel plates, held a fixed distance ", Cell[BoxData[ FormBox["d", TraditionalForm]]], " apart at a constant potential difference, calculate the dependance of the \ potential difference between one plate and a point between them at, say, ", Cell[BoxData[ FormBox["z", TraditionalForm]]], "." }], 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RowBox[{"\[DifferentialD]", "q"}], TraditionalForm]]], " is the infinitessimal element of charge and ", Cell[BoxData[ FormBox[ RowBox[{"d", "(", "x", ")"}], TraditionalForm]]], " is it's distance from the point ", Cell[BoxData[ FormBox["x", TraditionalForm]]], " where we wish to measure the potential." }], "Text", CellChangeTimes->{{3.39966741512825*^9, 3.3996674251176033`*^9}, { 3.399667467328555*^9, 3.3996676457319546`*^9}, {3.399667680902662*^9, 3.39966776157478*^9}, {3.3996678033569746`*^9, 3.3996678040134892`*^9}, { 3.399668101354689*^9, 3.3996682918741465`*^9}, {3.3996685896374507`*^9, 3.399668589715615*^9}}, FontSize->14], Cell[TextData[{ "Our first order of business is going to be writing ", Cell[BoxData[ FormBox[ RowBox[{"\[DifferentialD]", "q"}], TraditionalForm]]], " in a useful form. Since we're working with plates, we're going to need to \ integrate over a flat 2D area, so our charge density is going to be a surface \ charge, call it \[Sigma], and we'll need to pick an infinitessimal area. A \ simple one to use is a ring of radius ", Cell[BoxData[ FormBox["r", TraditionalForm]]], " and thickness ", Cell[BoxData[ FormBox[ RowBox[{"\[DifferentialD]", "r"}], TraditionalForm]]], ", giving" }], "Text", CellChangeTimes->{{3.39966741512825*^9, 3.3996674251176033`*^9}, { 3.399667467328555*^9, 3.3996676457319546`*^9}, {3.399667680902662*^9, 3.39966776157478*^9}, {3.3996678033569746`*^9, 3.3996678040134892`*^9}, { 3.399668101354689*^9, 3.3996683970922503`*^9}, {3.3996684525856237`*^9, 3.3996685309018993`*^9}, {3.399668649055235*^9, 3.399668780678032*^9}}, FontSize->14], Cell[BoxData[ RowBox[{ RowBox[{"\[DifferentialD]", "q"}], "=", RowBox[{"2", "\[Pi]", " ", "\[Sigma]", " ", "r", RowBox[{"\[DifferentialD]", "r"}]}]}]], "Input", CellChangeTimes->{{3.3996678061549644`*^9, 3.3996679122446938`*^9}, { 3.3996680465034356`*^9, 3.399668048473012*^9}, {3.3996684110984497`*^9, 3.3996684147719545`*^9}, {3.3996685347161083`*^9, 3.3996685381082573`*^9}, { 3.3996685958904433`*^9, 3.399668636033683*^9}, {3.3996688272459254`*^9, 3.399668838579091*^9}}, FontSize->14], Cell[TextData[{ "The next task is to turn ", Cell[BoxData[ FormBox[ RowBox[{"d", "(", "x", ")"}], TraditionalForm]]], " into something useful. Since this is supposed to be the distance between \ the charge element and the point we want to measure the potential of. It's \ convenient to take the origin of our coordinate system to be directly under \ the point whose potential we want to measure because then " }], "Text", CellChangeTimes->{{3.39966741512825*^9, 3.3996674251176033`*^9}, { 3.399667467328555*^9, 3.3996676457319546`*^9}, {3.399667680902662*^9, 3.39966776157478*^9}, {3.3996678033569746`*^9, 3.3996678040134892`*^9}, { 3.399668101354689*^9, 3.3996683970922503`*^9}, {3.3996684525856237`*^9, 3.3996685309018993`*^9}, {3.3996687855709105`*^9, 3.3996688155376005`*^9}, { 3.399668962368433*^9, 3.399669065867295*^9}, {3.39966924146135*^9, 3.3996692489178205`*^9}}, FontSize->14], Cell[BoxData[ RowBox[{ RowBox[{"d", "[", "z_", "]"}], ":=", SqrtBox[ RowBox[{ SuperscriptBox["z", "2"], "+", SuperscriptBox["r", "2"]}]]}]], "Input", CellChangeTimes->{{3.399669265612816*^9, 3.399669304552204*^9}}, FontSize->14], Cell["and we have", "Text", CellChangeTimes->{{3.39966741512825*^9, 3.3996674251176033`*^9}, { 3.399667467328555*^9, 3.3996676457319546`*^9}, 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3.3996693731925325`*^9}}, FontSize->14], Cell[BoxData[ RowBox[{ RowBox[{"V", "[", "z_", "]"}], ":=", RowBox[{ FractionBox["\[Sigma]", RowBox[{"2", SubscriptBox["\[Epsilon]", "0"]}]], RowBox[{"\[Integral]", FractionBox[ RowBox[{"r", RowBox[{"\[DifferentialD]", "r"}]}], SqrtBox[ RowBox[{ SuperscriptBox["z", "2"], "+", SuperscriptBox["r", "2"]}]]]}]}]}]], "Input", CellChangeTimes->{{3.3996678061549644`*^9, 3.3996679122446938`*^9}, { 3.3996680465034356`*^9, 3.399668048473012*^9}, {3.399668583149845*^9, 3.399668583228009*^9}, {3.399669260876313*^9, 3.3996692634556*^9}, { 3.3996693667208605`*^9, 3.399669405285164*^9}}, FontSize->14], Cell[TextData[{ "for an infinite plate ", Cell[BoxData[ FormBox["r", TraditionalForm]]], " goes from zero to \[Infinity]" }], "Text", CellChangeTimes->{{3.39966741512825*^9, 3.3996674251176033`*^9}, { 3.399667467328555*^9, 3.3996676457319546`*^9}, {3.399667680902662*^9, 3.39966776157478*^9}, {3.3996678033569746`*^9, 3.3996678040134892`*^9}, { 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3.399669468704523*^9}}, FontSize->14], Cell[TextData[{ "Now we ask ", StyleBox["Mathematica", FontSlant->"Italic"], " to integrate" }], "Text", CellChangeTimes->{{3.39966741512825*^9, 3.3996674251176033`*^9}, { 3.399667467328555*^9, 3.3996676457319546`*^9}, {3.399667680902662*^9, 3.39966776157478*^9}, {3.3996678033569746`*^9, 3.3996678040134892`*^9}, { 3.399668101354689*^9, 3.3996683970922503`*^9}, {3.3996684525856237`*^9, 3.3996685309018993`*^9}, {3.3996687855709105`*^9, 3.3996688155376005`*^9}, { 3.399668962368433*^9, 3.399669065867295*^9}, {3.39966924146135*^9, 3.3996692489178205`*^9}, {3.39966934961939*^9, 3.3996693532772894`*^9}, { 3.399669419776095*^9, 3.3996694352518544`*^9}, {3.3996698488002644`*^9, 3.3996698513170485`*^9}, {3.3996807346877017`*^9, 3.3996807529155464`*^9}}, FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ SubsuperscriptBox["\[Integral]", "o", "\[Infinity]"], RowBox[{ FractionBox["r", SqrtBox[ RowBox[{ SuperscriptBox["z", "2"], "+", SuperscriptBox["r", 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Let's try it without explicit integration limits and see what's \ happening\ \>", "Text", CellChangeTimes->{{3.39966741512825*^9, 3.3996674251176033`*^9}, { 3.399667467328555*^9, 3.3996676457319546`*^9}, {3.399667680902662*^9, 3.39966776157478*^9}, {3.3996678033569746`*^9, 3.3996678040134892`*^9}, { 3.399668101354689*^9, 3.3996683970922503`*^9}, {3.3996684525856237`*^9, 3.3996685309018993`*^9}, {3.3996687855709105`*^9, 3.3996688155376005`*^9}, { 3.399668962368433*^9, 