Category Archives: Advanced EM

Advanced Electromagentism (Phys 341)

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.

 

 

Share

John Loree Project Plan: 4/7

Sources and Resources:

1: Introduction to Electrodynamics by David J. Griffiths

2: http://web.mit.edu/mouser/www/railgun/physics.html

3: https://www.carroll.edu/library/thesisArchive/HarmonSFinal_2011.pdf

4: http://physics.wooster.edu/JrIS/Files/Rhoades_Web_Article.PDF

5: http://www.instructables.com/id/Rail-Gun-Linear-Accelerator/

6: Vassar College Laboratory technicians and resources as referred to by Professor Magnes

note: sources and citations may change as data is accrued and decided if needed

Models and Experiments:

The generated magnetic field in the railgun will be modeled using the biot-savart law, ampere’s law, and Faraday’s law. Upon creating and modeling the changing magnetic fields as a function of time, loop size and current, the force upon a slug will then be modeled using the Lorentz force law. As a result of these calculations, we can approximate the theoretical velocity at the end of the rail.

We then intend to build a functional  small scale railgun and test fire it. When testing, we will calculate the velocity upon leaving the rails, and the loss in energy of the projectile over its flight relative to the theoretical values calculated using the models from earlier in the project. Upon the calculation of relative efficiency and accuracy of theoretical models, we will compare our data & values to other railguns that have been produced.

Timeline:

April 16th: complete modeling of the magnetic field as a function of time using mathematica.

April 16th-23rd: construct small scale railgun for test fire with Elias Kim

April 20th: Model the force and velocities of the slug using mathematica.

April 21st: provide the theoretical velocity & range of our small scale railgun

April 22nd: post preliminary data on site using either LaTEX or Mathematica

April 27th: compare theoretical values of railgun to experimental values, upon calculation of relative efficiency of the two reactions, compare our model other railguns constructed.

April 30th and May 1st: prepare presentation and submit final blog

Collaborators:

I will be collaborating with Elias Kim in this project. While I am modeling the magnetic field, induction and force in the railgun, Elias will model the circuitry and current in the railgun. We will then collaborate to generate equations of motion, build the small scale gun and calculate its efficiency. We will regularly collaborate and discuss our projects. However as the project progresses, our roles may shift to split the work evenly between the two of us.

Share

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.

Share

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.

Share

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.

Share

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 $2Nlog_2(N)$.

Share

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.

Share

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.

Share

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.

Share

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.

Share