Project Plan – Modeling the Magnetic Field of Magnetic Dipoles using Mathematica

Sources and Resources

My main source will be Griffiths Introduction to Electrodynamics (Fourth Edition), especially pages 253-255, where the magnetic dipole is first introduced and some equations are derived.  Especially important is Eq. 5.88,

$\vec{B} = \frac{\mu_0 m}{4 \pi r^3} (2 cos \theta \hat{r} + sin \theta \hat{\theta})$

which gives the magnetic field of a pure dipole in spherical coordinates.  In this equation, $m$ is the magnetic dipole moment, as defined in Eq. 5.86 in Griffiths

$\vec{m} \equiv I \int d\vec{a}$

 Mathematica 9 will be my main tool for modeling and presenting my results.

Initial Plans

I plan to start with modeling a single magnetic dipole, whose magnetic field (in spherical coordinates) is given by Eq. 5.88.  Mathematica 9 only works in Cartesian coordinates for 3D vector fields, so the Transformedfield function in Mathematica is important for converting from spherical to Cartesian coordinates.  I will also perform this conversion by hand as a check.

One potential problem I notice already is the visibility of a 3D magnetic field model.  Perhaps it is just for ease in printing textbooks, but most magnetic fields shown in Griffiths are only shown in 2D with magnetic flux lines.  As I begin to make some models, I will experiment with the different visibilities when using 3D vs. 2D vector fields, as well as using vector arrows vs. magnetic flux lines.  It may be that a 3D vector field of arrows is too complicated to actually get a sense of what is going on.  Changes in colors, arrow sizes, and arrow density will also help visibility.  I plan to model a small loop at the origin to represent the magnetic dipole itself, but this may just crowd the model even more.

After the single magnetic dipole model is a bit more set, I will move on to superimposing many magnetic dipoles together to create bigger shapes, such as a bar magnet or horseshoe magnet.  I don’t understand much about how this will work in Mathematica, but the result will be the numeric approximation of what the magnetic fields of these bigger shapes look like.  I know that looping structures in Mathematica will be important, but I have to learn more about how these work and how they will work for my benefit.

Collaborators

I am collaborating with Ramy Abbady and Tewa Kpulun, whose projects are very similar to mine in that they are modeling electric and/or magnetic fields of relatively simple geometries (spheres, cylinders, etc.).  We will compare our techniques and presentation styles to see what the best way to present our results is, as well as support each other in learning some more intricate parts of Mathematica 9 coding.

Timeline

Week 1 (4/7-4/13):  I plan to begin modeling the magnetic field of a single dipole and trying out different presentation styles.

Week 2 (4/14-4/20):  I plan to finish my model of a single dipole and decide which presentation style works best.  I also plan on starting to work out Mathematica looping structures and how they can be used to model complex combinations of magnetic dipoles in my project.

Week 3 (4/21-4/27):  I plan on finishing my Mathematica looping education, and beginning to model bar magnets of different sizes and horseshoe configurations.  If at all successful, preliminary results of more complex configurations of magnetic dipoles will be posted to the blog.

Week 4 (4/28-5/4):  I plan on finishing my complex models, and beginning preparation of my finalized results for the blog.

Week 5 (5/5-5/11):  I plan on finishing my final results for the blog and reading and commenting on the results of my peers’ projects.

Project Plan: Modeling the E and B- fields of a Cylinder

Sources:

Introduction to Electrodynamics by David J. Griffiths

What am I Modeling:
I will be modeling the E and B fields for a simple cylinder and then I want to do the same thing for more complicated systems(i.e conductors, dielectrics, etc). I would love to finish my project by modeling the E and B fields for a coil.

Due Dates-
APRIL 14TH: Finish modeling the E and B Fields for a simple system
APRIL 21ST:Finish modeling for complex systems
APRIL 28TH: Finish modeling for coil
APRIL 29TH: Make sure that all animations work on my blog page
APRIL 30TH: Prepare my blog presentation
MAY 1ST: Submit final blog.

Collaborators:
I am collaborating with Ramy Abbady and Brian Deer. We have weekly meetings to talk about ways we could help each other and what we should expect the final product to be.