3.399669065867295*^9}, {3.39966924146135*^9, 3.3996692489178205`*^9}, {3.39966934961939*^9, 3.3996693532772894`*^9}, { 3.399669419776095*^9, 3.3996694352518544`*^9}, {3.3996698488002644`*^9, 3.399669893555285*^9}}, FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\[Integral]", RowBox[{ FractionBox["r", SqrtBox[ RowBox[{ SuperscriptBox["r", "2"], "+", SuperscriptBox["z", "2"]}]]], RowBox[{"\[DifferentialD]", "r"}]}]}]], "Input", CellChangeTimes->{{3.399669507722238*^9, 3.399669509473033*^9}}, FontSize->14], Cell[BoxData[ SqrtBox[ RowBox[{ SuperscriptBox["r", "2"], "+", SuperscriptBox["z", "2"]}]]], "Output", CellChangeTimes->{3.3996695162417393`*^9, 3.3996807836808968`*^9, 3.399718504480845*^9, 3.3997199949910727`*^9, 3.3997200873240633`*^9, 3.399804678740575*^9}, FontSize->14] }, Open ]], Cell[TextData[{ "Ah. What happens to this as ", Cell[BoxData[ FormBox[ RowBox[{"r", "\[Rule]", "\[Infinity]"}], TraditionalForm]]], "? Kaboom.\n\nFirst, do we believe ", StyleBox["Mathematica", FontSlant->"Italic"], "?" }], "Text", CellChangeTimes->{{3.39966741512825*^9, 3.3996674251176033`*^9}, { 3.399667467328555*^9, 3.3996676457319546`*^9}, {3.399667680902662*^9, 3.39966776157478*^9}, {3.3996678033569746`*^9, 3.3996678040134892`*^9}, { 3.399668101354689*^9, 3.3996683970922503`*^9}, {3.3996684525856237`*^9, 3.3996685309018993`*^9}, {3.3996687855709105`*^9, 3.3996688155376005`*^9}, { 3.399668962368433*^9, 3.399669065867295*^9}, {3.39966924146135*^9, 3.3996692489178205`*^9}, {3.39966934961939*^9, 3.3996693532772894`*^9}, { 3.399669419776095*^9, 3.3996694352518544`*^9}, {3.3996698488002644`*^9, 3.3996700395414486`*^9}, {3.3996709137917857`*^9, 3.3996709150579453`*^9}, { 3.399670951667263*^9, 3.3996709613901615`*^9}, {3.399718807218936*^9, 3.399718848232458*^9}}, FontSize->14], Cell[CellGroupData[{ Cell["Doing the integral instead of asking for it", "Subsection", CellChangeTimes->{{3.39966741512825*^9, 3.3996674251176033`*^9}, { 3.399667467328555*^9, 3.3996676457319546`*^9}, {3.399667680902662*^9, 3.3996677917117167`*^9}, {3.3996680269796324`*^9, 3.399668030168466*^9}, { 3.399677861113304*^9, 3.399677863927208*^9}, {3.399715155905615*^9, 3.399715163503224*^9}, {3.3997152545378113`*^9, 3.399715258488538*^9}, { 3.399718981392068*^9, 3.3997189904158707`*^9}}], Cell["We start with", "Text", CellChangeTimes->{{3.39966741512825*^9, 3.3996674251176033`*^9}, { 3.399667467328555*^9, 3.3996676457319546`*^9}, {3.399667680902662*^9, 3.39966776157478*^9}, {3.3996678033569746`*^9, 3.3996678040134892`*^9}, { 3.399668101354689*^9, 3.3996683970922503`*^9}, {3.3996684525856237`*^9, 3.3996685309018993`*^9}, {3.3996687855709105`*^9, 3.3996688155376005`*^9}, { 3.399668962368433*^9, 3.399669065867295*^9}, {3.39966924146135*^9, 3.3996692489178205`*^9}, {3.39966934961939*^9, 3.3996693532772894`*^9}, { 3.399669419776095*^9, 3.3996694352518544`*^9}, {3.3996698488002644`*^9, 3.3996700395414486`*^9}, {3.3996709137917857`*^9, 3.3996709150579453`*^9}, { 3.399670951667263*^9, 3.3996709613901615`*^9}, {3.399718807218936*^9, 3.399718848232458*^9}, {3.3997190055283403`*^9, 3.399719006983913*^9}, { 3.399719197304512*^9, 3.3997192070562696`*^9}}, FontSize->14], Cell[BoxData[ RowBox[{"\[Integral]", RowBox[{ FractionBox["r", SqrtBox[ RowBox[{ SuperscriptBox["r", "2"], "+", SuperscriptBox["z", "2"]}]]], RowBox[{"\[DifferentialD]", "r"}]}]}]], "Input", CellChangeTimes->{{3.399669507722238*^9, 3.399669509473033*^9}}, FontSize->14], Cell[TextData[{ "The first thing I note is that I've got a ", Cell[BoxData[ FormBox[ SqrtBox[ SuperscriptBox["r", "2"]], TraditionalForm]]], "term which is ... unpleasant. A good way to get rid of complicated things \ in integrals is to try substituion.\n\nLet ", Cell[BoxData[ FormBox[ RowBox[{"u", "=", SuperscriptBox["r", "2"]}], TraditionalForm]]], " then ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[DifferentialD]", "u"}], "=", RowBox[{"2", "r", RowBox[{"\[DifferentialD]", "r"}]}]}], TraditionalForm]]], " and we've got" }], "Text", CellChangeTimes->{{3.39966741512825*^9, 3.3996674251176033`*^9}, { 3.399667467328555*^9, 3.3996676457319546`*^9}, {3.399667680902662*^9, 3.39966776157478*^9}, {3.3996678033569746`*^9, 3.3996678040134892`*^9}, { 3.399668101354689*^9, 3.3996683970922503`*^9}, {3.3996684525856237`*^9, 3.3996685309018993`*^9}, {3.3996687855709105`*^9, 3.3996688155376005`*^9}, { 3.399668962368433*^9, 3.399669065867295*^9}, {3.39966924146135*^9, 3.3996692489178205`*^9}, {3.39966934961939*^9, 3.3996693532772894`*^9}, { 3.399669419776095*^9, 3.3996694352518544`*^9}, {3.3996698488002644`*^9, 3.3996700395414486`*^9}, {3.3996709137917857`*^9, 3.3996709150579453`*^9}, { 3.399670951667263*^9, 3.3996709613901615`*^9}, {3.399718807218936*^9, 3.399718848232458*^9}, {3.3997190055283403`*^9, 3.39971919146381*^9}, { 3.399719243696439*^9, 3.399719254976253*^9}, {3.399719525561194*^9, 3.399719532128578*^9}}, FontSize->14], Cell[BoxData[ RowBox[{ RowBox[{ FractionBox["1", "2"], RowBox[{"\[Integral]", RowBox[{ FractionBox[ RowBox[{"2", "r"}], SqrtBox[ RowBox[{ SuperscriptBox["r", "2"], "+", SuperscriptBox["z", "2"]}]]], RowBox[{"\[DifferentialD]", "r"}]}]}]}], "=", RowBox[{ RowBox[{ FractionBox["1", "2"], RowBox[{"\[Integral]", RowBox[{ FractionBox["1", SqrtBox[ RowBox[{"u", "+", SuperscriptBox["z", "2"]}]]], RowBox[{"\[DifferentialD]", "u"}]}]}]}], "=", RowBox[{ FractionBox["1", "2"], RowBox[{"\[Integral]", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"u", "+", SuperscriptBox["z", "2"]}], ")"}], RowBox[{ RowBox[{"-", "1"}], "/", "2"}]], RowBox[{"\[DifferentialD]", "u"}]}]}]}]}]}]], "Input", CellChangeTimes->{{3.399669507722238*^9, 3.399669509473033*^9}, { 3.3997192163145723`*^9, 3.399719303931158*^9}}, FontSize->14], Cell[TextData[{ "which is a little better but not much. Since substitution worked last time \ let ", Cell[BoxData[ FormBox[ RowBox[{"y", "=", RowBox[{"u", "+", SuperscriptBox["z", "2"]}]}], TraditionalForm]]], " then ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[DifferentialD]", "y"}], "=", RowBox[{"\[DifferentialD]", "u"}]}], TraditionalForm]]], " and we have" }], "Text", CellChangeTimes->{{3.39966741512825*^9, 3.3996674251176033`*^9}, { 3.399667467328555*^9, 3.3996676457319546`*^9}, {3.399667680902662*^9, 3.39966776157478*^9}, {3.3996678033569746`*^9, 3.3996678040134892`*^9}, { 