![\"3D](\"http:\/\/pages.vassar.edu\/magnes\/files\/2014\/04\/3Dvectorplot.jpg\")
<\/a>3D vector plot of the magnetic field of a magnetic dipole<\/p><\/div>\n
Even zoomed in and seen from a more right angle, this field is not helpful.<\/p>\n
<\/a>Same plot as above, but seen zoomed in and in the xz plane<\/p><\/div>\n
Since this approach didn’t work well, for reasons discussed more in my next Conclusion post, and my previous attempts at plotting the vector field at representative points was getting complicated as well, I decided to try to use this new transformation by substitution method to try Contour plots instead.<\/p>\n
I began by forming one expression for the magnetic field value at points in space, with no vector information. \u00a0This expression ends up being pretty manageable, and the contours of this can be made effectively using ContourPlot3D. \u00a0Some different results are shown below, taken from my Mathematica file “3D_contour_graphs”.<\/p>\n
<\/a>3D contour of my expression for the magnitude of the magnetic field of a magnetic dipole. This contour is for when the magnetic field equals 1<\/p><\/div>\n
<\/a>The same plot as above, but this time zoomed in and shown “cut in half” to see what happens on the inside of the symmetric circular outer portion<\/p><\/div>\n
These results are pretty promising, so I then added in a few more contours, to see how the different values looked relative to one another.<\/p>\n
<\/a>Similar plot to the above, but this time with three contours. From the inner to outer contours, these surfaces represent the places where the magnitude of the magnetic field equals 0.75, 0.5, and 0.28 respectively.<\/p><\/div>\n
This is pretty good, but where are these surfaces in relation to the small current loop that is supposed to be creating these magnetic field contours? \u00a0My next few contours include a small yellow ring at the origin, which indicates the placement of the loop, and also plots both positive and negative contours, which gives the whole picture both above and below the ring (which, looking at the original equation, should be symmetric).<\/p>\n
<\/a>The same three contours as above, along with their negative counterparts, and a small yellow ring representing the current loop that creates the magnetic field modeled.<\/p><\/div>\n
<\/a>A different view of the above graph, which allows one to look into the part of the contour that has been “cut open”.<\/p><\/div>\n
Now that there is a current loop in our models, what happens when the current in this loop is increased? \u00a0The above models were made with a magnetic dipole moment, $m=1$, but the below image increased the current so that $m=2$. Similar plot to above, but with a stronger magnetic field (m=2).<\/p><\/div>\n Putting images with $m=1$, $m=2$, and $m=3$ side by side shows that the same contours move farther away (and change curvature a bit) for models with larger m\u00a0<\/em>values, which is expected because larger\u00a0m<\/em> values indicate larger magnetic fields.<\/p>\n <\/p>\n From left to right, Contours with m = 1, m = 2, and m = 3.<\/p>\n This is all pretty good, but what are these contour surfaces actually telling us about the magnetic field? \u00a0These contour surfaces represent places where the magnitude of the magnetic field is at a constant value, so these shapes represent equipotential surfaces. \u00a0While equipotential surfaces are not often talked about when talking about magnetic fields, the general principle that field vectors are perpendicular to equipotential surfaces is true of magnetic fields as well as electric. \u00a0Therefore, these contours actually tell us something about the magnetic field of a dipole, albeit in an indirect manner.<\/p>\n To try and get a better picture of what the field lines look like around this current loop, I plotted these contour surfaces in 2D by removing any y dependence, so that the 2D contours plotted are shown in the xz plane. \u00a0This doesn’t lose much information again because of the $\\phi$ independence of the magnetic field of a dipole. \u00a0The result is shown below, which is taken from my Mathematica file “2D_contour_graphs”.<\/p>\n 2D contour, showing 20 different contour lines, both positive and negative.<\/p><\/div>\n This gives us a representation where the vector field lines may be easier to picture, and easier to compare to the lines shown in Griffiths’ “Introduction to Electrodynamics” 5.4.3 – pg. 255, which are shown in 2D. \u00a0It took some squinting, but I can see that lines perpendicular to these (and more) contour lines forms loops expected of a magnetic dipole. \u00a0I overlaid a few representative loops over my 2D contour image in paint to illustrate this idea.<\/p>\n
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