Capacitors and making complicated shapes simpler

by ayushkhaitan3437

The paper that I intend to blog about is: Parallel-Plate Capacitor by the Schwartz-Christoffel Transformation by Harlan B. Palmer.

I intend to read some research papers in various academic fields, and, perhaps amateurishly, blog about them. We could all be better about what is happening outside of our narrow fields of interest. Although I hope to be regular about this, it is likely that this is going to be a sporadic bunch of posts until my employer asks that I devote more time to my day job.

This is the paper by Harlan B. Palmer that I am referring to.

How do we measure the electrical “energy” (electrostatic potential energy) or electrical field strength of a capacitor? If we can somehow measure the flux density between the plates of the capacitor, we will have the answer. However, in such a calculation, we only consider the sides of the plates facing each other. We don’t consider the back parts of these plates. These also have charge, and hence affect the electrical field strength. The author contends that the true value of the capacitance, which depends on the electrostatic potential, might be “several thousand percent greater than indicated”.

Another complicating factor is “fringing”. Near the ends of the plates, the flux lines veer towards the center of the opposite plates, and don’t go straight up. This makes the flux lines more difficult to deal with mathematically.

Consider the figure given below:

We take a polygon containing the front and back sides of both plates. So ideally, if we can integrate the flux attached to all sides of the plates, we should be able to calculate capacitance. However, the flux lines are all twisty and non-uniform, the sides of the polygon seem complicated and difficult to deal with, and the whole enterprise looks very challenging.

What if we could “straighten” this messy polygon into a rectangle, such that all the flux lines pass through the sides of this rectangle uniformly? That would make life simple! This miraculous transformation is achieved by the Schwartz-Christoffel transformation.

The Schwartz-Christoffel Transformation takes the figure given below, and maps it to any polygon that is desired:

Moreover, this transformation is reversible. Hence, effectively we can map any polygon on the complex plane to any other polygon on the complex plane, and that too conformally. What this means is that we can preserve the Physics principles. The lines of electric flux should always be perpendicular to equipotential lines, and because this is the case in the complicated polygon, this will remain the case even in the simpler rectangle.

The Schwartz-Christoffel transformation looks like this:

Z=\int (\zeta-\zeta_a)^{m_a}(\zeta-\zeta_b)^{m_b}\dots (\zeta-\zeta_n)^{m_n}d\zeta

Here, the m_i‘s are telling Z what angle to turn by, in order to form the desired polygon. Also, we can determine the lengths of the sides of the polygons such that the flux lines attached to the plates indeed have uniform density.

What I think has happened is this. Look at the rectangle below:

If you take the top plate of the capacitor, peel out the front and back sides of this plate and lay them side by side, you’ll get the top side of this rectangle. Similarly, if you take the front and back sides of the lower plate of a capacitor and law them out side by side, you’ll get the bottom plate of this rectangle. Moreover, there is no “fringing” as you may imagine that the thin polygon in Figure 1b has been straightened out. Hence, the flux lines are now uniform. This will now lead to easy calculations without error.

References

  1. Parallel-Plate Capacitor by the Schwartz-Christoffel Transformation, by Harlan B. Palmer