3.399668101354689*^9, 3.3996683970922503`*^9}, {3.3996684525856237`*^9, 3.3996685309018993`*^9}, {3.3996687855709105`*^9, 3.3996688155376005`*^9}, { 3.399668962368433*^9, 3.399669065867295*^9}, {3.39966924146135*^9, 3.3996692489178205`*^9}, {3.39966934961939*^9, 3.3996693532772894`*^9}, { 3.399669419776095*^9, 3.3996694352518544`*^9}, {3.3996698488002644`*^9, 3.3996700395414486`*^9}, {3.3996709137917857`*^9, 3.3996709150579453`*^9}, { 3.399670951667263*^9, 3.3996709613901615`*^9}, {3.399718807218936*^9, 3.399718848232458*^9}, {3.3997190055283403`*^9, 3.39971919146381*^9}, { 3.399719243696439*^9, 3.399719254976253*^9}, {3.399719314937681*^9, 3.3997193732798853`*^9}}, FontSize->14], Cell[BoxData[ RowBox[{ RowBox[{ FractionBox["1", "2"], RowBox[{"\[Integral]", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"u", "+", SuperscriptBox["z", "2"]}], ")"}], RowBox[{ RowBox[{"-", "1"}], "/", "2"}]], RowBox[{"\[DifferentialD]", "u"}]}]}]}], "=", RowBox[{ FractionBox["1", "2"], RowBox[{"\[Integral]", RowBox[{ SuperscriptBox["y", RowBox[{ RowBox[{"-", "1"}], "/", "2"}]], RowBox[{"\[DifferentialD]", "y"}]}]}]}]}]], "Input", CellChangeTimes->{{3.399669507722238*^9, 3.399669509473033*^9}, { 3.3997192163145723`*^9, 3.399719303931158*^9}, {3.399719382627432*^9, 3.3997194039696693`*^9}}, FontSize->14], Cell["This I can do, giving", "Text", CellChangeTimes->{{3.39966741512825*^9, 3.3996674251176033`*^9}, { 3.399667467328555*^9, 3.3996676457319546`*^9}, {3.399667680902662*^9, 3.39966776157478*^9}, {3.3996678033569746`*^9, 3.3996678040134892`*^9}, { 3.399668101354689*^9, 3.3996683970922503`*^9}, {3.3996684525856237`*^9, 3.3996685309018993`*^9}, {3.3996687855709105`*^9, 3.3996688155376005`*^9}, { 3.399668962368433*^9, 3.399669065867295*^9}, {3.39966924146135*^9, 3.3996692489178205`*^9}, {3.39966934961939*^9, 3.3996693532772894`*^9}, { 3.399669419776095*^9, 3.3996694352518544`*^9}, {3.3996698488002644`*^9, 3.3996700395414486`*^9}, {3.3996709137917857`*^9, 3.3996709150579453`*^9}, { 3.399670951667263*^9, 3.3996709613901615`*^9}, {3.399718807218936*^9, 3.399718848232458*^9}, {3.3997190055283403`*^9, 3.39971919146381*^9}, { 3.399719243696439*^9, 3.399719254976253*^9}, {3.399719314937681*^9, 3.3997193732798853`*^9}, {3.399719411609655*^9, 3.399719425546577*^9}}, FontSize->14], Cell[BoxData[ RowBox[{ RowBox[{ FractionBox["1", "2"], RowBox[{"\[Integral]", RowBox[{ SuperscriptBox["y", RowBox[{ RowBox[{"-", "1"}], "/", "2"}]], RowBox[{"\[DifferentialD]", "y"}]}]}]}], "=", RowBox[{ SuperscriptBox["y", RowBox[{"1", "/", "2"}]], "=", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"u", "+", SuperscriptBox["z", "2"]}], ")"}], RowBox[{"1", "/", "2"}]], "=", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{ SuperscriptBox["r", "2"], "+", SuperscriptBox["z", "2"]}], ")"}], RowBox[{"1", "/", "2"}]], "=", SqrtBox[ RowBox[{ SuperscriptBox["r", "2"], "+", SuperscriptBox["z", "2"]}]]}]}]}]}]], "Input", CellChangeTimes->{{3.399669507722238*^9, 3.399669509473033*^9}, { 3.3997192163145723`*^9, 3.399719303931158*^9}, {3.399719382627432*^9, 3.399719503441689*^9}}, FontSize->14] }, Open ]], Cell[CellGroupData[{ Cell["Now that we believe the integral ...", "Subsection", CellChangeTimes->{{3.39966741512825*^9, 3.3996674251176033`*^9}, { 3.399667467328555*^9, 3.3996676457319546`*^9}, {3.399667680902662*^9, 3.3996677917117167`*^9}, {3.3996680269796324`*^9, 3.399668030168466*^9}, { 3.399677861113304*^9, 3.399677863927208*^9}, {3.399715155905615*^9, 3.399715163503224*^9}, {3.3997152545378113`*^9, 3.399715258488538*^9}, { 3.399718912069803*^9, 3.399718952160014*^9}}], Cell["\<\ In one sense we've got the answer to the integration as you can see what \ happens if we put in integration limits. Still, it's nice to get the \ computer to verify things even when they're pretty clear. So let's work \ around the infinity for the moment and try the integration with a dummy upper \ limit\ \>", "Text", CellChangeTimes->{{3.39966741512825*^9, 3.3996674251176033`*^9}, { 3.399667467328555*^9, 3.3996676457319546`*^9}, {3.399667680902662*^9, 3.39966776157478*^9}, {3.3996678033569746`*^9, 3.3996678040134892`*^9}, { 3.399668101354689*^9, 3.3996683970922503`*^9}, {3.3996684525856237`*^9, 3.3996685309018993`*^9}, {3.3996687855709105`*^9, 3.3996688155376005`*^9}, { 3.399668962368433*^9, 3.399669065867295*^9}, {3.39966924146135*^9, 3.3996692489178205`*^9}, {3.39966934961939*^9, 3.3996693532772894`*^9}, { 3.399669419776095*^9, 3.3996694352518544`*^9}, {3.3996698488002644`*^9, 3.3996700395414486`*^9}, {3.3996709137917857`*^9, 3.3996709150579453`*^9}, { 3.399670951667263*^9, 3.3996709613901615`*^9}, {3.3997188617380533`*^9, 3.399718862568511*^9}, {3.3997189567044764`*^9, 3.3997189583364687`*^9}, { 3.399719588865593*^9, 3.399719627704274*^9}}, FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Assuming", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"x", ">", "0"}], ",", RowBox[{"r", ">", "0"}], ",", RowBox[{"z", ">", "0"}], ",", RowBox[{"Element", "[", RowBox[{ RowBox[{"{", RowBox[{"x", ",", "z", ",", "r"}], "}"}], ",", "Reals"}], "]"}]}], "}"}], ",", RowBox[{ SubsuperscriptBox["\[Integral]", "0", "x"], RowBox[{ FractionBox["r", SqrtBox[ RowBox[{ SuperscriptBox["z", "2"], "+", SuperscriptBox["r", "2"]}]]], RowBox[{"\[DifferentialD]", "r"}]}]}]}], "]"}]], "Input", CellChangeTimes->{{3.399669539768069*^9, 3.399669543222763*^9}, { 3.399669574080573*^9, 3.3996695779886045`*^9}, {3.3996696216804147`*^9, 3.399669642236687*^9}, {3.3996697014825525`*^9, 3.3996697316995363`*^9}, { 3.3996697999183817`*^9, 3.3996698173170137`*^9}}, FontSize->14], Cell[BoxData[ RowBox[{ RowBox[{"-", "z"}], "+", SqrtBox[ RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["z", "2"]}]]}]], "Output", CellChangeTimes->{3.3996695696410503`*^9, 3.399669606736094*^9, 3.3996697339036713`*^9, 3.399669818770808*^9, 3.39971850770687*^9, 3.399720000039028*^9, 3.399720091735879*^9, 3.3998046886325607`*^9}, FontSize->14] }, Open ]], Cell["Putting all the other stuff back in", "Text", CellChangeTimes->{{3.39966741512825*^9, 3.3996674251176033`*^9}, { 3.399667467328555*^9, 3.3996676457319546`*^9}, {3.399667680902662*^9, 3.39966776157478*^9}, {3.3996678033569746`*^9, 3.3996678040134892`*^9}, { 3.399668101354689*^9, 3.3996683970922503`*^9}, {3.3996684525856237`*^9, 3.3996685309018993`*^9}, {3.3996687855709105`*^9, 