(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 3416225, 120188]*) (*NotebookOutlinePosition[ 3417280, 120224]*) (* CellTagsIndexPosition[ 3417236, 120220]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{Cell[TextData[ "ScalarFields\n(Electric Potential)"], "Title", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Version 2.3\n2/18/92\nJohn B. Schneider"], "Subsubtitle", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Introduction"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ " This Notebook examines electric potential for static systems consisting \ of point charges. Static electric potential is one of the simplest fields in \ electromagnetics. It does not depend on time and, since it is a scalar \ field, the value at each point is merely a number (as opposed to a vector). \ Electric potential is also quite easy to interpret physically. Most people \ can imagine moving a small positive test charge through a potential field. \ If the charge has moved to a position where the potential is greater than \ where it started, then work has been done by you. You can, of course, think \ of this in the opposite way. If you move a small test charge in a potential \ field and you have to do work to move the charge, then the potential of that \ charge has increased. Likewise, if the charge drags you around \[Dash] the \ charge is doing work on you \[Dash] then the charge will lose some of its \ potential.\n\n Despite the fact that the static electric potential is so \ simple to understand, it can still be difficult to visualize. It is \ important that you have a good feeling for static, scalar fields before we \ consider time-varying or vector fields. We will examine all the common ways \ that are used to depict these fields. Several functions are used in this \ notebook that are not predefined Mathematica functions. These functions are \ defined in the subsection entitled \[Trademark]Initialization Cells,\ \[Integral] at the end of this NoteBook."], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Surface Plots"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Introduction"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ " We will begin by considering the electric potential about a single \ point charge, a monopole, located at the origin. The potential field for \ this charge exists everywhere in the space about the charge. The challenge \ is to represent this potential in such a way as to get a clear picture of its \ behavior. Probably the best way to do this is to employ surface plots. With \ a surface plot, the positions where the potential is determined are in a \ given plane \[Dash] say the plane define by ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " = 0. We then plot the potential as a function of the two variables which \ fix our position in that particular plane. If we were considering the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" = 0 plane, then specifing the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["z", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " values would uniquely determine a point, and we plot our potential at \ that point as a height above (or below) a plane.\n \n After the \ monopole, we will examine the potential about a dipole and a quadrupole. \ There is a short animated sequence at the end of the dipole subsection which \ illustrates how the potential varies as the evaluation plane (the plane over \ which the potential is measured) is moved.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Monopole"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ " The following Mathematica commands generates a surface plot of the \ potential about a single positive charge. In these commands, the function ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Monopole[x,y,z]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" gives the electric potential at a point (", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[",", Evaluatable->False, AspectRatioFixed->True], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[",", Evaluatable->False, AspectRatioFixed->True], StyleBox["z", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ ") when the point charge is located at the origin. We restrict ourselves \ to the plane that has an ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" value of zero, and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["z", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" each vary over the range \[PlusMinus]4 to 4 (", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["=0, \[PlusMinus]4 ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[LessEqual] ", Evaluatable->False, AspectRatioFixed->True], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[LessEqual] ", Evaluatable->False, AspectRatioFixed->True], StyleBox["4, \[PlusMinus]4 ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[LessEqual] ", Evaluatable->False, AspectRatioFixed->True], StyleBox["z", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[LessEqual]", Evaluatable->False, AspectRatioFixed->True], StyleBox[" 4). We are not concerned with units yet so we will say that ", Evaluatable->False, AspectRatioFixed->True], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["z ", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ "vary over the range \[PlusMinus]4 anythings to +4 anythings. If we were \ to pay attention to units, that would not change the form of the output \ \[Dash] we would just have to be more careful in our labeling. If you want \ more details about the following Mathematica command, you can double click on \ the cell directly below the plot. To see a diagrammatical representation of \ the monopole and the evaluation plane, click on the cell below the \ explanation cell. ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Note:", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[ " If you don't know how to use the following command to generate output, \ you should review the Introduction NoteBook!", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "charge = Pi Epsilon0;\nlocation = {0.,0.,0.};\nPlot3D[ \ Monopole[charge,location,{0,y,z}], {y,-4,4}, {z,-4,4},\n ClipFill -> None, \ Lighting -> True, PlotPoints -> 31,\n PlotRange -> {Automatic, Automatic, \ {-.2,.2}},\n BoxRatios -> {1,1,1}, AxesLabel -> {\"y\",\"z\",\"V\"},\n \ PlotLabel -> \"Fig. 1\"]"], "Input", AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "Double click on this cell for information about the Mathematica statement \ used to generate Fig. 1."], "Help", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[" The first line sets the variable ", Evaluatable->False, AspectRatioFixed->True], StyleBox["charge", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " to a convenient value \[Dash] this gives the strength of the point \ charge. The second line sets the variable ", Evaluatable->False, AspectRatioFixed->True], StyleBox["location", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " to the origin. The third line contains a plotting command that plots ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Monopole", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" as a funtion of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["z", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[". All the entries in the command with an arrow (", Evaluatable->False, AspectRatioFixed->True], StyleBox["->", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ ") specify options.