Project Plan (revised)

C. elegans Diffraction Pattern Modeling

Sources & Resources

  1. Introduction to Electrodynamics, D. Griffiths
  2. A Student’s Guide to Fourier Transforms, J. F. James (chapters 1 and 9)
  3. Past work by Professor J. Magnes and her assistants
  4. C. elegans nematodes, property of Professor K. Susman, photos taken by her laboratory assistants
  5. Introduction to Optics, 3rd edition, Pedrotti & Pedrotti & Pedrotti (chapter 11)

Necessary steps & what I plan to model

  • Take photos of the C. elegans with the Insight Camera (directly from the microscope)
  • Manipulate the photos (remove noise) using Mathematica
  • Take the Fourier transform of the photos to discover the diffraction pattern
  • ( |Fourier Transform|2 = the diffraction pattern)

Collaborators

I will be working alone, but with Professor K. Susman’s worms and equipment, and Professor J. Magnes’ old work as a reference. I also will be receiving some training in using the microscope’s camera attachment (most likely from one of Professor Susman’s lab assistants).

Tentative Timeline

Starting with week1= April 6-12 (Sunday to Saturday):

Week1: I plan to take photos (if I get permission from Professor Susman), and work on an algorithm for generating reliably noiseless photos. I plan to become comfortable with Fourier transforms on Mathematica, and take some preliminary transformation models.

Week2: From now on, it is going to progress in a semi-continuous fashion. This week I plan to continue working with the Fourier transforms on Mathematica, making improvements to the algorithm and recording the changes. Keeping in mind the goal of this project, it is important to take note of the models in the context of electromagnetic waves. It is also important to keep careful track of the data, and make sure the images are clear and easy to see.

Week3: By now I expect to have some good images. I will continue to improve on them, and possibly will begin resorting to doing the matrix multiplication “by hand” on Mathematica. I hope to have a very good understanding of Fourier transforms of images by the end of this project.

Week4: Now that I hopefully have good images corresponding to several shapes of C. elegans, I hopefully will be able to begin adding at least a small volume of genuinely good data to the Diffraction Symmetries Library.

Week5: In the final week, I will make some final touches to the library. Even though it is primarily about the data, I hope to polish the presentation and add any necessary comments. 

Some Preliminary Information

It is important to discuss what I already have knowledge of: I already know how to grow worms (how to transfer them to new E. coli food dishes so they can reproduce). I have a general idea how to take Fourier Transforms by hand, and what they generate. I do not know how to take a Discrete Fourier Transform of an image, and I do not know how to do it with matrix multiplication. I also know very little about electromagnetic Fraunhofer diffraction.

Some Essential Relationships

The crux of my thesis:

(1)   \begin{equation*}\left|FT|\right^2=Diffraction Pattern\end{equation*}

Basic Fourier Transform equation:

(2)   \begin{equation*} F(t)= \int_{-\infty}^{\infty} \Phi (v) {e }^{2 \pi i v t } dv \end{equation*}

(3)   \begin{equation*} F(t) \leftrightarrow \Phi (v) \end{equation*}

Discrete Fourier Transform in matrix form:

(4)   \begin{equation*}\begin{bmatrix} A(0)\\ A(1)\\ A(2)\\ ...\\ A(N-1) \end{bmatrix} = \begin{bmatrix} 1&1&1&...&1\\ 1&e^(\frac{2 \pi i}{n})&e^(\frac{4 \pi i}{n})&...&e^(2(N-1)\frac{\pi i}{n})\\ 1&e^(\frac{4 \pi i}{n}) &e^(\frac{8 \pi i}{n}) &...& e^(4(N-1)\frac{\pi i}{n})\\ ... & ... & ... & .... &...\\ 1 & ... & ... &...& e^((N-1)^2(\frac{\pi i}{n})) \end{bmatrix} \begin{bmatrix} a(0)\\ a(1)\\ a(2)\\ ...\\ a(N-1) \end{bmatrix} \end{equation*}

(It can be noted that this matrix multiplication requires $N^2$ multiplications. The Fast Fourier Transform (FFT) method, which can only be run by computing machines, reduces the number of multiplications from $N^2$ to $2Nlog_2(N)$.

Electrodynamics and Einstein’s first paper on relativity.