3.3996688155376005`*^9}, { 3.399668962368433*^9, 3.399669065867295*^9}, {3.39966924146135*^9, 3.3996692489178205`*^9}, {3.39966934961939*^9, 3.3996693532772894`*^9}, { 3.399669419776095*^9, 3.3996694352518544`*^9}, {3.3996698488002644`*^9, 3.399669939795403*^9}, {3.39967007805863*^9, 3.3996701497460136`*^9}}, FontSize->14], Cell[BoxData[ RowBox[{ RowBox[{"V", "[", "z_", "]"}], ":=", RowBox[{ FractionBox["\[Sigma]", RowBox[{"2", SubscriptBox["\[Epsilon]", "0"]}]], RowBox[{"(", RowBox[{ RowBox[{"-", "z"}], "+", SqrtBox[ RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["z", "2"]}]]}], ")"}]}]}]], "Input", CellChangeTimes->{{3.3996678061549644`*^9, 3.3996679122446938`*^9}, { 3.3996680465034356`*^9, 3.399668048473012*^9}, {3.399668583149845*^9, 3.399668583228009*^9}, {3.399669260876313*^9, 3.3996692634556*^9}, { 3.3996693667208605`*^9, 3.399669405285164*^9}, {3.3996694448968506`*^9, 3.399669468704523*^9}, {3.3996701656279516`*^9, 3.3996701705207167`*^9}}, FontSize->14], Cell[TextData[{ "This is the answer for a finite disk of radius ", Cell[BoxData[ FormBox["x", TraditionalForm]]], " but that's not what we want and it blows up for an infinite disk, which, \ if you think about it, makes sense. An infinite disk with a finite charge \ density contains an infinite amount of charge. It's not unreasonable, though \ I suppose not always necessary, that an infinite amount of charge produces an \ infinite potential." }], "Text", CellChangeTimes->{{3.39966741512825*^9, 3.3996674251176033`*^9}, { 3.399667467328555*^9, 3.3996676457319546`*^9}, {3.399667680902662*^9, 3.39966776157478*^9}, {3.3996678033569746`*^9, 3.3996678040134892`*^9}, { 3.399668101354689*^9, 3.3996683970922503`*^9}, {3.3996684525856237`*^9, 3.3996685309018993`*^9}, {3.3996687855709105`*^9, 3.3996688155376005`*^9}, { 3.399668962368433*^9, 3.399669065867295*^9}, {3.39966924146135*^9, 3.3996692489178205`*^9}, {3.39966934961939*^9, 3.3996693532772894`*^9}, { 3.399669419776095*^9, 3.3996694352518544`*^9}, {3.3996698488002644`*^9, 3.399669939795403*^9}, {3.39967007805863*^9, 3.3996701314255247`*^9}, { 3.3996701817600145`*^9, 3.3996702350490723`*^9}}, FontSize->14], Cell[TextData[{ "So what do we do? Well, we're supposed to be working on two parallel \ plates and we've only figured out one of them, so let's work on the other.\n\n\ Right off we're pretty close to done. The equation's going to have to be \ similar since the other plate is ... a plate too. But there are two \ differences. If the plates are a distance ", Cell[BoxData[ FormBox["d", TraditionalForm]]], " apart and we're looking at a point that's ", Cell[BoxData[ FormBox["z", TraditionalForm]]], " above one, it must be ", Cell[BoxData[ FormBox[ RowBox[{"d", "-", "z"}], TraditionalForm]]], " below the other and the charge on the second plate must be of oposite \ sign.\n\nSo we'll call our first potential ", Cell[BoxData[ FormBox[ SubscriptBox["V", "b"], TraditionalForm]]], " for bottom and the other ", Cell[BoxData[ FormBox[ SubscriptBox["V", "t"], TraditionalForm]]], " for top." }], "Text", CellChangeTimes->{{3.39966741512825*^9, 3.3996674251176033`*^9}, { 3.399667467328555*^9, 3.3996676457319546`*^9}, {3.399667680902662*^9, 3.39966776157478*^9}, {3.3996678033569746`*^9, 3.3996678040134892`*^9}, { 3.399668101354689*^9, 3.3996683970922503`*^9}, {3.3996684525856237`*^9, 3.3996685309018993`*^9}, {3.3996687855709105`*^9, 3.3996688155376005`*^9}, { 3.399668962368433*^9, 3.399669065867295*^9}, 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3.3996708248950143`*^9}, {3.3996751210765905`*^9, 3.3996751252659864`*^9}, { 3.3996751827608185`*^9, 3.3996751830578284`*^9}}, FontSize->14], Cell[TextData[{ "This is the potential at a point ", Cell[BoxData[ FormBox["z", TraditionalForm]]], " between two finite plates of radius ", Cell[BoxData[ FormBox["x", TraditionalForm]]], " a distance ", Cell[BoxData[ FormBox["d", TraditionalForm]]], " apart. How do we deal with the infinite part?" }], "Text", CellChangeTimes->{{3.39966741512825*^9, 3.3996674251176033`*^9}, { 3.399667467328555*^9, 3.3996676457319546`*^9}, {3.399667680902662*^9, 3.39966776157478*^9}, {3.3996678033569746`*^9, 3.3996678040134892`*^9}, { 3.399668101354689*^9, 3.3996683970922503`*^9}, {3.3996684525856237`*^9, 3.3996685309018993`*^9}, {3.3996687855709105`*^9, 3.3996688155376005`*^9}, {3.399668962368433*^9, 3.399669065867295*^9}, { 3.39966924146135*^9, 3.3996692489178205`*^9}, {3.39966934961939*^9, 3.3996693532772894`*^9}, {3.399669419776095*^9, 3.3996694352518544`*^9}, { 3.3996698488002644`*^9, 3.399669939795403*^9}, {3.39967007805863*^9, 3.3996701314255247`*^9}, {3.3996701817600145`*^9, 3.399670444830204*^9}, { 3.3996704777669625`*^9, 3.399670489444125*^9}, {3.39967058412502*^9, 3.3996706083065186`*^9}, {3.399670854548047*^9, 3.399670880512096*^9}, { 3.399670982446048*^9, 3.399671012193197*^9}, 3.39967513337904*^9, { 3.39967518812263*^9, 3.3996752579212418`*^9}, {3.3996753842791185`*^9, 3.3996754879698343`*^9}, {3.399715230913288*^9, 3.399715232039846*^9}}, FontSize->14] }, Open ]], Cell[CellGroupData[{ Cell["Going to \[Infinity]", "Subsection", CellChangeTimes->{{3.39966741512825*^9, 3.3996674251176033`*^9}, { 3.399667467328555*^9, 3.3996676457319546`*^9}, {3.399667680902662*^9, 3.3996677917117167`*^9}, {3.3996680269796324`*^9, 3.399668030168466*^9}, { 3.399677861113304*^9, 3.399677863927208*^9}, {3.399715155905615*^9, 3.399715163503224*^9}, {3.3997152545378113`*^9, 3.399715258488538*^9}}], Cell[TextData[{ "Well, from a 'proof by eyeball' we can see what's going to happen when ", Cell[BoxData[ FormBox["x", TraditionalForm]]], ", the radius of the plate, gets big: the two square root terms are going to \ cancel. However, it pays to try to prove it. You can't do science by \ hunches alone (though you can't do it without them either).