\n\n The options only need to be specified if you \ want to change their default values. For the above plot, the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["ClipFill", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " option tells Mathematica to leave the truncated values blank. Setting \ the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Lighting", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" option to ", Evaluatable->False, AspectRatioFixed->True], StyleBox["True", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " shades the plot as if it were illuminated by a real light source. If \ this was omitted or set to ", Evaluatable->False, AspectRatioFixed->True], StyleBox["False", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ ", the shading would be determined by the height of the surface. The ", Evaluatable->False, AspectRatioFixed->True], StyleBox["PlotPoints", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " option tells Mathematica how many points to use along the two independent \ variable axes. The ", Evaluatable->False, AspectRatioFixed->True], StyleBox["PlotRange", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " option is set so that Mathematica will use what it \[Trademark]thinks\ \[Integral] are the best ranges for the two independent axes and the range of \ the dependent variable is set to \[PlusMinus].2 to .2 (for reasons of \ comparison with plots that we will generate later). The ", Evaluatable->False, AspectRatioFixed->True], StyleBox["BoxRatios", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " option sets all sides of the cube which contains the surface plot to have \ equal lengths. The ", Evaluatable->False, AspectRatioFixed->True], StyleBox["AxesLabel", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["PlotLabels", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " option does the obvious \[Dash] they provide a labels for the axes and \ the plot.\n\n Note that Mathematica has a little bit of difficulty \ plotting the truncated edge of the function. The edge of the \ \[Trademark]volcano\[Integral] should be a circle, but since we are using a \ finite number of points to plot this function, some jaggedness is present in \ the plot. The jaggedness could be further reduced by increasing the number \ of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["PlotPoints,", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " but this would come at the cost of increased computation time. The \ number of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["PlotPoints", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " chosen here more than adequately illustrates the form of the function. \ On some of the following plots the jaggedness where the function is truncated \ becomes more pronounced, and you may want to increase the number of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["PlotPoints", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" \[Dash] feel free to do so, but be prepared to wait.\n ", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData[ "Double click on this cell for a diagrammatical representation of the \ monopole and the evaluation plane."], "Help", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Show[ EvaluationPlane[0.], axes, mpoint, \n Lighting -> False,\n \ AxesLabel -> {\"x\",\"y\",\"z\"}]"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox[ " In this sketch, the evaluation plane is shown as the shaded plane which \ runs through the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["z", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " axes. The charge is represented as a sphere at the origin. For the \ purpose of illustration, this sphere is given a radius which is clearly \ non-zero. For a true point charge, the charge is completely contained in a \ sphere of vanishingly small radius.\n ", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[{ StyleBox[ " Notice that Fig. 1 looks something like a volcano. In fact, the \ surface plot should go to infinity at the origin. Since a function \ obviously can't be plotted all the way to infinity, the plot is truncated at \ a \[Trademark]reasonable\[Integral] value.\n \n Given this suface \ plot, you can get a better feel for what potential represents. Think of \ yourself as a positive test charge. Your potential is given by your height \ above the \[Trademark]ground\[Integral] plane. This means that if you want \ to increase your potential, no matter where you are in this plane, you would \ have to move toward the center of the plane (meaning the origin in this \ case). You would have to climb up the hill, doing work. If you want to \ decrease your potential, you can let the field do the work by moving away \ from the center and just slide down the hill. Sliding down the hill means \ that your potential decreases as you move out from the center, while in \ coordinate space you would be moving away from the origin and staying in the \ ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " = 0 plane. You must keep in mind that the diagram above is just for \ points in the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " = 0 plane. If you move out of this plane, a new surface plot may pertain \ to the description of your potential.\n\n To illustrate what would happen \ if you were in the plane ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " = 2, go to the Mathematica command which generated Fig. 1, and change the \ value of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" in the variable ", Evaluatable->False, AspectRatioFixed->True], StyleBox["location", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" to 2 (so that the second line reads ", Evaluatable->False, AspectRatioFixed->True], StyleBox["location = {2.,0.,0.};", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ "). After you have done this, hit the Enter key (not the Return key); wait \ for the output; after giving the output the once over, return here and read \ on.