Category Archives: Spring 2014

Project Plan: Electric & Magnetic Field Modeling

Sources:

  • Introduction to Electrodynamics by David J. Griffiths, Fourth Edition: Chapter 5

Model:

I will be creating models of the magnetic fields of a bar magnet, sphere and a cylinder. Each of these fields will be first be derived for continuous distributions and then modeled on Mathematica using its Vector Field plot function in 2 and 3 dimensions. Time-permitting, I will also model the magnetic fields for distributions that are not continuous (perhaps with varying current densities dependent on space J(r)). Example systems can be found in Griffiths’ Introduction to Electrodynamics exercises.

Timeline:

April 7-April 14: Complete project proposal and begin derivations

April 14-April 21: Alter project proposal as needed and complete derivations

April 21- April 28: Post derivations and begin Mathematica modeling

April 28-May 5: Final Mathematica modeling and combination with Peter’s Modeling

Collaborators:

I will be working with Peter Florio, who will derive and model the electric fields of the same distributions. We plan to compare our results side-by-side to observe the similarities and differences between our models and to notice the parallels in our derivations.

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Project Plan: Modeling Electric and Magnetic Fields

What will be modeled:

I will be modeling the electric fields of a bar magnet, cylinder and sphere. The derivations of the field geometries will be shown in a step by step process and then will be modeled with Mathematica. More complex systems taken from examples in Introduction to Electrodynamics may be used an modeled as well, time permitting.

Timeline:

April 7-April 14: Complete project proposal and begin derivations

April 14-April 21: Alter project proposal as needed and completed derivations

April 21- April 28: Post derivations and begin Mathematica modeling

April 28-May 5: Final Mathematica modeling and combination with Cedric’s Modeling

Collaborators:

I will be working with Cedric Chang who will derive and model the magnetic fields of the same objects that I model the electric fields of. At the end, both of our models will be combined to display both electric and magnetic fields of the shapes in question.

Sources:

Introduction to Electrodynamics by David J. Griffiths, Third Edition

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Project Proposal: Measuring and Modeling Physical RCL Circuits.

This project is based on an initial interest in the RLC circuit lab in 114 where students put together parts of a radio and demonstrated how inductance and capacitance was used to tune a radio. My end goal is to compare measurements across components of a real RLC circuit (radio) to the ideal values one might expect to find based on equations from our textbook that model these circuits. I would begin with a simple LC circuit, maybe constructed out of components that would later be found in the radio circuit (or whatever components are available to make the simplest LC circuit I can make measurements on). I would collect data such as potential across various components. An important part of the analysis would be proposing possible/probable causes of any discrepancy between expected values and experimental values. I would use an oscilloscope to make the measurements. Ideally I would use a computer oscilloscope and a laptop and set up in room 203A.

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Project Plan: Modeling Electromagnetic Fields for Spherical Objects

Sources

I will be utilizing Introduction to Electrodynamics, 4th Edition, by David J. Griffiths. Specifically, I will begin with Gauss’s Law, as defined by Griffiths on page 69:

$ \oint \! \textbf{E} \cdot \mathrm{d} \textbf{a} = \frac{1}{\epsilon_0} Q_{enc} $

Further, I will utilize the formula for the electric field of a point charge below (found on Griffiths page 72), which can be generalized for a spherical object:

$\textbf{E} = \frac{1}{4 \pi \epsilon_0} \frac{q}{r^2} \mathbf{\hat{r}} $

I will additionally work with the magnetic field for the spherical object. Griffiths (page 263) gives the average magnetic field due to uniform current over a sphere as:

$ \textbf{B}_{ave} = \frac{\mu_0}{4 \pi} \frac{\textbf{m}}{R^3}$

Where m is the total dipole moment of the sphere and R is the radius of the sphere.

I will be using Mathematica 9 as my modeling tool.

