In the beginning, I set out to model the electric and magnetic field of any objects shaped like a cylinder. I wanted to demonstrate the direction and intensity for a conductor, a dielectric, and a coil. I did not get to model the E and B fields for all those different types of cylindrical objects, however I developed a method for modeling on mathematica. With the “Help” tab I was able to model the direction and intensity for a conducting cylinder.
To find the electric field I chose to look at the E and B-fields of a long cylinder first, where this cylinder had a uniform charge density. Using Gauss’ Law I derived the E field (inside the long cylinder) and started looking at different functions on mathematica, to determine which one best demonstrated the direction and magnitude of the electric field. I first converted the coordinates from cylindrical to cartesian using the TransformedField function. After playing around with the VectorPlot and VectorPlot3D I was able to show (using the Show function) the electric field of a long cylinder. The E-field for the long cylinder, from my previous post, demonstrate the electric field within the long cylinder, where the electric field magnitude is directly proportional to the radius of the Gaussian cylinder. So, it makes sense that the arrows are getting thicker instead of shrinking in size.
To find the magnetic field I also chose the same long cylinder, only this time, with a uniform current distributed on the surface of the cylinder. With this induced current flowing in the negative X direction, into the screen, I used the right hand rule to find that the magnetic field is in the positive phi direction. Using Ampere’s Law I came up with the equation for both inside and outside of the cylinder. As you can see in my previous post, the B-field equals zero inside of the cylinder and the B-field outside is inversely proportional to the radius of the cylinder. This is so because all of the current is on the surface of the long cylinder and as you move further away from long cylinder the intensity decreases.
I then used the same procedures, kept the charge density for the E-Field and uniform current for the B-field the same. However, the equations are different now because of the parameters I placed on the finite cylinder. This cylinder has a length of 30cm with a radius of 10cm. After putting that into mathematica, similar results to the long cylinder were produced.
My main focus for this project was to learn how to use mathematica because deriving the E and B fields for a cylinder are things we learn in class. Mathematica is frustrating at first but after figuring out the proper functions I was able to model what we learn in class.
Hi Tewa,
Now that I’ve opened up your two mathematica files, I feel like I can see your whole process. At first your blog posts are a little difficult to follow because of how quickly you move along. I can see this is because a great deal of your actual work (/trial and error) is hiding inside your mathematica files. Because of this I have a suggestion on your blog organization and presentation. It looks like a significant purpose and end result of this project was your increased familiarity and ability to apply the use of mathematica to physical concepts learned in class. I feel like the majority of your learning is evident inside the mathematica files, but your stripping down of the blog posts to the end/correct results hides much of this. I would definitely appreciate seeing some of this on your blog (though it is wonderfully presented within the files themselves).
Like I said, sometimes the fact that much of the math was either done by hand or in mathematica leaves me a little rushed along on the blog posts.
It is nice to see models of things we have been learning about for so long but never seen in 3D. Your growing familiarity with mathematica and how to manipulate these equations into useful models is evident over your blog. One thing I would suggest is if you are going to have a scale to the plots, label them. A good number of your models have numbers on the sides but I’m not sure what the units are or what they even mean. (I say this knowing how frustrated I was trying to get labels right on mathematica…)
As far as the physics goes, these models definitely look good, especially in the “infinite wire/cylinder” scenarios. One thing I wonder about reading your post is how you could use the power of mathematica to try and model fringe effects when the cylinder gets shorter. My understanding of the equations is that the field of a “long” cylinder should not look exactly the same as a finite/short cylinder. I can’t tell if trying to do this with mathematica would actually be harder than by hand due to its temperamental nature and learning curve, but it seems like there is a possibility it would be able to help model something that is very difficult to accurately portray.
-Matteo