\n\n Notice that the volcano has been replaced by a smooth bump. \ The potential no longer goes to infinity at the origin because the charge is \ not located at the center of this plane. When you are at the \[Trademark]", Evaluatable->False, AspectRatioFixed->True], StyleBox["center\[Integral]", Evaluatable->False, AspectRatioFixed->True], StyleBox[" of the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " = 2 plane, you are at the point (2,0,0) but the charge is at the origin \ (0,0,0) \[Dash] you are still a distance of 2 away from the charge. You \ should also notice that, once again, this plot tells us that no matter where \ we are in this plane, if we want to increase our potential, then we must move \ towards the center. But, keep in mind, that this only tells us how things \ behave in the plane defined by ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x ", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ "= 2. It doesn't necessarily tell us the best direction in general in \ which to move to increase our potential. It only tells us the best direction \ in this particular plane. In fact, for the simple case of a single positive \ charge at the origin, it should be clear that moving directly toward the \ charge at the origin increases potential most rapidly. We will consider this \ more when we look at electric fields. ", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Dipole"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ " A static electric dipole consists of two charges that have the same \ magnitude and opposite sign that are a fixed distance apart. We will \ consider the case where there is a positive charge +Q at the point (0,0,1) \ and a negative charge \[PlusMinus]Q at the point (0,0,\[PlusMinus]1). \n \ \n In the following Mathematica commands, the function ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Dipole[x,y,z]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" serves the same role as the function ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Monopole[x,y,z]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " did in Fig. 1. The dipole function gives the potential at a point (", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[",", Evaluatable->False, AspectRatioFixed->True], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[",", Evaluatable->False, AspectRatioFixed->True], StyleBox["z", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ ") about a dipole. We will start by generating a surface plot of the \ potential in the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " = 0 plane. If you want more information on the Mathematica commands that \ generate this plot, go back to the cell following Fig. 1 (in \ \[SZ]Surface_Plots.Monopole). If you want the diagrammatical representation \ of the physical distribution of charge and the evaluation plane, double click \ on the cell following the Fig. 2.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "charge = Pi Epsilon0;\nlocation1 = {0,0,1};\nlocation2 = {0,0,-1};\n\ Plot3D[Dipole[ charge, location1, location2, {0,y,z}],\n {y,-4,4}, \ {z,-4,4},\n ClipFill -> None, Lighting -> True, PlotPoints -> 31,\n \ PlotRange -> {Automatic, Automatic, {-.2,.2}},\n BoxRatios -> {1,1,1}, \ AxesLabel -> {\"y\",\"z\",\"V\"},\n PlotLabel -> \"Fig. 2\"]"], "Input", AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "Double click on this cell for a diagrammatical representation of the dipole \ and the evaluation plane."], "Help", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Show[ EvaluationPlane[0.], axes, dpoints, \n Lighting -> False, AxesLabel \ -> {\"x\",\"y\",\"z\"}]"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox[" The evaluation plane is the shaded portion of the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " = 0 plane. The point charges are represented by spheres along the z \ axis. The positive charge is black and is above the negative charge (which \ is \[Trademark]dark\[Integral] gray) as seen from this angle.\n", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[{ StyleBox[ " There are a couple of important points about potential fields which are \ illustrated by this plot that could not have been easily demonstrated with \ the surface plots obtained from the monopole. First, you should see that \ there are now two singular points (points where the function blows up). As \ you approach the singularity at the positive charge, the potential goes to \ positive infinity; as you approach the negative charge, the potential goes to \ negative infinity. This points out that \[Trademark]zero \ potential\[Integral] is a relative value.\n \n Again, imagine \ yourself as a positive test charge. Even if you found yourself at a point of \ zero potential, you can still loose even more potential. For these plots, \ the potential is referenced to the potential found at point infinitely far \ from the point charges. If you were infinitely far away from these charges, \ we would say that you had zero potential. However, there are points close to \ the charges that also have zero potential. If you started infinitely far \ away and moved to one of these points, you would have gotten as much energy \ out of the system as you put into it. Also, in some cases, there is a path \ along which all points are at zero potential. This path is a line of \ equipotential at a potential of zero. In Fig. 2 there is an equipotential \ line at zero potential that bisects (at a right angle) the line joining the \ two charges. As a matter of fact, this line corresponds to the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" axis \[Dash] all points on the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " axis are at zero potential. This is visible in Fig. 2 as the straight \ line that runs from left to right, between the two singularities.\n \n \ The fact that the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" axis is an equipotential line means that once you are on the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" axis, you can move to any other point on the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" axis without expending any energy. On the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " axis, the negative charge attracts you towards the origin just as much as \ the positive charge repels you. Since you are being pulled in one direction \ just as hard as you are being pushed in the opposite direction, you can move \ about freely along that direction. Be careful not to confuse the attraction \ and repulsion of the charges along the y axis with the forces in another \ direction. If you move off the y axis, you will either gain or loose \ potential.\n \n If you were at the origin, you would have zero \ potential. You could have gotten there in one of two ways. You might have \ gone from infinitely far away, all the way to the origin, by moving just \ along the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " axis. In this case you never would have expended any energy. \ Alternatively, you might have arrived at the origin by moving along any path \ which includes some segments not on along the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " axis. In this case, along some stretches you will have to work to move \ against the field (go up hill) and along other segments the field will do \ work on you (go down the hill). But, no matter which route you take, when \ you arrive at the origin, you would have gotten as much energy out of the \ system as you put in.\n \n When you are at the origin, what will \ happen if you just let yourself go? If you were a positive test charge (and \ you have never been anything but), you would go sailing towards the negative \ charge at the point (0,0,\[PlusMinus]1). Despite the fact that you have zero \ potential, you still have plenty of enery to loose on your way down to a \ potential of negative infinity. You would go down the potential \ \[Trademark]hole\[Integral] that is quite visible in Fig. 2.