Some features of real capacitors are sometimes ignored for pedagogical purposes, notably fringing fields. I intend to model a capacitor and its features, such as energy storage and capacitance, while making as few assumptions as possible, for example by paying attention to fringing effects (though assumptions, approximations, and numerical techniques will still be used as needed).
Methods and Comments:
Mathematica 9 will be used as the only computational tool to model a 3-dimensional capacitor. First, I intend to model the electric field of a rectangular conducting plate with uniform surface density σ. Then, using the principle of electric field superposition, I will find the combined field of this plate and that of an oppositely charged, geometrically identical plate offset by a small distance normal to the plane (i.e. a parallel plate capacitor). These fields will be based on Coulomb’s Law for a surface charge density:
In addition, 3 approximations will be used: the simple capacitor field approximation to be quickly sure that the above results are correct, an electric field built using Mathematica’s NIntegrate function, which may be needed if the default Integrate fails to evaluate efficiently, and an approximation of my own design I call the “grid model,” whereby a charged sheet is replaced by a fine grid of hundreds or thousands of appropriately charged point charges. This may be necessary for oddly-shaped capacitors, and if it is sufficiently accurate, to “fill in the gaps” where other methods take too long and/or are inaccurate. Mathematica’s Timing function will be particularly useful for determining the efficiency of methods.
After the electric field computations have been done, the effects of a dielectric sandwiched between the plates can be analyzed using the principles of Griffiths, chapter 4. For example, once the electric field at a set of points in the capacitor has been found, one can, assuming linear dielectrics, apply to find the electric displacement, and hence the energy stored in the configuration from the equation . This can be compared to stated values by capacitor manufacturers.
My exact-value method based on Coulomb’s law works for a charged rectangular sheet in the xy-plane. At a point directly above the plate’s center at a distance reasonable for a capacitor (1% the width), the integral takes 9 seconds to evaluate; not unreasonable if time and ambition are budgeted. However, at a point half-way between the center and a corner, the program evaluated after 10 minutes, demonstrating the complexity of a seemingly simple integral. Using the grid method for a configuration of 40,000 point charges, however, yielded results that were 99.998% accurate in less than three seconds; the accuracy was barely lower for a 10,000 charge grid, which took half a second to evaluate. The grid method was also much more accurate than the infinite plane approximation taught in introductory physics, accounting for the finite dimensions of the sheet.
– Week 1: Further refine, test, and expand my existing models to work for more arbitrarily shaped plates.
– Week 2: Model the properties of simple parallel plate capacitors of rectangular and circular shapes.
– Week 3: Model the properties of more complicated capacitors, for example rolled up capacitors or variable capacitors.
– Week 4: Possibly account for more complicated properties of dielectrics, such as non-linearity. If this proves unmanageable, I may simply extend the work of week three.
– Week 5: Compare more thoroughly my models to real capacitors based on information from manufacturers and/or other sources. Also, this week, for which no computational work is budgeted, can be used as a safety net if previous weeks do not run as smoothly as I anticipate.
– Griffiths “Introduction to Electrodynamics”, 4th Edition (this will be the primary source of information and equations that will be translated to Mathematica)
– Mathematica 9, Student Edition, including its extensive documentation.
– A relatively new edition of the “CRC Handbook of Chemistry and Physics” for values of the permittivities of different dielectric materials. Not yet procured.
– I am considering a resource on real capacitors, their energy storing capabilities, dimensions, and dielectric media to compare my values to the real objects I am modeling. However, at this early point in the project, this has not yet been procured nor will it likely be needed until week 3. Further, Griffiths has permittivity data, which may be sufficient.
– Through V-Apps, I may need to access more powerful computers than my 2011 laptop for intense computational work.