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.