Is there a mathematical way to determine what the electric field between to electrode of different shapes? I messed up an experiment for my physics II lab 'Electric and Potential Fields'. We were supposed to measure the field strength at eight different points, but instead I only did the very center. Is there a way to determine what these values should be at any point considering the shape of the electrode?
I'm guessing I could just use Coulomb's law, but I'm not sure what to use for the charge and the distance. There are three different electrode configurations (lab manual):


*

*rectangular;  

*concentric;  

*circular.


Could I just use the distance from the closest electrode to find out the charge at a given point? 
Would I use $E = \frac{kq}{r^2}$ or $E = \frac{kQ}{R}$? I'm guessing the latter.
 A: Yes there is, but this may not help you much in your lab, as the point of the lab seems to be for you to experimentally observe the equipotential lines (this experiment fascinated me when I did it at age fourteen). The calculation method is to solve Laplace's equation $\nabla^2 V = 0$ for the potential field $V(\vec{r})$ given the boundary conditions that (1) the electrode surfaces are at constant potentials (with the difference between them being the voltage applied across the electrodes) and 2 that the potential approaches a constant value as the distance from the electrodes becomes infinitely large. This can be done numerically - there are many EM field analysis programs around, or one can use the method of complex potentials for many of the shapes your lab instructions give. If you solve Laplace's equation for the concentric electrodes, for example, you find that the complex potential $\Omega=\log z$ gives the right shape of lines. Any mathematically sound method - including guessing - that finds a solution to Laplace's equation for the given boundary conditions is correct, for Laplace's equation comes with very strong uniqueness of solution theorems.
Once you have $V(\vec{r})$, you calculate $\vec{E}(\vec{r}) = -\nabla V(\vec{R})$. 
