# Fringing Field Effect on 2D Capacitor

I am running a relatively «simple» (or so I thought) 2D electrostatic simulation of a parallel plate capacitor where I would like to see the effects of the fringing fields.

Case 1: air exclusively between two plates

In my initial simulation, I limited the air exclusively to the region between the two plates, and the results matched very nicely with the ideal equations that I extracted from my finite element method software for capacitance, electrostatic energy, electrostatic force, etc.

Case 2: extended the air domain to completely encompass the two capacitor plates

As you can see in Case 2, I extended the air domain to completely encompass the two capacitor plates and ran a parametric sweep on the dimensions of this «air block» to see the effects of the fringing fields. I expected the capacitance and electrostatic energy to both increase slightly and asymptotically reach a steady-state value because, at some point, the additional contributions of the fringing fields become negligible.

I confirmed this to be the case for the capacitance and electrostatic energy, but for the electrostatic force, I had some difficulty interpreting my simulation results.

As you can see in my schematic for Case 2 and the corresponding plots of electrostatic force v. air block dimensions, the force increases at the beginning, then drops sharply, and somewhat stabilizes.

What I am having difficulty understanding is that the final value is less than the case where we don’t take into account the fringing fields. From what I have seen in literature and what I have seemed to remember in my courses is that when we take fringing fields into account, the electrostatic force should increase, not decrease.

Case 3: limited the air block only to the edges of the bottom plate

With that in mind, I decided to simulate what I thought would be a more «realistic» scenario, i.e., Case 3, where I have limited the air block only to the edges of the bottom plate.

My rationale was that very rarely do we see two suspended plates like in Case 2, just floating in the air. Usually, the bottom plate is large (in comparison to the bottom) or rests atop some other material so that the electric field doesn’t have as easy access to the bottom side. When I simulate, the results I obtain for the electrostatic force seem to make more sense in that the final force is indeed bigger than the ideal case with no fringing field.

I realize that this is artificially constraining the field lines and that this behavior is expected, but I wanted to know if this result/behavior in Case 2 and my interpretation is reasonable or whether there is something fundamentally wrong with my simulation or approach.

Thank you in advance for all of your help and I look forward to discussing with you!