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.