\n\nFirst, \ mechanically:" }], "Text", CellChangeTimes->{{3.39966741512825*^9, 3.3996674251176033`*^9}, { 3.399667467328555*^9, 3.3996676457319546`*^9}, {3.399667680902662*^9, 3.39966776157478*^9}, {3.3996678033569746`*^9, 3.3996678040134892`*^9}, { 3.399668101354689*^9, 3.3996683970922503`*^9}, {3.3996684525856237`*^9, 3.3996685309018993`*^9}, {3.3996687855709105`*^9, 3.3996688155376005`*^9}, {3.399668962368433*^9, 3.399669065867295*^9}, { 3.39966924146135*^9, 3.3996692489178205`*^9}, {3.39966934961939*^9, 3.3996693532772894`*^9}, {3.399669419776095*^9, 3.3996694352518544`*^9}, { 3.3996698488002644`*^9, 3.399669939795403*^9}, {3.39967007805863*^9, 3.3996701314255247`*^9}, {3.3996701817600145`*^9, 3.399670444830204*^9}, { 3.3996704777669625`*^9, 3.399670489444125*^9}, {3.39967058412502*^9, 3.3996706083065186`*^9}, {3.399670854548047*^9, 3.399670880512096*^9}, { 3.399670982446048*^9, 3.399671012193197*^9}, 3.39967513337904*^9, { 3.39967518812263*^9, 3.3996752579212418`*^9}, {3.3996753842791185`*^9, 3.3996754879698343`*^9}, {3.399715236648093*^9, 3.399715238368033*^9}, { 3.399715281144229*^9, 3.3997153084315243`*^9}}, FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Assuming", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"x", ">", "0"}], ",", RowBox[{"d", ">", "0"}], ",", RowBox[{"z", ">", "0"}], ",", RowBox[{"Element", "[", RowBox[{ RowBox[{"{", RowBox[{"x", ",", "d", ",", "z"}], "}"}], ",", "Reals"}], "]"}]}], "}"}], ",", RowBox[{"Limit", "[", RowBox[{ RowBox[{"d", "-", RowBox[{"2", "z"}], "+", SqrtBox[ RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["z", "2"]}]], "-", SqrtBox[ RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox[ RowBox[{"(", RowBox[{"d", "-", "z"}], ")"}], "2"]}]]}], ",", RowBox[{"x", "\[Rule]", "\[Infinity]"}]}], "]"}]}], "]"}]], "Input", CellChangeTimes->{{3.3996752644556456`*^9, 3.399675359564061*^9}}, FontSize->14], Cell[BoxData[ RowBox[{"d", "-", RowBox[{"2", " ", "z"}]}]], "Output", CellChangeTimes->{3.3996753693500366`*^9, 3.399718513564705*^9, 3.399720003867524*^9, 3.399720096317893*^9, 3.399804694435604*^9}, FontSize->14] }, Open ]], Cell[TextData[{ "That's good. We got the same answer as we did by eye. But how do you know \ the machine did it right? Perhaps more importantly, you're not going to be \ allowed to use ", StyleBox["Mathematica", FontSlant->"Italic"], " on any tests. So let's do the limit by hand.\n\nThe first thing I note is \ that the leading ", Cell[BoxData[ FormBox[ RowBox[{"d", "-", RowBox[{"2", "z"}]}], TraditionalForm]]], " doesn't matter. We're only interested in the parts that contain ", Cell[BoxData[ FormBox["x", TraditionalForm]]], ". What is" }], "Text", CellChangeTimes->{{3.39966741512825*^9, 3.3996674251176033`*^9}, { 3.399667467328555*^9, 3.3996676457319546`*^9}, {3.399667680902662*^9, 3.39966776157478*^9}, {3.3996678033569746`*^9, 3.3996678040134892`*^9}, { 3.399668101354689*^9, 3.3996683970922503`*^9}, {3.3996684525856237`*^9, 3.3996685309018993`*^9}, {3.3996687855709105`*^9, 3.3996688155376005`*^9}, {3.399668962368433*^9, 3.399669065867295*^9}, { 3.39966924146135*^9, 3.3996692489178205`*^9}, {3.39966934961939*^9, 3.3996693532772894`*^9}, {3.399669419776095*^9, 3.3996694352518544`*^9}, { 3.3996698488002644`*^9, 3.399669939795403*^9}, {3.39967007805863*^9, 3.3996701314255247`*^9}, {3.3996701817600145`*^9, 3.399670444830204*^9}, { 3.3996704777669625`*^9, 3.399670489444125*^9}, {3.39967058412502*^9, 3.3996706083065186`*^9}, {3.399670854548047*^9, 3.399670880512096*^9}, { 3.399670982446048*^9, 3.399671012193197*^9}, 3.39967513337904*^9, { 3.39967518812263*^9, 3.3996752579212418`*^9}, {3.3996753842791185`*^9, 3.3996754879698343`*^9}, {3.3996755198915076`*^9, 3.399675645202244*^9}, { 3.399715316704092*^9, 3.399715334607716*^9}}, FontSize->14], Cell[TextData[{ Cell[BoxData[ FormBox[ RowBox[{ UnderscriptBox["lim", RowBox[{"x", "\[Rule]", "\[Infinity]"}]], RowBox[{"(", RowBox[{ SqrtBox[ RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["z", "2"]}]], "-", SqrtBox[ RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox[ RowBox[{"(", RowBox[{"d", "-", "z"}], ")"}], "2"]}]]}], ")"}]}], TraditionalForm]]], "?\n\nFirst we break it up\n\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ UnderscriptBox["lim", RowBox[{"x", "\[Rule]", "\[Infinity]"}]], SqrtBox[ RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["z", "2"]}]]}], "-", RowBox[{ UnderscriptBox["lim", RowBox[{"x", "\[Rule]", "\[Infinity]"}]], SqrtBox[ RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox[ RowBox[{"(", RowBox[{"d", "-", "z"}], ")"}], "2"]}]]}]}], TraditionalForm]]], "\n\nbut this is about as far as we can go by just looking for simple rules \ for limits. Both of these go to \[Infinity] giving us\n\n", Cell[BoxData[ FormBox[ RowBox[{"\[Infinity]", "-", "\[Infinity]"}], TraditionalForm]]], "\n\nwhich is undefined. \n\nTo see why, consider that ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ UnderscriptBox["lim", RowBox[{"x", "\[Rule]", "\[Infinity]"}]], RowBox[{"(", "x", ")"}]}], "-", RowBox[{ UnderscriptBox["lim", RowBox[{"x", "\[Rule]", "\[Infinity]"}]], RowBox[{"(", RowBox[{"x", "+", "1"}], ")"}]}]}], TraditionalForm]]], " gives the same ", Cell[BoxData[ FormBox[ RowBox[{"\[Infinity]", "-", "\[Infinity]"}], TraditionalForm]]], " result but the limit is clearly ", Cell[BoxData[ FormBox[ RowBox[{"-", "1"}], TraditionalForm]]], ". The moral of the story is that ", Cell[BoxData[ FormBox[ RowBox[{"\[Infinity]", "-", "\[Infinity]"}], TraditionalForm]]], " can be anything, hence it's 'undefined'. Another example of the \ unfortunateness of ", Cell[BoxData[ FormBox[ RowBox[{"\[Infinity]", "-", "\[Infinity]"}], TraditionalForm]]], " more relevent to our situation is\n\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ UnderscriptBox["lim", RowBox[{"x", "\[Rule]", "\[Infinity]"}]], SqrtBox[ RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["z", "2"]}]]}], "-", RowBox[{ UnderscriptBox["lim", RowBox[{"x", "\[Rule]", "\[Infinity]"}]], SqrtBox[ RowBox[{ SuperscriptBox["x", "314"], "+", SuperscriptBox[ RowBox[{"(", RowBox[{"d", "-", "z"}], ")"}], "2"]}]]}]}], TraditionalForm]]], "\n\nClearly here the limit should be ", Cell[BoxData[ FormBox[ RowBox[{"-", "\[Infinity]"}], TraditionalForm]]], ".\n\nThe issue is that the limit of the difference of two terms can be \ effected by how fast the two terms grow. By eye we can see that in our case \ they grow at the same speed.\n\nIn general, however, this may not be obvious. \ One solution is to look at the limit of the ratio of the two terms. This \ technique will capture the problem of relative growth rates, since if one is \ growing much faster than the other the limit of the ratio will either go to \ zero or \[Infinity].