In an effort to better understand what an electric field is - and thus gain some notion of what an oscillating electric field could be - I decided to start with the the force exerted by static charges.

In our world, this is represented by

F = q1 * q2 / (4 * pi * r^2 * E0)

Where F is the force, q1 and q1 are the charges, and E0 is some ridiculously complicated constant called the permitivity of free space.  Since I'm pretty far from understanding what E0 is under than "some unchanging number", I thought I would look at the others.

The "4 * pi * r^2" refers to the surface area of a sphere.  So basically this means that the force is divded up evenly around the sphere defined by your radius.  But in Array World, there are no spheres, just diamonds, so we'll need to find the surface area of a diamond for this.

I started with the circumference.  In our world, circumference is 2 * pi * r.  In Array World, there's a couple of ways you could decide how to define this, and each yields different results.  In one method, you'd figure out how long it would take light to propogate along each side of the diamond, then multiply this by 4.  If you do this, each side takes 2r updates (dx + dy), for a total of 8r.  On the other hand, maybe we're more interested in how many cells are taken up by the perimeter of the diamond, in which case it comes out to 4r (a figure I got by counting them).  Immediately you can see a difference between Array World and our world, because in our world we'd expect to get the same result with either measurement.

I'm a bit reluctant to decide between either one at this point - if somehow the force is generated by something orbiting the center charge, we'd want to use the former version

But on a hunch I'd like to go with the latter method, since I picture each cell as counting for some equal portion of the force.  By the same reasoning, I then found the surface area through the brute force method of taking one of my earlier diamond bitmaps and extending it in the z direction, then counting the cubes.  This result seemed to be described by 4 * r^2 + 2.

The +2 is a bit perplexing - and in, fact, it seems to come about because I drew my diamonds with a blank center pixel.  It goes away if I draw the diamonds as not centered on a pixel, but symetrical around an intersection of grid lines.  This may be the more common case, since my single pixel represents something akin to a Planck unit, and we know that electrons (from which coulomb's equation is derived) are gigantic compared to it.  I'm inclined to discard it for the time being, but keep it in mind in case I come across some margin of error that can't be explained any other way.

So then the Array World version of coulomb's law looks like it will just be q1 * q2 / ((4 * r^2) * E0).

Of course, AW has a different E0 as well - if it has one at all  - since it's also defined in terms of Pi, which we don't have an analog for.  My inclination is to decide that E0 is 1, and determine the values of charges from there, rather than vice versa.

So let's go with

F = q1 * q2 / (4 * r^2).