# Motivating Capacitance

#### Prerequisites

In thinking about how charges are going to behave and move around in electrical systems such as a cell, and to understand how to model neural systems, we have to take into account a number of foothold principles.

1. The electric field is very strong (compared to, say, gravity) and everything is made of of electric chargesAs a result, introducing an unbalanced charge into a system of objects can result in a lot of charge moving around.
2. We use the concepts of electric field and potential produced by these charges make things become a lot simpler. It would be impossible to keep track of all the charges and sum their forces.
3. Since opposite charges attract each other strongly, matter tends to be neutral, even locally on a molecular scale. But one important mechanism in living systems it separating charges, for example on opposite sides of a membrane, we need to figure out how to describe separated charges.
4. Since separated charges attract each other strongly, are a kind of storage place for electric energy.

If charges can be separated and kept separate, they act like a kind of stretched spring, storing energy that can be released later.

Since electrical potential energy is charge times electric potential, the storage of energy corresponding to a separation of charge will create a potential differences. The way those potential differences are related to the charges that create them is called the capacitance, $C$. We define it by the equation

$$Q = C\Delta V$$

where $\Delta V$ is the electric potential difference created by the separation of a charge into $Q$ and $-Q$. This is an anchor equation for understanding the relation of charge and energy in an electrical system. $C$ is kind of a "figure of merit", telling us how much work we have to do (as represented by the voltage) in order to get a certain amount of energy (charge separation).

Capacitance is a critical concept both in building electric circuits and creating circuit models of neuronal function.

The unit of capacitance is equal to the Farad = 1 Coulomb/Volt.

When unbalanced charges are placed on a conductor, the way they distribute themselves has a lot to do with the shape of the conductor. The charges will move around until it reaches the state with the lowest free energy. We'll consider two toy model examples that can serve as the basis for modeling more complex systems.

## A toy model that illustrates the concept: a conducting sphere

To see how this works in equations, consider a simple model. How much work does it take to build up a sphere of positive charge? Imagine we have a small conducting sphere and we want to carry charge to it. We'll imagine that the charges are being brought from very far away so that we can ignore the interaction with the negative charges that are left behind. (The negative charges that create the neutrality are "at infinity".)

The first positive charge we bring up takes no work, but once the sphere has some charge on it, it will repel any later charges we try to bring and we'll have to do work to push the charge up to the sphere. (Once it gets onto the conducting sphere, it will stay there.)

As a result, we will be adding potential energy to the system of charges (remember the Work-Energy theorem!) and so the electric potential of the charged sphere will rise as we add charges. By how much?

Take a conducting sphere of radius $R$ and put a charge, $Q$ on it. Since the particles of which the charge is made up are of the same sign, they will repel and try to get as far apart as possible. As a result, the will all go to the surface of the sphere and spread out as uniformly as possible. We know from our discussion of spherical charges (A simple electric model: a sphere of charge) that the electric field outside the charge will be the same as for a point charge: $E = k_CQ/r^2$. This means that the electric potential at the surface of the conductor (taking the potential at infinity to be 0) will be $V = k_CQ/R$. So the potential of the conductor is proportional to how much charge we've put on it.

By our definition of capacitance, for the case of a single sphere compared to infinity, we get $C = R/k_C$.

## A more productive model: Two parallel sheets of charge

While the sphere of charge model is sometimes useful (when we have some partially isolated charges), but a more common situation is that a small amount of charge is separated a small distance, leaving close amounts of positive and negative charge. An important example in biology is a cell membrane, which separates charges on the inside and outside of the membrane resulting in a potential difference across it. This potential difference plays a critical role in the cell's managing of ions.

The most useful model for this situation is the potential difference created between two surfaces by equal and opposite charges placed on those two surfaces. This model serves as the basis for our construction of a fundamental circuit element and for our treatment of membranes and neurons. This is worked out in these webpages

## More general situations

We put in the delta in our defining equation to emphasize that we are describing a difference in potentials. For the conducting sphere, the difference is the potential at the surface of the sphere compared to the potential at infinity, but since we can take the potential at infinity to be 0, there, the $\Delta$ doesn't matter. In most cases, it will be very important!

These distributions of charge store electric energy by keeping apart charges that would prefer to get together. (You can see this from the 1st law of thermodynamics, since it takes work to move the charges from one plate to another in order to separate them.) In a typical two-sided capacitor that has a positive charge $Q$ on one side and a negative charge $-Q$ on the other, the energy stored in the capacitor is

energy stored = $½QΔV$

where $ΔV$  is the difference in the potential between the two sides. (It's not the "$qV$" we would expect, but only half. (To see why, go to the page: The energy stored in a capacitor.)

Joe Redish 2/18/16

Article 631