\n\nFor our case we have\n\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ UnderscriptBox["lim", RowBox[{"x", "\[Rule]", "\[Infinity]"}]], FractionBox[ SqrtBox[ RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["z", "2"]}]], SqrtBox[ RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox[ RowBox[{"(", RowBox[{"d", "-", "z"}], ")"}], "2"]}]]]}], "=", RowBox[{ RowBox[{ UnderscriptBox["lim", RowBox[{"x", "\[Rule]", "\[Infinity]"}]], SqrtBox[ FractionBox[ RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["z", "2"]}], RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox[ RowBox[{"(", RowBox[{"d", "-", "z"}], ")"}], "2"]}]]]}], "=", RowBox[{ SqrtBox[ RowBox[{ UnderscriptBox["lim", RowBox[{"x", "\[Rule]", "\[Infinity]"}]], RowBox[{"(", FractionBox[ RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["z", "2"]}], RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox[ RowBox[{"(", RowBox[{"d", "-", "z"}], ")"}], "2"]}]], ")"}]}]], "=", SqrtBox[ RowBox[{ UnderscriptBox["lim", RowBox[{"x", "\[Rule]", "\[Infinity]"}]], RowBox[{"(", RowBox[{ FractionBox[ SuperscriptBox["x", "2"], RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox[ RowBox[{"(", RowBox[{"d", "-", "z"}], ")"}], "2"]}]], "+", FractionBox[ SuperscriptBox["z", "2"], RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox[ RowBox[{"(", RowBox[{"d", "-", "z"}], ")"}], "2"]}]]}], ")"}]}]]}]}]}], TraditionalForm]]], "\n\nbut now we're getting somewhere because\n\n", Cell[BoxData[ FormBox[ RowBox[{ SqrtBox[ RowBox[{ UnderscriptBox["lim", RowBox[{"x", "\[Rule]", "\[Infinity]"}]], RowBox[{"(", RowBox[{ FractionBox[ SuperscriptBox["x", "2"], RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox[ RowBox[{"(", RowBox[{"d", "-", "z"}], ")"}], "2"]}]], "+", FractionBox[ SuperscriptBox["z", "2"], RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox[ RowBox[{"(", RowBox[{"d", "-", "z"}], ")"}], "2"]}]]}], ")"}]}]], "=", RowBox[{ SqrtBox[ RowBox[{ RowBox[{ UnderscriptBox["lim", RowBox[{"x", "\[Rule]", "\[Infinity]"}]], RowBox[{"(", FractionBox[ SuperscriptBox["x", "2"], RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox[ RowBox[{"(", RowBox[{"d", "-", "z"}], ")"}], "2"]}]], ")"}]}], "+", RowBox[{ UnderscriptBox["lim", RowBox[{"x", "\[Rule]", "\[Infinity]"}]], RowBox[{"(", FractionBox[ SuperscriptBox["z", "2"], RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox[ RowBox[{"(", RowBox[{"d", "-", "z"}], ")"}], "2"]}]], ")"}]}]}]], "=", RowBox[{ SqrtBox[ RowBox[{"1", "+", "0"}]], "=", "1"}]}]}], TraditionalForm]]], "\n\nwhich means the two terms are indeed equal in the limit of infinite \ radius. You are invited to consider the ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", "2"], " ", RowBox[{"vs", ".", " ", SuperscriptBox["x", "314"]}]}], TraditionalForm]]], " case above." }], "Text", CellChangeTimes->{{3.39966741512825*^9, 3.3996674251176033`*^9}, { 3.399667467328555*^9, 3.3996676457319546`*^9}, {3.399667680902662*^9, 3.39966776157478*^9}, {3.3996678033569746`*^9, 3.3996678040134892`*^9}, { 3.399668101354689*^9, 3.3996683970922503`*^9}, {3.3996684525856237`*^9, 3.3996685309018993`*^9}, {3.3996687855709105`*^9, 3.3996688155376005`*^9}, {3.399668962368433*^9, 3.399669065867295*^9}, { 3.39966924146135*^9, 3.3996692489178205`*^9}, {3.39966934961939*^9, 3.3996693532772894`*^9}, {3.399669419776095*^9, 3.3996694352518544`*^9}, { 3.3996698488002644`*^9, 3.399669939795403*^9}, {3.39967007805863*^9, 3.3996701314255247`*^9}, {3.3996701817600145`*^9, 3.399670444830204*^9}, { 3.3996704777669625`*^9, 3.399670489444125*^9}, {3.39967058412502*^9, 3.3996706083065186`*^9}, {3.399670854548047*^9, 3.399670880512096*^9}, { 3.399670982446048*^9, 3.399671012193197*^9}, 3.39967513337904*^9, { 3.39967518812263*^9, 3.3996752579212418`*^9}, {3.3996753842791185`*^9, 3.3996754879698343`*^9}, {3.3996755198915076`*^9, 3.3996757745619955`*^9}, {3.399675811533095*^9, 3.3996766327379627`*^9}, { 3.39968083556616*^9, 3.3996808668161273`*^9}, {3.3997153911117363`*^9, 3.399715426607394*^9}, {3.399715599547814*^9, 3.39971561921563*^9}, { 3.399715651253188*^9, 3.39971589092759*^9}, {3.3997159875997953`*^9, 3.3997162237195663`*^9}, {3.399719783856763*^9, 3.3997198227447567`*^9}}, FontSize->14] }, Open ]], Cell[CellGroupData[{ Cell["The 'answer'", "Subsection", CellChangeTimes->{{3.39966741512825*^9, 3.3996674251176033`*^9}, { 3.399667467328555*^9, 3.3996676457319546`*^9}, {3.399667680902662*^9, 3.3996677917117167`*^9}, {3.3996680269796324`*^9, 3.399668030168466*^9}, { 3.399677861113304*^9, 3.399677863927208*^9}, {3.399715155905615*^9, 3.399715163503224*^9}, {3.399715950691345*^9, 3.3997159526954412`*^9}}], Cell[TextData[{ "So where were we? We were trying to figure out what this was for large ", Cell[BoxData[ FormBox["x", TraditionalForm]]], ":" }], "Text", CellChangeTimes->{{3.39966741512825*^9, 3.3996674251176033`*^9}, { 3.399667467328555*^9, 3.3996676457319546`*^9}, {3.399667680902662*^9, 3.39966776157478*^9}, {3.3996678033569746`*^9, 3.3996678040134892`*^9}, { 3.399668101354689*^9, 3.3996683970922503`*^9}, {3.3996684525856237`*^9, 3.3996685309018993`*^9}, {3.3996687855709105`*^9, 3.3996688155376005`*^9}, {3.399668962368433*^9, 3.399669065867295*^9}, { 3.39966924146135*^9, 3.3996692489178205`*^9}, {3.39966934961939*^9, 3.3996693532772894`*^9}, {3.399669419776095*^9, 3.3996694352518544`*^9}, { 3.3996698488002644`*^9, 3.399669939795403*^9}, {3.39967007805863*^9, 3.3996701314255247`*^9}, {3.3996701817600145`*^9, 3.399670444830204*^9}, { 3.3996704777669625`*^9, 3.399670489444125*^9}, {3.39967058412502*^9, 3.3996706083065186`*^9}, {3.399670854548047*^9, 3.399670880512096*^9}, { 3.399670982446048*^9, 3.399671012193197*^9}, 3.39967513337904*^9, { 3.39967518812263*^9, 3.3996752579212418`*^9}, {3.3996753842791185`*^9, 3.3996754879698343`*^9}, {3.3996755198915076`*^9, 3.399675645202244*^9}, { 3.399676702632211*^9, 3.3996767331005383`*^9}}, FontSize->14], Cell[BoxData[ RowBox[{ RowBox[{"V", "[", "z_", "]"}], ":=", RowBox[{ FractionBox["\[Sigma]", RowBox[{"2", SubscriptBox["\[Epsilon]", "0"]}]], RowBox[{"(", RowBox[{"d", "-", RowBox[{"2", "z"}], "+", SqrtBox[ RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["z", "2"]}]], "-", SqrtBox[ RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox[ RowBox[{"(", RowBox[{"d", "-", "z"}], ")"}], "2"]}]]}], ")"}]}]}]], "Input", CellChangeTimes->{{3.3996678061549644`*^9, 