As many of you know probably know, Albert Einstein’s initial inspiration for his theory of relativity stemmed from his considerations of Maxwell’s equations. The connection is so important that the first paper he published on relativity was titled “On the electrodynamics of moving bodies (1905).” In particular he considered the relative motion of a magnet and a conductor. In the conductor’s rest frame the interaction between the two is electric, yet in the magnet’s frame they interact magnetically. I can model this problem very easily and then go about showing how Einstein resolved the seemingly paradoxical relativistic nature of classical electrodynamics. From there I can continue into discussing the field transformations and showing how these effect real life problems.

Project Proposal: Earth’s Van Allen Belts

Earth’s Van Allen Radiation Belts are areas of concentrated charged particles surrounding the planet due in part to its naturally generated magnetic field. These particles come from the Sun’s solar wind and are divided into two layers whose borders are the magnetic field lines. While these radiation belts are not unique to Earth, they are of utmost importance to us when considering safe space travel and the placement of satellites. The goal of my project is to use Mathematica and publicly supplied data to model the Earth’s magnetic field. Next, that model will be used to define the location of Earth’s Van Allen Belts. These experimental values will then be compared to the observed values of the belts in order to determine the accuracy of the model which can then hopefully be altered to determine the equivalent belts on other planets.

Modeling Electric and Magnetic Fields

For my project, I will work with Cedric Chang and study the electric and magnetic fields of a bar magnet, cylinder, and sphere. I will specifically be modeling the electric fields and will begin by deriving the Electric fields for each geometry using Gauss’s Law. Then I will use Mathematica to model the vector fields of each in three dimensions. Problems from David Griffiths’ Introduction to Electrodynamics, Third Edition may be used as examples of these kinds of geometries. These will be combined with the magnetic field of each object from Cedric and compared.

E&M Scenarios Reimagined in Relativistic Reference Frames

While this class has covered the electric and magnetic fields of stationary objects, it has not addressed the concept of objects in motion. In this project, I will first derive the relativistic transformation equations for electric and magnetic fields. Then, I will apply these equations to a number of scenarios that have already arisen in the stationary form, such as a parallel plate capacitor, a bar magnet, a solenoid, and a current-carrying wire. These scenarios will then be assessed at varying speeds.

Project Proposal: Modeling Capacitors

I intend to model the capacitance, energy storage capabilities, and fringing fields of capacitors. The assumption of infinite area will be discarded in favor of using analytic approaches whenever possible; for example, one could start with Coulomb’s Law for a surface charge density:

where  is the separation vector between a source point and the field point. From here, one can theoretically model the electric field of a sheet of arbitrary shape and charge density. I will start with a uniform charge density and a simply shaped sheet and work up to more complicated, realistic arrangements of positive and negative plates (i.e. capacitors, though not necessarily parallel plate capacitors only).

When a capacitor with a vacuum between the plates has been satisfactorily described, the effects of dielectrics placed between the plates could be modeled. Exotic dielectric media could be explored. If feasible, these realistic simulated capacitors could be incorporated into an RLC circuit model. As stated above, analytic solutions will be preferred, with approximations used only when the exact answer is computationally impractical. Approximations can, however, be used often as reality checks, as well as to compare the accuracy and computational requirements of approximations with those of the analytic solution.

Project Proposal: Modeling Optical Filters

When analyzing the brightness of a star, or a mode of a laser beam we observe the effects of that object. In this case we gather photons, and we use tools to gather as much radiation as possible. This data is then transmitted onto a screen to see a representation of what is actually going on. When an electric signal is sent to an oscilloscope, are we actually seeing the signal? A spectrum of light must enter through a series of optical and electrical things before being displayed, and those things can and do distort the image. These are optical filters. Sometimes this is done intentionally to block out certain frequencies, but other times the distortion is unavoidable. By understanding the convolution of electromagnetic waves one can isolate the desired data from the signal presented. I will model using Mathematica different spectra and examine how convolution and deconvolution work as means of setting up usable data.

Project Proposal: Electric & Magnetic Field Modeling

For my project, I will work with Peter Florio to model the electric and magnetic fields of a bar magnet, sphere and cylinder. I will be looking at the magnetic fields of these distributions in 3-space and modeling their vector fields using Mathematica and Maxwell’s Equations. Problems given in David Griffiths’ Introduction to Electrodynamics will be used as specific examples of these kinds of charge distributions. My results will then be compared to the electric fields of the same distributions found by Peter.