Plan of Action

I will begin by using the equations above to start with modeling the electric field of a point charge. From there, I will model the electric field for a hollow spherical object. I will create a manipulatable object in Mathematica for changes in radius and charge. I will then move on to modeling the average magnetic field for a spherical object, and attempt to create a manipulatable object akin to the one for electric fields. Next, I will model the electric and magnetic fields for concentric spherical objects, with the goal of ultimately coming up with a very liberal approximation for modeling the magnetic field of the Earth, if the Earth is thought of as several concentric spheres (due to the crust, mantle, and outer/inner cores). However, this will only occur if time permits, as will a preliminary examination of dielectrics.

Timeline

Week 1 (4/6-4/12): Work on the simplest case of a point charge, and learn to work within Mathematica

Week 2 (4/13-4/19): Work to create manipulatable object for electric field of sphere, and begin working on modeling the average magnetic field for a spherical object with uniform current density

Week 3 (4/20-4/26): Model electric and magnetic fields of concentric spherical objects, submit preliminary results on Tuesday on blog

Week 4 (4/27-5/3): Wrap up, submit final data and conclusion on Wednesday, dielectrics if time permits

Collaborators

I am working with Brian Deer, who is focusing on bar magnets, and Tewa Kpulun, who is focusing on cylindrical objects. We will be meeting weekly to discuss our progress, share Mathematica-related insights, and help each other in whatever ways we can.

 

 

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Project Plan: Modeling E and B Fields of a Toroidal Conductor to Explore Tokamak Plasmas

Goal: 

Model the electric and magnetic fields (and D and H) of a solid toroidal conductor with a current flowing through it.  Originally, I intended to model the current as a volume current and vary the aspect ratio of the torus and determine the effects on the fields, but my preliminary research has shown that it is more accurate to model the current as several helically wrapped linear currents, similar to a toroidal solenoid.  I will vary the number of turns (N) and observe the effects of this change.  I will make my decision of initial values based on the values of current tokamak safety factors (safety factor, q{r), describes the ratio of the number of times a given magnetic field line wraps around the torus in the toroidal direction to the number of times it wraps around the torus in the poloidal direction).  Ideally, I will eventually model the tokamak as a series of concentric tori, since these linear helical currents exist throughout the volume of the plasma.  Initially, I will model it as one current along the outside of the plasma, surrounding a conductor

Tentative Methods:

  • Determine the safety factor of, as well typical current through, a tokamak reactor, such as JET (the Joint European Torus).
  • Using the above values, use Maxwell’s Equations to derive expressions for E and B of a torus (expanding upon Griffiths 3rd Ed. Example 5.10); expand this to account for D and H since the interior plasma can be magnetized.
  • Consider how these quantities change as the safety factor of the torus is changed
  • Use Mathematica to model these fields as N changes.

Resources:

  • Griffiths Introduction to Electrodynamics, 3rd Edition
  • Journal article (to be determined – for  JET specifications)

General Notes:

I think that the most difficult part of this will be in creating the model in Mathematica, and getting it to do what I want.  I feel relatively confident about the ease of determining the values to use, and about deriving expression for E and B, though those are not trivial calculations.  Once I have the expressions, it will be relatively simple to vary q(r).

 Schedule:

7 April – 13 April: Research tokamak properties and determine current and size values.  Begin work to derive expressions for B.  Update Project Plan to account for comments.

14 April – 20 April (Tuesday 15 April: Updated Project Plan): Check expressions for B, and then find other fields.  Begin work on building Mathematica model.

21 April – 27 April  (Tuesday  22 April: Preliminary Results): Work on fixing issues with the Mathematica model, and make sure that it works and looks as desired.  Draft conclusion/interpretation of results.  For results, have expressions for all necessary quantities and have a first draft of model.

28 April – 4 May (Thursday 1 May: Begin Presentations): Refine model, and fix any remaining issues.  Elaborate on interpretation of results.  Begin reviewing classmate’s project.

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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.

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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 $2Nlog_2(N)$.

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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.

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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.

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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.

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