3.3996679122446938`*^9}, { 3.3996680465034356`*^9, 3.399668048473012*^9}, {3.399668583149845*^9, 3.399668583228009*^9}, {3.399669260876313*^9, 3.3996692634556*^9}, { 3.3996693667208605`*^9, 3.399669405285164*^9}, {3.3996694448968506`*^9, 3.399669468704523*^9}, {3.3996701656279516`*^9, 3.3996701705207167`*^9}, { 3.3996704507860155`*^9, 3.399670451083023*^9}, {3.3996705563796716`*^9, 3.3996705633199058`*^9}, {3.399670614418331*^9, 3.3996706290491896`*^9}, { 3.3996706611871285`*^9, 3.3996706640632887`*^9}, {3.3996707433143444`*^9, 3.3996708248950143`*^9}, {3.3996751210765905`*^9, 3.3996751252659864`*^9}, { 3.3996751827608185`*^9, 3.3996751830578284`*^9}}, FontSize->14], Cell["and the answer is now known to be", "Text", CellChangeTimes->{{3.39966741512825*^9, 3.3996674251176033`*^9}, { 3.399667467328555*^9, 3.3996676457319546`*^9}, {3.399667680902662*^9, 3.39966776157478*^9}, {3.3996678033569746`*^9, 3.3996678040134892`*^9}, { 3.399668101354689*^9, 3.3996683970922503`*^9}, {3.3996684525856237`*^9, 3.3996685309018993`*^9}, {3.3996687855709105`*^9, 3.3996688155376005`*^9}, {3.399668962368433*^9, 3.399669065867295*^9}, { 3.39966924146135*^9, 3.3996692489178205`*^9}, {3.39966934961939*^9, 3.3996693532772894`*^9}, {3.399669419776095*^9, 3.3996694352518544`*^9}, { 3.3996698488002644`*^9, 3.399669939795403*^9}, {3.39967007805863*^9, 3.3996701314255247`*^9}, {3.3996701817600145`*^9, 3.399670444830204*^9}, { 3.3996704777669625`*^9, 3.399670489444125*^9}, {3.39967058412502*^9, 3.3996706083065186`*^9}, {3.399670854548047*^9, 3.399670880512096*^9}, { 3.399670982446048*^9, 3.399671012193197*^9}, 3.39967513337904*^9, { 3.39967518812263*^9, 3.3996752579212418`*^9}, {3.3996753842791185`*^9, 3.3996754879698343`*^9}, {3.3996755198915076`*^9, 3.399675645202244*^9}, { 3.399676702632211*^9, 3.399676746560379*^9}}, FontSize->14], Cell[BoxData[ RowBox[{ RowBox[{"V", "[", "z_", "]"}], ":=", RowBox[{ FractionBox["\[Sigma]", RowBox[{"2", SubscriptBox["\[Epsilon]", "0"]}]], RowBox[{"(", RowBox[{"d", "-", RowBox[{"2", "z"}]}], ")"}]}]}]], "Input", CellChangeTimes->{{3.3996678061549644`*^9, 3.3996679122446938`*^9}, { 3.3996680465034356`*^9, 3.399668048473012*^9}, {3.399668583149845*^9, 3.399668583228009*^9}, {3.399669260876313*^9, 3.3996692634556*^9}, { 3.3996693667208605`*^9, 3.399669405285164*^9}, {3.3996694448968506`*^9, 3.399669468704523*^9}, {3.3996701656279516`*^9, 3.3996701705207167`*^9}, { 3.3996704507860155`*^9, 3.399670451083023*^9}, {3.3996705563796716`*^9, 3.3996705633199058`*^9}, {3.399670614418331*^9, 3.3996706290491896`*^9}, { 3.3996706611871285`*^9, 3.3996706640632887`*^9}, {3.3996707433143444`*^9, 3.3996708248950143`*^9}, {3.3996751210765905`*^9, 3.3996751252659864`*^9}, { 3.3996751827608185`*^9, 3.3996751830578284`*^9}, {3.3996767542517166`*^9, 3.399676755267849*^9}}, FontSize->14] }, Open ]], Cell[CellGroupData[{ Cell["Potential vs. Potential Difference", "Subsection", CellChangeTimes->{{3.39966741512825*^9, 3.3996674251176033`*^9}, { 3.399667467328555*^9, 3.3996676457319546`*^9}, {3.399667680902662*^9, 3.3996677917117167`*^9}, {3.3996680269796324`*^9, 3.399668030168466*^9}, { 3.399677861113304*^9, 3.399677863927208*^9}, {3.399715155905615*^9, 3.399715163503224*^9}}], Cell[TextData[{ "But we're not really done. You can never ", StyleBox["measure", FontSlant->"Italic"], " 'the potential', you can only ever measure a ", StyleBox["potential difference", FontSlant->"Italic"], " (which Halliday & Resnick probably state in one sentence somewhere in the \ book and then seldom, if ever, mention again. Introductory physics textbooks \ have the problem of stuffing 400 years of understanding into 1,000 pages. \ The good ones invariably accomplish this by making every sentence count. \ This means you need to read the book carefully. Important things may well be \ said only one time.).\n\nIgnoring the potential-difference-only bit for the \ moment, we can check our answer to see how well it does.\n\nWe know what the \ potential difference should be from one plate to the next, i.e. between ", Cell[BoxData[ FormBox[ RowBox[{"z", "=", RowBox[{ RowBox[{"0", " ", "and", " ", "z"}], "=", "d"}]}], TraditionalForm]]], ". 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What we want is not ", Cell[BoxData[ FormBox[ RowBox[{"V", "(", "z", ")"}], TraditionalForm]]], " but ", Cell[BoxData[ FormBox[ RowBox[{"\[CapitalDelta]V", "(", "z", ")"}], TraditionalForm]]], " since we can only ever measure potential ", StyleBox["differences:", FontSlant->"Italic"] }], "Text", CellChangeTimes->{{3.39966741512825*^9, 3.3996674251176033`*^9}, { 3.399667467328555*^9, 3.3996676457319546`*^9}, {3.399667680902662*^9, 3.39966776157478*^9}, {3.3996678033569746`*^9, 3.3996678040134892`*^9}, { 3.399668101354689*^9, 3.3996683970922503`*^9}, {3.3996684525856237`*^9, 3.3996685309018993`*^9}, {3.3996687855709105`*^9, 3.3996688155376005`*^9}, {3.399668962368433*^9, 3.399669065867295*^9}, { 3.39966924146135*^9, 3.3996692489178205`*^9}, {3.39966934961939*^9, 3.3996693532772894`*^9}, {3.399669419776095*^9, 3.3996694352518544`*^9}, { 3.3996698488002644`*^9, 3.399669939795403*^9}, {3.39967007805863*^9, 3.3996701314255247`*^9}, {3.3996701817600145`*^9, 3.399670444830204*^9}, { 3.3996704777669625`*^9, 3.399670489444125*^9}, {3.39967058412502*^9, 3.3996706083065186`*^9}, {3.399670854548047*^9, 3.399670880512096*^9}, { 3.399670982446048*^9, 3.399671012193197*^9}, 3.39967513337904*^9, { 3.39967518812263*^9, 3.3996752579212418`*^9}, {3.3996753842791185`*^9, 3.3996754879698343`*^9}, {3.3996755198915076`*^9, 3.399675645202244*^9}, { 3.399676702632211*^9, 3.399676746560379*^9}, {3.3996767820312023`*^9, 3.399677061983385*^9}, {3.399677119840378*^9, 3.399677132330985*^9}, { 3.399677247075737*^9, 3.3996774300889263`*^9}, {3.399677484347191*^9, 3.3996775023874426`*^9}, {3.399677549332741*^9, 3.399677615271891*^9}, { 3.399677735894576*^9, 3.3996777672696056`*^9}, {3.3996784832518454`*^9, 3.3996785029179077`*^9}, {3.399681155804068*^9, 3.39968115721102*^9}, { 3.3996814499039345`*^9, 3.3996814541873217`*^9}, {3.399716595768035*^9, 3.399716615335778*^9}}, FontSize->14], Cell[BoxData[ RowBox[{ RowBox[{"\[CapitalDelta]V", "[", "z_", "]"}], ":=", RowBox[{ RowBox[{ RowBox[{"V", "[", "z", "]"}], "-", RowBox[{"V", "[", "0", "]"}]}], "=", RowBox[{ RowBox[{ RowBox[{ FractionBox["\[Sigma]", RowBox[{"2", SubscriptBox["\[Epsilon]", "0"]}]], RowBox[{"(", RowBox[{"d", "-", RowBox[{"2", "z"}]}], ")"}]}], "-", RowBox[{ FractionBox["\[Sigma]", RowBox[{"2", SubscriptBox["\[Epsilon]", "0"]}]], "d"}]}], "=", RowBox[{ RowBox[{ FractionBox["\[Sigma]", RowBox[{"2", SubscriptBox["\[Epsilon]", "0"]}]], RowBox[{"(", RowBox[{"d", "-", RowBox[{"2", "z"}], "-", "d"}], ")"}]}], "=", RowBox[{ RowBox[{"-", " ", FractionBox[ RowBox[{"2", "\[Sigma]", " ", "z"}], RowBox[{"2", SubscriptBox["\[Epsilon]", "0"]}]]}], "=", RowBox[{"-", " ", FractionBox[ RowBox[{"\[Sigma]", " ", "z"}], SubscriptBox["\[Epsilon]", "0"]]}]}]}]}]}]}]], "Input", CellChangeTimes->{{3.3996678061549644`*^9, 3.3996679122446938`*^9}, { 3.3996680465034356`*^9, 3.399668048473012*^9}, {3.399668583149845*^9, 3.399668583228009*^9}, {3.399669260876313*^9, 3.3996692634556*^9}, { 3.3996693667208605`*^9, 3.399669405285164*^9}, {3.3996694448968506`*^9, 3.399669468704523*^9}, {3.3996701656279516`*^9, 3.3996701705207167`*^9}, { 3.3996704507860155`*^9, 3.399670451083023*^9}, {3.3996705563796716`*^9, 3.3996705633199058`*^9}, {3.399670614418331*^9, 3.3996706290491896`*^9}, { 3.3996706611871285`*^9, 3.3996706640632887`*^9}, {3.3996707433143444`*^9, 3.3996708248950143`*^9}, {3.3996751210765905`*^9, 3.3996751252659864`*^9}, { 3.3996751827608185`*^9, 3.3996751830578284`*^9}, {3.3996767542517166`*^9, 3.399676755267849*^9}, {3.3996770771472006`*^9, 3.3996770879182*^9}, { 3.399677141335478*^9, 3.399677230801992*^9}, {3.3996776272935143`*^9, 3.399677728797285*^9}}, FontSize->14], Cell[BoxData[ RowBox[{ RowBox[{"\[CapitalDelta]V", "[", "z_", "]"}], ":=", RowBox[{"-", " ", FractionBox[ RowBox[{"\[Sigma]", " ", "z"}], SubscriptBox["\[Epsilon]", "0"]]}]}]], "Input", CellChangeTimes->{{3.3996678061549644`*^9, 3.3996679122446938`*^9}, { 3.3996680465034356`*^9, 3.399668048473012*^9}, {3.399668583149845*^9, 3.399668583228009*^9}, {3.399669260876313*^9, 3.3996692634556*^9}, { 3.3996693667208605`*^9, 3.399669405285164*^9}, {3.3996694448968506`*^9, 3.399669468704523*^9}, {3.3996701656279516`*^9, 3.3996701705207167`*^9}, { 3.3996704507860155`*^9, 3.399670451083023*^9}, {3.3996705563796716`*^9, 3.3996705633199058`*^9}, {3.399670614418331*^9, 3.3996706290491896`*^9}, { 3.3996706611871285`*^9, 3.3996706640632887`*^9}, {3.3996707433143444`*^9, 3.3996708248950143`*^9}, {3.3996751210765905`*^9, 3.3996751252659864`*^9}, { 3.3996751827608185`*^9, 3.3996751830578284`*^9}, {3.3996767542517166`*^9, 3.399676755267849*^9}, {3.3996770771472006`*^9, 3.3996770879182*^9}, { 3.399677141335478*^9, 3.399677230801992*^9}, {3.3996776272935143`*^9, 3.399677728797285*^9}, {3.3996777862790904`*^9, 3.3996777868262386`*^9}}, FontSize->14], Cell["\<\ which again is the Right Answer (though I don't think it actually appears in \ the textbook. It follows pretty directly, however, from the same section in \ chapter 25 mentioned above). The careful student will note that there's a minus sign difference between \ what we have and what can be derived from the section in chapter 25. I leave \ you to ponder why that is and why it doesn't make what we did wrong.\ \>", "Text", CellChangeTimes->{{3.39966741512825*^9, 3.3996674251176033`*^9}, { 3.399667467328555*^9, 3.3996676457319546`*^9}, {3.399667680902662*^9, 3.39966776157478*^9}, {3.3996678033569746`*^9, 3.3996678040134892`*^9}, { 3.399668101354689*^9, 3.3996683970922503`*^9}, {3.3996684525856237`*^9, 3.3996685309018993`*^9}, {3.3996687855709105`*^9, 3.3996688155376005`*^9}, {3.399668962368433*^9, 3.399669065867295*^9}, { 3.39966924146135*^9, 3.3996692489178205`*^9}, {3.39966934961939*^9, 3.3996693532772894`*^9}, {3.399669419776095*^9, 3.3996694352518544`*^9}, { 3.3996698488002644`*^9, 3.399669939795403*^9}, {3.39967007805863*^9, 3.3996701314255247`*^9}, {3.3996701817600145`*^9, 3.399670444830204*^9}, { 3.3996704777669625`*^9, 3.399670489444125*^9}, {3.39967058412502*^9, 3.3996706083065186`*^9}, {3.399670854548047*^9, 3.399670880512096*^9}, { 3.399670982446048*^9, 3.399671012193197*^9}, 3.39967513337904*^9, { 3.39967518812263*^9, 3.3996752579212418`*^9}, {3.3996753842791185`*^9, 3.3996754879698343`*^9}, {3.3996755198915076`*^9, 3.399675645202244*^9}, { 3.399676702632211*^9, 3.399676746560379*^9}, {3.3996767820312023`*^9, 3.399677061983385*^9}, {3.399677119840378*^9, 3.399677132330985*^9}, { 3.399677247075737*^9, 3.3996774300889263`*^9}, {3.399677484347191*^9, 3.3996775023874426`*^9}, {3.399677549332741*^9, 3.399677615271891*^9}, { 3.399677735894576*^9, 3.399677839946493*^9}, {3.399716677711891*^9, 3.399716712759802*^9}, {3.399716746744306*^9, 3.399716804951582*^9}}, FontSize->14] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Solution from Gauss' Law (the short way)", "Section", CellChangeTimes->{{3.39966741512825*^9, 3.3996674251176033`*^9}, { 3.399667467328555*^9, 3.3996676457319546`*^9}, {3.399667680902662*^9, 3.3996677917117167`*^9}, {3.3996680269796324`*^9, 3.399668030168466*^9}, { 3.399677861113304*^9, 3.399677863927208*^9}, {3.399677959834436*^9, 3.3996779689014597`*^9}}], Cell["\<\ See section 23-8 for the field inside two parallel plates, it's only a few \ paragraphs of thinking plus a few lines of math. Add to this eqn 25-7 \ relating the potential and the field and you're done. Gauss is your friend.\ \>", "Text", CellChangeTimes->{{3.39966741512825*^9, 3.3996674251176033`*^9}, { 3.399667467328555*^9, 3.3996676457319546`*^9}, {3.399667680902662*^9, 3.39966776157478*^9}, {3.3996678033569746`*^9, 3.3996678040134892`*^9}, { 3.399668101354689*^9, 3.3996683970922503`*^9}, {3.3996684525856237`*^9, 3.3996685309018993`*^9}, {3.3996687855709105`*^9, 3.3996688155376005`*^9}, {3.399668962368433*^9, 3.399669065867295*^9}, { 3.39966924146135*^9, 3.3996692489178205`*^9}, {3.39966934961939*^9, 3.3996693532772894`*^9}, {3.399669419776095*^9, 3.3996694352518544`*^9}, { 3.3996698488002644`*^9, 3.399669939795403*^9}, {3.39967007805863*^9, 3.3996701314255247`*^9}, {3.3996701817600145`*^9, 3.399670444830204*^9}, { 3.3996704777669625`*^9, 3.399670489444125*^9}, {3.39967058412502*^9, 3.3996706083065186`*^9}, {3.399670854548047*^9, 3.399670880512096*^9}, { 3.399670982446048*^9, 3.399671012193197*^9}, 3.39967513337904*^9, { 3.39967518812263*^9, 3.3996752579212418`*^9}, 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