\n \n \ As in the case of the monopole, the plot in Fig. 2 is only for one particular \ plane, the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " = 0 plane. Feel free to change the values in the equation for Fig. 2 to \ answer whatever other questions you might have about the potential field for \ a dipole. ", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Moving the evaluation plane"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ " If you felt no need to go back and change the values in the equation \ that generated Fig. 2, hopefully that was because you felt that you already \ understood everything, and not because you were being lazy. Well, whether \ you are on top of things or you are lazy, we will now piece together some of \ concepts involved here and generate a simple animated sequence to test your \ understanding of these static fields.\n \n We will consider the same \ dipole as used in Fig. 2 except that now the evaluation plane will be moved. \ The evaluation plane will be kept parallel to the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" = 0 plane but the location where it intersects the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " axis will vary. Specifically, we will move the evaluation plane between \ the planes ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" = 0 and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " = \[PlusMinus]2 in small enough steps that we can observe gradual \ changes in electric potential with this movement.\n \n Before you \ jump ahead to the animation, you should try to predict how the field varies \ with the motion of the evaluation plane. Also, to make sure you understand \ the physical layout which underlies this animation, be sure you first look at \ the animation in the cell for the diagrammatical representation. ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Note:", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[ " If you don't know how to control an animation, review the material in \ the Introduction NoteBook.\n \n An explanation of the Mathematica \ command used to generate the animations is available below.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "Double click on this cell for a diagrammatical representation of the moving \ of the evaluation plane"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ " The following command generates the frames for the animation that shows \ the movement of the evaluation plane. The positive charge is black and the \ negative charge is gray. To animate this, select the frames and hit Command \ y."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "For[x = 0, x >= -2., x -= 0.25,\n Show[ EvaluationPlane[x], axes, dpoints, \ \n Lighting -> False,\n PlotRange -> {{-4,4}, {-4,4}, {-4,4}},\n \ BoxRatios -> {1,1,1}, AxesLabel -> {\"x\",\"y\",\"z\"}] ]"], "Input", AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData[ "Double click on this cell for the electric potential as the evaluation plane \ is moved"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ " Since the frames in this animation take so long to generate, they have \ already been generated for you. You can view the animation by selecting the \ frames and then typing Command y. Feel free to vary the command that \ generated the frames and generate a new set. However, if you do this, you \ should be prepared to wait."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "charge = Pi Epsilon0;\nlocation1 = {0,0,1};\nlocation2 = {0,0,-1};\nFor[x = \ 0, x >= -2., x -= 0.25,\n Plot3D[Dipole[charge, location1, location2, \ {x,y,z}],\n {y,-4,4}, {z,-4,4},\n ClipFill -> None, Lighting -> True, \ PlotPoints -> 31,\n PlotRange -> {Automatic, Automatic, {-.2,.2}},\n \ BoxRatios -> {1,1,1}, AxesLabel -> {\"y\",\"z\",\"V\"}]]"], "Input", AspectRatioFixed->True], Cell[CellGroupData[{Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 1.08079 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations -0.026052 1.101837 0.027546 1.101837 [ [(-4)] 0.10149 0.24974 0.980657 1 Msboxa [(-2)] 0.22831 0.19616 0.888921 1 Msboxa [(0)] 0.36251 0.13949 0.797184 1 Msboxa [(2)] 0.50475 0.07947 0.705448 1 Msboxa [(4)] 0.65575 0.01581 0.613711 1 Msboxa [(y)] 0.31959 0.08564 0.797184 1 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The syntax of the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["For", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" command is ", Evaluatable->False, AspectRatioFixed->True], StyleBox["For[start_condition,test,increment,expression]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[". In these animations, the starting condition is ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x ", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["= 0 and the test is \[PlusMinus]2.0 ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[LessEqual] ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x\[Dash] the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["For", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" command will stop executing when ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " is no longer greater than or equal to \[PlusMinus]2.0. The increment \ command is ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x\[PlusMinus]=.25", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[". This is a shorthand notation for ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x ", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["= ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x ", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ "\[PlusMinus] 0.25 (this is the same notation used in the programming \ language C). The expression which the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["For", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" command executes is ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Plot3D", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[". In this case, the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["For", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" command starts with ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " = 0; it executes the plotting command; it then does the increment \ command, setting x = \[PlusMinus]0.25; finally, the test is done to see if \ execuation of the command should stop. Since x = \[PlusMinus]0.25 is \ greater than \[PlusMinus]2.0 the test is satisfied and the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["For", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " command cycles through the instructions again. It now increments ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " so that it equals \[PlusMinus]0.5. The plot is then redrawn with this ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" value and so on.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]]}, Open]], Cell[CellGroupData[{Cell[TextData["Quadrupole"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ " Hopefully, you now have a crystal clear understanding of surface plots \ of electric potential about a system of point charges. However, at the risk \ of overkill, we will go just one step further and discuss the quadrupole. A \ quadrupole consists of two positive and two negative charges, all with the \ same magnitude. These charges are placed at the corners of a square, with \ the positive charges on the corners linked by one diagonal, and the negative \ charges on the other. \n\n The command ", Evaluatable->False, AspectRatioFixed->True], StyleBox["NPoles", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " is used to plot the quadrupole potential. This command allows evaluation \ of the elctric potential at any point in space given any distribution of \ point charges. The previous commands that were used to find potential, ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Monopole", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Dipole", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[", were actually special cases of the command ", Evaluatable->False, AspectRatioFixed->True], StyleBox["NPoles", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[". ", Evaluatable->False, AspectRatioFixed->True], StyleBox["NPoles", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " is explained in greater detail below.