Motivating simple electric models

Though much biochemistry and lots of cellular biology are the result of electrical forces, there are very few biological situations where forces between a small number of charges is what's going on. And since Coulomb's law tells us that only the force between individual point charges has a simple $1/r^2$ dependence, for any complex system, we have to add up the results of lots of different charges. Adding all those vectors can be a mess. Even if we use electric potential energy instead of electric force ($1/r$ dependence and no vectors) it can still be very complicated. Well, sometimes that's just what you have to do. And computers can help (see, for example, the problem: The water-coat potential). But often, we can use one of a few simple models as an approximation. This can give us a good starting point for understanding a complicated situation. 

There are three models that we can analyze in a pretty straightforward way (or, be more complicated and actually carry out the integrals). They are (1) an infinite uniform line of charge; (2) an infinite uniform plane of charge; (3) a spherical charge distribution.

Though of course we never have "infinite" distributions of charge, letting these become infinite makes the calculations easier and suppresses "edge effects" — changes that occur when we get to the end of a finite line or sheet of charge. And we will see just what the conditions are that let us treat a finite system as if it went on forever.

These idealized (toy) model distributions can provide convenient starting points for modeling complex distributions of charge. 

Joe Redish 2/14/12