I have an analogy that tracks Q X Q / R^2. Will replace the vacuum with a 2D matrix of 100 1 M resistors somewhat like a screen door. Will solder a copper wire along the outside of the matrix of the resister grid. This will be the Faraday cage walls. Now it works. If I place 1 electron in the middle of this matrix of resistors it will have no reason to move up , down , left or right as there is no voltage gradient in the matrix to cause the electron to move even if it is almost touching the Faraday wall. A vacuum is an insulator not a resistor so it is not the best analogy but at least it is tracking Coulomb's law inside a sphere.
Yes!Your analogy is smack on the money, and more valid and powerful than you could imagine. What you are describing is known as FEA, Finite Element Analysis. Of course, while you describe a 2D array of 'resistors', the full solution is a 3-D matrix of such 'resistors', AKA finite elements. As you say, a vacuum is not literally resistive, but it turns out that the math for solving the electric potential at all points in space, for any structure of conductive objects in 3D space, is identical as if you were solving the voltage at every point in the situation where the vacuum is occupied by a solid resistive material such as carbon, which in turn is (in effect) modelled as a matrix of resistors, exactly as you describe. As you would realize, solving for the voltage at every node in a huge 3D array of resistors, where there could be millions of nodes, is computationally intensive, though conceptually simple.
The FEA method is extraordinarily powerful. I invented and developed it 25 years ago for solving magnetic problems, writing my own FEA code to run on the first 8086 PC's, years before commercial magnetic modelling applications were generally available. I developed an iterative algorithm for efficiently solving the huge matrices, being the FEA 'engine', and then added a (primitive by today's standards) graphical user interface so I could view the 3D model from any angle, and display 2D slices on any plane. My application at that time was for designing ultra-high-efficiency brushless electric motors, and my FEA software allowed me to calculate the flux density at every point within an electric motor consisting of windings, magnetic materials and air. By observing the change in flux as the armature is rotated, one can calculate torque, speed, efficiency etc. The fantastic thing about FEA is it's complete generality, so soon I was using the same software to model and design complex systems of magnetic coils used in physics research.
Next I modified the code to solve electrostatic problems as well, exactly as you describe, and went on to design all manner of 'ion optics' for physics research, specialized electron guns, ion lenses and accelerators and so on. The commercial electrostatic FEA modelling package was (and still is) called SIMION, with similar capabilities. As FEA gives you the electric potential at every point in 3D space, it also gives you the E-field at every point in space, being just the difference in potential at 2 points on the matrix, divided by the distance between these 2 points. Then I added to capability for 'ray tracing' of charged particle(s), which is actually quite easy, as the force on the particle at any point is just the E-field times the charge, so it is just a matter of iteratively applying Newton's laws of motion to trace out the path of a charged particle(s). With Simion, the user inputs a particular physical design of electrodes, and can then ray-trace to observe the result. The user can then manually modify the design check the result, and repeat so as to hone and optimize the design, a slow and laborious process. I added more code to my FEA application to automate that process, so that many thousands of designs can be tried with the design being progressively optimized automatically, a process that would take hundreds of man years with the commercially available software. As a result, I have designed some of the most perfect ion imaging lenses ever built for the physics research group that I work with. FEA is a lot of fun.
FEA can solve almost any 3D field problem, or any similar problem where a physical quantity 'flows' as a result of a driving force. Another example where I have applied my code to good effect is thermal modelling. The math for all this FEA stuff is essentially the same, its just a matter of adding different front ends to the FEA engine to solve different types of problem. I can model any arrangement of thermal materials, define the heat sources, and then calculate and display the temperature at every point in the model, a lot of fun.
Another application is calculating the resistance of any arbitrary 3D shape, consisting of any number of different materials. I collaborate with electronic manufacturers on lots of stuff, one very large resistor manufacturer is interested in my code for calculating the resistance of complicated shapes of resistive foil, for example in the very large shunt that they built for me. At present, they 'guestimate' the effect of square bends etc in the zig-zag shape, then fine tune the design later to get exactly the right resistance. With FEA, you can accurately calculate the resistance of ANY shape.
Another really big application is calculation of stress and strain in mechanical engineering structures, and it goes on and on.
Some of your ideas may not work out, but your 'resistor matrix' ides is more relevant and powerful than you could have imagined.