\n \n The following is \ a plot of the electric potential about a quadrupole when the evaluation plane \ is taken to be a segment of the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " = 0 plane. Double click on the cell following the plot of the electric \ potential to see a diagrammatical representation of the quadrupole \ configuration. ", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "charge = Pi Epsilon0;\nlocation1 = {0,1,1};\nlocation2 = {0,1,-1};\n\ Plot3D[NPoles[{charge,-charge,charge,-charge},\n \ {location1,location2,-location1,-location2},{0,y,z}],\n {y,-4,4}, {z,-4,4},\ \n ClipFill -> None, Lighting -> True, PlotPoints -> 31,\n PlotRange -> \ {Automatic, Automatic, {-.2,.2}},\n BoxRatios -> {1,1,1}, AxesLabel -> {\"y\ \",\"z\",\"V\"},\n PlotLabel -> \"Fig. 3\"]"], "Input", AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "Diagrammatical representation of the quadrupole and the evaluation plane."], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Show[ EvaluationPlane[0.], axes, qpoints, \n Lighting -> False, AxesLabel \ -> {\"x\",\"y\",\"z\"}]\n"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox[ " The positive charges are located at the upper-right and lower-left \ corners of the square formed by the four charges and are black. The negative \ charges are gray. The evaluation plane is the shaded portion of the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x ", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ "= 0 plane. As noted earlier, the charges are represented by spheres with \ non-zero radius, even though we are considering point charges.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["More information about the command NPoles."], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[" The function ", Evaluatable->False, AspectRatioFixed->True], StyleBox["NPoles", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " calculates the potential at an observation point given a distribution of \ point charges. The syntax is:\n\n ", Evaluatable->False, AspectRatioFixed->True], StyleBox["NPoles[{list_of_charges},{list_of_locations},{x0,y0,z0}]\n", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" \nwhere ", Evaluatable->False, AspectRatioFixed->True], StyleBox["list_of_charges", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " is a comma separated list of the strength of the charges in the system, ", Evaluatable->False, AspectRatioFixed->True], StyleBox["list_of_locations", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " gives the location of the charges as a comma separated list of ordered \ triplets (each triplet is itself a list of comma separated numbers \ corresponding to the location of the charge in cartesian coordinates), and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["{x0,y0,z0}", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" is the observation point.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[{ StyleBox[ " The potential about a quadrupole has one additional feature that was \ not present in the potential about the monopole or the dipole. If you look \ at the center of Fig. 3 (which corresponds to the origin), you see that the \ potential there is zero. Unlike the line of zero potential we saw along the \ ", Evaluatable->False, AspectRatioFixed->True], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " axis for the dipole, if you were located exactly at the origin, you would \ stay there. There is no net force trying to move you. However, you would be \ in a highly unstable position \[Dash] a slight movement in any direction \ other than along the line joining the two positive charges, and you would \ tend to fall down one of the holes. The origin is known as a saddle point. \ It is neither a minimun nor a maximum, but the directional derivatives there \ are zero. It is as if the origin was an infinitesimally small flat spot \ between two mountains and two canyons.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]]}, Open]], Cell[CellGroupData[{Cell[TextData["Contour and Density Plots"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Introduction"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ " Contour and and density plots are another way in which potential fields \ are often displayed. These plots are used primarily to give you a clearer \ picture of the equipotential lines. This information is contained within the \ surface plots \[Dash] they are merely the line at a constant height. \ However, since the three-dimensional surface plot must be displayed in two \ dimensions, it is generally quite difficult to see just where the \ equipotential lines are. Contour and density plots are generally evaluated \ over the same regions for which you would use a surface plot \[Dash] that is \ to say a plane. Contour plots trace several lines of constant potential (or \ constant height from the surface plots) in the plane being considered.\n \ \n Density plots assign a color or shade to each point in the evaluation \ plane based upon the potential at that point. Density plots are good for \ giving a overall picture of the behavior of the field. However, there are \ some difficulties with density plots that can often limit their usefulness. \ The problems arise when you want to consider some of the details of the \ behavior of the field. It is difficult for the eye to judge differences in \ values by looking at change in shade or color. Also, rates of change are \ hard to ascertain from changes in colors. To get around this, density plots \ are usually accompanied by a key which tells you which shades or colors \ correspond to which values. Nevertheless, density plots are rarely used to \ examine any of the fine details of a field.\n \n We will start \ with the dipole since the monople is just too uninteresting for these plots. \ We will also examine the quadrupole using these plots. "], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Dipole"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Contour Plot"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[" Again, ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Dipole[x,y,z]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " will be used to describe the function being plotted. However, now we are \ displaying the result using a different Mathematica command. Instead of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Plot3D", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True], StyleBox["ContourPlot", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " will be used. The configuration of charge is the same as that used in \ Fig. 2.