While I have made a lot of progress since the start of this project, I am disappointed that I don’t have more time to work on it because I made so much progress right at the end, finally figuring out techniques that worked well in Mathematica. If I had more time I would try to make some contours of multiple dipoles, and see how things cancel out. Admittedly this would be a lot harder to picture the perpendicular field lines, and the 3D structure would probably become more necessary. Its also possible that the contour technique I used here might not work well at all for multiple dipoles, but it would have been good to check it out a bit at least.

That being said, I did make some pretty cool 3D graphs of magnetic equipotential surfaces that scale as expected with current intensity. The 2D case helped greatly to show how these contour surfaces represent the magnetic field in a general way, partly because of the greater number of contours that could be shown at once in a plot. This 2D case is actually more helpful in my mind because the field of a single dipole is so symmetric in a lot of ways that 3D images actually just confuse the issue and get in the way of each other. If I got the chance to model multiple dipoles, I likely would have tried to keep their dipole moments (vectors normal to the center of the current loops) in the same plane (xz most likely, for consistency). This way, at least in some regions, the field would still be independent of $\phi$, and thus the analysis of such fields would be simpler.

The 3D graphical limitations of Mathematica could be seen on some of my contour plots, especially those with the current loop shown as well, where the surface disappeared around the xy plane. I expect this is due to the near horizontal nature of the contour at this point, and the rendering issues that come into play when plotting such an extreme surface. The limitations (or just my inability to utilize Mathematica to its full extent) of VectorPlot3D and similar functions are pretty obvious in my case as they didn’t work well at all for the vector functions that I was trying to plot.

Despite these limitations and my Mathematica difficulties, I think that I managed to convey a good amount of information about the fields (and the less generally useful equipotential surfaces) of a single magnetic dipole. Hopefully I also laid the groundwork for students in this class in the future to work on modeling magnetic dipoles, specifically multiple dipoles in complex configurations. I also learned a lot about the benefits, frustrations, and limitations that Mathematica brings to the table as a computational scientific tool.