Conclusion

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.

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One thought on “Conclusion

  1. nimongillo

    The first thing I would like to mention is I’m happy that I am getting to comment on your project as we had a lot of overlap. It is interesting to see the different methods we took as we tried to complete the same general goal. I can especially relate with wanting more time to work on the project. It was really only towards the end of my data work that I was able to get the commands working in a way that I wanted too. I said this in my own conclusion but to reiterate; I feel like Mathematica is an incredibly helpful program but it has a steep learning curve due to the copious amount of commands that can be used to accomplish only slightly different requests. I really wish we had known how similar our projects would end up being as a collaboration probably would have helped us ascend that curve quickly and maybe even made it even further in our projects than we expected.
    You did end up with some really cool 3D graphs in the end however. I really like the idea of using contours for modeling the field! That was a really clever way to breach the VectorPlot3D problems that I know I encountered as well. Furthermore it also made for a really nice visual of the shape of the field at different radii. I was modelling the Van Allen Belts in my project and this would have been a great way to show the the boundaries of these areas existing at particular magnetic field lines through their particular magnitudes. While this may have just been a method to compensate for the difficulty of working with the VectorPlot3D and TransformedField commands, I think it was a great way to represent the data.
    As for changing the initial equation from a Spherical to Cartesian coordinate system , my suggestion would still be the TransformedField command. I too used the equation of a magnetic field supplied by our textbook and using this command, it properly evaluated the equation unlike the CoordinateTransform command. In terms of the vector arrows I too had problems with the way Mathematica rendered the vectors so I wouldn’t worry about that too much. It would be really interesting to fiddle around with those settings to see how much they could be edited however.

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