\n ", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "charge = Pi Epsilon0;\nlocation1 = {0,0,1};\nlocation2 = {0,0,-1};\n\ ContourPlot[Dipole[ charge, location1, location2, {0,y,z} ],\n {y,-4,4}, \ {z,-4,4},\n PlotPoints -> 45,\n ContourShading -> False,\n Contours -> \ 13,\n AxesLabel -> {\"y\",\"z\"},\n PlotLabel -> \"Fig. 4\"]"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox[ " For these contour lines, we have not bothered to label the potential to \ which each line corresponds. Mathematica not providing any simple means of \ accomplishing this. Usually a countour plot will provide information about \ the potential that each line represents. In Fig. 4, the line which cuts the \ plot is approximately the zero potential line which is the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" axis. The contour lines above the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " axis correspond to positive voltages, and those below correspond to \ negative voltages. Notice that the lines are more closely spaced around the \ charges where the potential changes most rapidly.\n \n The \ information which can be read directly from the contour plot is the behavior \ of the equipotential lines. If you were a test charge and you moved along \ one of these lines, you would neither gain nor loose energy. The line you \ move on would be at a constant height (or depth) on the surface plots. ", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Density Plot"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ " Density plots can be very useful tools when examing the \[Trademark]big \ picture\[Integral] of the behavior of a field, but these plots generally do \ not give information about detailed behavior. Limitations of the device on \ which a density plot is displayed can also inhibit its usefulness. \n \ \n Shown below is the density plot for the dipole of Figs. 2 and 4. "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "charge = Pi Epsilon0;\nlocation1 = {0,0,1};\nlocation2 = {0,0,-1};\n\ DensityPlot[Dipole[ charge, location1, location2, {0,y,z} ], \n {y,-4,4}, \ {z,-4,4},\n PlotPoints -> 45,\n ColorFunction -> Hue,\n Mesh -> False,\n\ AxesLabel -> {\"y\",\"z\"},\n PlotLabel -> \"Fig. 5\"]"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox[ " In Fig. 5, you can see some of the difficulties determining detail \ about a field from its density plot. Even if you were provided with a key \ that described the relationship between shade and field strength, it might be \ rather difficult to say were the zero potential line was. However, you can \ easily see approximately where the charges are.\n\n Figure 5 was generated \ from the same dipole function that we have used for all the other plots of \ dipole potential. The only thing that has changed is that the Mathematica \ function ", Evaluatable->False, AspectRatioFixed->True], StyleBox["DensityPlot", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " is now being used. This is significant in that one underlying equation \ describes all the dipole plots you are seeing \[Dash] despite the immense \ differences in the displayed form. (Feel free to look at the last section \ of this NoteBook, where all the Mathematica definitions of the \ electromagnetic equations are stored. The equations had to be translated \ into forms that Mathematica understands and they loose some of there elegence \ in the translation. Therefore, you are probably better off just looking in \ an introductory electromagnetics text book if you need a refresher on the \ form of some of these equations.)", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Combination Plots"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ " Of course, it is possible to generate a combination contour-density plot \ as shown here."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "charge = Pi Epsilon0;\nlocation1 = {0,0,1};\nlocation2 = {0,0,-1};\n\ ContourPlot[Dipole[ charge, location1, location2, {0,y,z} ],\n {y,-4,4}, \ {z,-4,4},\n PlotPoints -> 45,\n ColorFunction -> Hue,\n Contours -> 13,\ \n AxesLabel -> {\"y\",\"z\"}]"], "Input", AspectRatioFixed->True]}, Open]]}, Open]], Cell[CellGroupData[{Cell[TextData["Quadrupole"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Contour Plot"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ " Figure 6 shows the contour plot for the quadrupole. The system of charge \ is the same as used in Fig. 3."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "charge = Pi Epsilon0;\nlocation1 = {0,1,1};\nlocation2 = {0,1,-1};\n\ ContourPlot[NPoles[{charge,-charge,charge,-charge},\n \ {location1,location2,-location1,-location2},{0,y,z}],\n {y,-4,4}, {z,-4,4},\ \n PlotPoints -> 45,\n ContourLevels -> 11,\n AxesLabel -> \ {\"y\",\"z\"},\n PlotLabel -> \"Fig. 6\"]"], "Input", AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Density Plot"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ " The density plot for a quadrupole is shown in Fig. 7."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "charge = Pi Epsilon0;\nlocation1 = {0,1,1};\nlocation2 = {0,1,-1};\n\ DensityPlot[NPoles[{charge,-charge,charge,-charge},\n \ {location1,location2,-location1,-location2},{0,y,z}],\n {y,-4,4}, {z,-4,4},\ \n PlotPoints -> 45,\n Mesh -> False,\n PlotLabel -> \"Fig. 7\"]"], "Input", AspectRatioFixed->True]}, Open]]}, Open]]}, Open]], Cell[CellGroupData[{Cell[TextData["Parametric Plots (Spherical Coordinates)"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Introduction"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ " Parametric plots display the potential as a function of one or two \ parameters. These parameters indicate the evaluation point. The surface \ plots that we have already seen can be thought of as parametric plots. The \ parameters were the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["z", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" values of the evaluation point \[Dash] the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " value was fixed. Points in space can also be described by spherical \ coordinates. The location is be determined by a distance from the origin and \ two angles (see the NoteBook CoordinateSystems). If the distance from the \ origin is constant, the set of points described by varying the two angles is \ a sphere. We might be interested in the intensity of the fields over this \ sphere (note intensity is a magnitude \[Dash] it implies absolute value). \ The intensity of the field as a function of the two angles can be displayed \ as a distance from the origin. In cartesian coordinates we fixed the value \ of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" and varied ", Evaluatable->False, AspectRatioFixed->True], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["z", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ ". The field at any point was shown as a height above a plane. In \ spherical coordinates, the radius from the origin to the evaulation point is \ fixed and the two angles are varied. We display the intensity at the \ evaluation point as a distance from the origin (note that this distance is \ unrelated to the distance from the origin to the evaluation point).