# Preliminary Data: Capacitors

Summary of Progress up to this Point:

I have found the energy storage capability of a simple, slightly physically impractical, square parallel plate capacitor. Compared to the idealized model taught in introductory physics, this capacitor stores 94.34% the energy, the discrepancy likely caused by the finite dimensions of the plates. The accuracy of my model is likely high based on a comparison of the methods used and a sampling of values produced by the exact, theoretical model.

Details:

Recall Coulomb’s Law for a charged surface: $\large \vec{E}(\vec{r}) =\frac{1}{4 \pi \epsilon_0} \int \frac{\sigma\left(\vec{r'}\right)}{\mathfrak{r}^{2}}\hat{\mathfrak{r}} \: da'$. In Cartesian coordinates for a flat plate lying in the xy-plane at z=0, this can be more explicitly written as:

$\vec{E}(x,y,z)=\frac{1}{4 \pi \epsilon_0}\int_{y'_i}^{y'_f}\int_{x'_i}^{x'_f}\sigma\frac{(x - x', y - y', z - z')}{((x - x')^2 + (y - y')^2 + (z - z')^2)^{3/2}}\mathrm{d}x'\mathrm{d}y'$

This complicated multi-variable vector integral expression is actually executed correctly by Mathematica, giving the field at a point very close to the middle of a rectangular charged plate, or of any point fairly far away from the plate in a short time (~10 seconds at the most on my personal computer). However, at points very close to the plate at an arbitrary location (for example, half-way between the center and a corner) the program took several minutes to evaluate; not a practical outcome. Therefore, my method of splitting the plane into thousands of tiny point charges (which is computationally efficient everywhere) needed to be examined for accuracy.

This method is built on the simpler form of Coulomb’s Law for a point charge: $\vec{E}(\vec{r})=\frac{1}{4 \pi \epsilon_0}\frac{q}{r^2}\hat{r}$ but is applied across thousands of point charges that form a grid superimposed over the plane being studied (the limit as the number of tiny charges approaches infinity is the exact expression). This method is extremely accurate at a distance, but up close it breaks down as the quantization of charge is noticeable, unfortunate given that capacitors are being studied. Therefore, the field produced by two slightly different arrangements was examined. For example, a grid of 100×100 points was calculated and averaged with a field produced by a 101×101 grid. This was done because a point that is very near a charge for one configuration is commensurable far away from the nearest charges for the other configuration. Averaging these out produces a value very close (only 0.04% too high) to the field of the theoretical model.

Once the model was justified, the points being tested were selected. Only points half-way between plates were evaluated because getting to close to the grid resulted in inaccurate values for the E-field, despite the otherwise successful attempt above to “smooth out” the point charges. Sampling this narrow range was justified because the exact method showed that the field varies by less than a percent across the small separation gap between plates. The values for the charge placed on the plates was taken from a problem in Knight.

To save time for the heavy computation, symmetry was invoked. Recognizing that only one quadrant of the area between plates needed to be evaluated, and that only one plate needed to be examined (because plates are identical aside from their charge sign), the computation was performed in one eighth the time previously expected. Nevertheless, the computation of the electric field at 256 points close to the plate took 10 minutes and 18 seconds when run through the relatively fast and efficient servers available to Vassar students through VApps.

To find the energy stored within, I had to reverse engineer eq. 2.45 (Griffiths, 4th Ed.). Instead of using the integral that describes the energy in a capacitor $W=\frac{\epsilon_0}{2}\int E^2 (\vec{r})\mathrm{d}\tau$, I instead broke this integral into a sum:

$W=\frac{\epsilon_0}{2}\sum_{i}E^2 (\vec{r}_i)\Delta V$

where ΔV was the area in the immediate vicinity of the point sampled (for all 1024 of the evenly-spaced points, this was simply 1/1024 the total volume). This was necessary because I only have a quantized set of point at which I know the electric field, rather than an integrable function for the electric field.

In summary for the data gathered thus far, the energy stored in a 2 meter by 2 meter parallel plate capacitor with a separation of 2 centimeters is 0.2664 mJ. The basic approximation, where $\vec{E}_{ppc}=\frac{\sigma}{\epsilon_0}\hat{n}$, resulted in 0.2824 mJ.

Next Steps:

As of yet, no visual aids have been developed due to the numerical nature of the early stages of the project. As the first heavy computation has been done and shown to be a success, time can now be devoted to the following:

• Illustrating the fringing field at the very edge of a capacitor. This is likely one of the causes of the reduced energy storage observed compared to the infinite approximation, thus it would be an important concept to visualize.
• Running a few more capacitor simulations and plotting the effect of plate separation on energy storage.

The math behind calculating related values (such as capacitance or the effect of an added dielectric) is fairly straightforward and can be saved until the end.

References:

• Griffiths, Electrodynamics 4th Ed.
• Knight, Physics for Scientists and Engineers 2nd Ed

Mathematica:

• Main File: https://www.dropbox.com/s/2ab6445v9s95xvz/ModelingCapacitors_Kachelein_PHYS341_2014.nb
• Data File: https://www.dropbox.com/s/47apmk39swj0ucn/ModelingCapacitors_Kachelein_PHYS341_2014_GIANT%20DATA%20FOLDER.nb