\n \n \ In this section we will consider the spherical parametric plots of the \ potential field about a dipole, quadrupole and octupole. In each of these \ cases the distance from the origin to the evaluation point is held fixed at 4 \ m. The monopole will not be shown because it is not very interesting. If we \ were to measure the potential a fixed distance away from a single charge (a \ monopole), what would we find? The potential would not vary as the angles \ specifying the evaulation point changed. The result would just be a sphere.\n\ \n Unfortunately the following plots take some time to generate. \ Unless you are patient, you probably don't want to tweak a parameter and run \ these yourself.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Dipole"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ " The parametric plot of the electric potential for a dipole is shown \ below. Remeber that the evaluation point is a fixed distance from the origin \ and the evaluation point is changed by changing the angles ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Theta]", Evaluatable->False, AspectRatioFixed->True], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Phi]", Evaluatable->False, AspectRatioFixed->True], StyleBox[ ". If a line were drawn from the origin to the evaluation point, it would \ form an angle ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Theta]", Evaluatable->False, AspectRatioFixed->True], StyleBox[" with the positive ", Evaluatable->False, AspectRatioFixed->True], StyleBox["z", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" axis. If this line were projected into the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["xy", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" plane, it would form an angle ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Phi]", Evaluatable->False, AspectRatioFixed->True], StyleBox[" with the positive ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " axis. (See the NoteBook CoordinateSystems for a further description of \ spherical coordinates.)\n \n In the plot below, the origin is at the \ center of the cube. The labels on the side of the cube show you the \ orientation of the axes but they are not the axes themselves \[Dash] the \ \[Trademark]real\[Integral] axes intersect at the center of the cube. The \ distance from the origin to the surface of the object in a particular \ direction indicates how strong the field is in that same direction, but keep \ in mind that we are evaluating the strength of this field 4m from the origin. \ You should be able to see that the magnitude of the potential is greatest on \ the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["z", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" axis (the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["z", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " axis contains the points on the surface most distant from the origin). \ In the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["xy", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " plane the potential goes to zero and hence the distance from the origin \ is zero in that plane. The reason the magnitude of the potential is greatest \ on the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["z", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " axis is because those points are as close to one charge as possible while \ being as distant from the opposite charge as possible. \n \n The \ reason the absolute value of the potential is used is because it is confusing \ to plot negative distances from the origin in spherical coordinates. If you \ are curious about why the absolute value is needed, feel free to delete the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Abs[", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" from the following command (you will also have to remove the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" associate with this command so that the \[Trademark]", Evaluatable->False, AspectRatioFixed->True], StyleBox["]]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[",\[Integral] at the end of the seventh line reads \[Trademark]", Evaluatable->False, AspectRatioFixed->True], StyleBox["],", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ "\[Integral]). 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FontSlant->"Italic"], StyleBox[" is the distance from the origin, and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Theta]", Evaluatable->False, AspectRatioFixed->True], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Phi]", Evaluatable->False, AspectRatioFixed->True], StyleBox[ " specify a direction from the origin. The syntax of the command is \n \ \n", Evaluatable->False, AspectRatioFixed->True], StyleBox[ "SphericalPlot3D[\n \ r[theta,phi],{theta,start,stop,step},{phi,start,stop,step}\n ]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox["\n \n The command ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Abs", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " takes the absolute value of its argument. This is done to insure the \ distance from the origin is non-negative. ", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[{ StyleBox[" Even though ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Theta]", Evaluatable->False, AspectRatioFixed->True], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Phi]", Evaluatable->False, AspectRatioFixed->True], StyleBox[ " are spherical coordinates that determine where the potential will be \ evaluated, we are not restricted to plotting the output only in spherical \ coordinates. We can create a surface plot where the two independent \ variables are ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Theta]", Evaluatable->False, AspectRatioFixed->True], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Phi]", Evaluatable->False, AspectRatioFixed->True], StyleBox[", and the dependent variable is the potential. 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Notice that it is harder to get a feel for the distribution of \ potential over the sphere in which we are interested.\n ", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "charge = Pi Epsilon0;\nlocation1 = {0,0,1};\nlocation2 = {0,0,-1};\nradius = \ 4.;\nPlot3D[\n Abs[Dipole[ charge, location1, location2,\n \ {radius*xloc[t,p], radius*yloc[t,p], radius*zloc[t,p]} ]],\n {t,0,Pi},{p,0,2 \ Pi},\n AxesLabel -> {\"theta\",\"phi\",\"Potential\"},\n PlotLabel -> \ \"Fig. 9\"]"], "Input", PageBreakAbove->True, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ " This surface plot differs from those shown previously in that the \ magnitude of the potential is plotted. You can see that the potential is \ independent of phi and it obtains its maximum value at theta equal 0 or ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Pi]", Evaluatable->False, AspectRatioFixed->True], StyleBox[ " (these points correspond to points on the postive and negative ", Evaluatable->False, AspectRatioFixed->True], StyleBox["z", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" axis.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Quadrupole"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ " The parametric plot for a quadrupole is shown in Fig. 10. Owing to the \ four charges in the quadrupole, there are four \[Trademark]lobes\[Integral] \ in the distribution of potential. 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Use options in the Actions Preferences dialog box to control when Unformatted text is generated.\ \>", "\<\ -Graphics3D-\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[ " Figure 11 is a surface plot presentation of the information contained in \ Fig. 10. Notice that it is easier to ascertain the physical distribution of \ potential from Fig. 10 than it is in Fig. 11."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "charge = Pi Epsilon0;\nlocation1 = {0,1,1};\nlocation2 = {0,1,-1};\nradius = \ 4.;\nPlot3D[\n Abs[NPoles[{charge,-charge,charge,-charge},\n \ {location1,location2,-location1,-location2},\n {radius*xloc[t,p], \ radius*yloc[t,p], radius*zloc[t,p]}]],\n {t,0,Pi},{p,0,2 Pi},\n PlotPoints \ -> 25,\n AxesLabel -> {\"theta\",\"phi\",\"Potential\"},\n PlotLabel -> \ \"Fig. 11\"]"], "Input", AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Octupole"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ " An octupole consists of eight charges that are located at the corners of \ a cube. Like charges appear on the diagonals of any face and the other \ diagonal will have charges of the opposite sign. A diagrammatical view of \ the distribution of charges about the origin is available in a cell below. \ The spherical parametric plot for a octupole is shown in Fig. 12. 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Use options in the Actions Preferences dialog box to control when Unformatted text is generated.\ \>", "\<\ -Graphics3D-\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData[ "Diagrammatical representation of the octupole."], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Show[ opoints, axes, AxesLabel -> {\"x\",\"y\",\"z\"}]\n"], "Input", AspectRatioFixed->True], Cell[TextData[ " The positive charges are black and the negative charges are gray. \ There is a positive charge at (1,1,1). The points that are one segment away \ from this charge are negative \[Dash] the charges at (1,\[PlusMinus]1,1), (\ \[PlusMinus]1,1,1), and (1,1,\[PlusMinus]1) are negative. You are viewing \ this diagramatical representation at the same angle used in the display of \ the parametric plot."], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[ " Figure 13 shows the surface plot representation of Fig. 12. However, in \ this case the absolute value is not taken so you can see which angular \ directions are associated with which types of charge. The positive \ \[Trademark]bumps\[Integral] correspond to the angular direction closest to a \ positive charge whereas the negative \[Trademark]bumps\[Integral] indicate \ the direction closest to a negative charge. "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "charge = Pi Epsilon0;\nradius = 4.;\nPlot3D[\n NPoles[\n { charge, \ -charge, charge, -charge,\n -charge, charge, -charge, charge},\n \ {{1,1,1},{1,-1,1},{-1,-1,1},{-1,1,1},\n \ {1,1,-1},{1,-1,-1},{-1,-1,-1},{-1,1,-1}},\n {radius*xloc[t,p], \ radius*yloc[t,p], radius*zloc[t,p]}],\n {t,0,Pi},{p,0,2 Pi},\n PlotPoints \ -> 30,\n AxesLabel -> {\"theta\",\"phi\",\"Potential\"},\n PlotLabel -> \ \"Fig. 13\"]"], "Input", PageBreakAbove->True, AspectRatioFixed->True]}, Open]]}, Open]], Cell[CellGroupData[{Cell[TextData["Initialization Cells"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "The initialization cells do the following:\n\t* Define values for constants \ and give default values to some variable used in the\n\t equations.\n\t* \ Define EM functions used in this section.\n\t* Define some graphics \ primitives used in the illustrative graphics."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["The ElectricPotential package."], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "This is a Mathematica package. It defines commands in a manner that allows \ the user to receive information about the command (via ?Command), and it \ keeps local variables hidden."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "BeginPackage[\"ElectricPotential`\"]\n\nDipole::usage = \ \"Dipole[q,p01,p02,p1] give the potential\nat the point p1 = {x1,y1,z1} due \ to a charge of q at point p01 =\n{x01,y01,z01} and a charge of -q at the \ point p02 = {x01,y02,z02}.\"\n\nDistance::usage = \"Distance[p0,p1] gives the \ distance between the\npoint p1 = {x0,y0,z0} and p2 = {x1,y1,z1}.\"\n\n\ Epsilon0::usage = \"Epsilon0 is the permittivity of freespace =\n8.5837 \ 10^-12.\"\n\nPotential::usage = \"Potential[q,r] gives the potential a \ distance r\naway from a charge of q (in Coulombs). The permittivity of free\n\ space is assumed.\"\n\nMonopole::usage = \"Monopole[q,p0,p1] give the \ potential\nat the point p1 = {x1,y1,z1} due to a charge of q at point p0 =\n\ {x0,y0,z0}.\"\n\nNPoles::usage = \ \"NPoles[{q1,q2,...,qn},{p01,p02,...,p0n},{x1,y1,z1}]\ngive the potential at \ the point p1 = {x1,y1,z1} due to discrete charges\nq1, q2, ... qn located at \ points p01, p02, ... p0n, respectively.\"\n\n\nBegin[\"`Private`\"]\n\n\ Epsilon0 = 8.8537 10^-12\n\nPotential[q_Real,r_Real] := \n N[q/(4 Pi \ Epsilon0 r)]\n\nDistance[{x_,y_,z_},{x0_,y0_,z0_}] := \n Sqrt[(x-x0)^2 + \ (y-y0)^2 + (z-z0)^2 ]\n\n(* Monopole calls NPoles with the proper syntax for \ a single\n charge. Monopole provides a slightly more intuitive syntax\n \ than NPoles. *)\nMonopole[q_Real, p0_List, p1_List] := \n \ NPoles[{q},{p0},p1]\n\n(* Dipole calls NPoles with the proper syntax for two \ charges of\n equal magnitude and opposite signs. Dipole provides a \ slightly\n more intuitive syntax than NPoles. *)\nDipole[q_Real,p01_List, \ p02_List, p1_List] := \n NPoles[{q,-q},{p01,p02},p1]\n \t\n\ NPoles[qList_List, p0List_List, p1_List] :=\n Block[{i},\n \ N[Apply[Plus,\n Table[Potential[qList[[i]], Distance[ p0List[[i]], p1 \ ]],\n {i,1,Length[qList]}]\n ]\n ]\n ]\n \t\nEnd[(* \ \"`Private`\" *)]\n\nEndPackage[(* \"ElectricPotential`\" *)]"], "Input", InitializationCell->True, AspectRatioFixed->True], Cell[OutputFormData["\<\ \"ElectricPotential`\"\ \>", "\<\ ElectricPotential`\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True], Cell[OutputFormData[ "\<\ \"Dipole[q,p01,p02,p1] give the potential at the point p1 = {x1,y1,z1} \ due to a\\ charge of q at point p01 = {x01,y01,z01} and a charge of -q at the point \ p02 =\\ {x01,y02,z02}.\"\ \>", "\<\ Dipole[q,p01,p02,p1] give the potential at the point p1 = {x1,y1,z1} due \ to a charge\\ of q at point p01 = {x01,y01,z01} and a charge of -q at the point p02 =\\ {x01,y02,z02}.\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True], Cell[OutputFormData[ "\<\ \"Distance[p0,p1] gives the distance between the point p1 = {x0,y0,z0} \ and p2 =\\ {x1,y1,z1}.\"\ \>", "\<\ Distance[p0,p1] gives the distance between the point p1 = {x0,y0,z0} and \ p2 =\\ {x1,y1,z1}.\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True], Cell[OutputFormData[ "\<\ \"Epsilon0 is the permittivity of freespace = 8.5837 10^-12.\"\ \>", "\<\ Epsilon0 is the permittivity of freespace = 8.5837 10^-12.\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True], Cell[OutputFormData[ "\<\ \"Potential[q,r] gives the potential a distance r away from a charge of \ q (in\\ Coulombs). 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