Over the last two weeks, you have read about the electric field and electric potential that is created by a single point charge. While a single point charge is definitely the simplest case, most charged objects that you will run into are not single point charges. These notes will talk about the electric field and electric potential due to multiple charges - starting with the next most complicated case: two point charges that are near each other. The idea of adding together the electric field or electric potential for multiple charges is called the superposition of the field (for both vector and scalar fields), and it generalizes to any number of charges as you will read soon.

If you place two charges next to each other (with nothing else around them), one of two things will happen: either (1) they will move towards each other (opposite signs of charge) or (2) they will move away from one another (same sign of charge). Sometimes it makes sense to think about this movement of the two charges (i.e., when thinking about forces or changes in energy); other times, it can be difficult to think of two moving objects simultaneously (i.e., when trying to describe their field or potential).

To simplify the situation, we will usually make some sort of assumption. For example, we often assume that the charge(s) are fixed in place (something is holding them at a particular location, but we don't care what that something is). Or we will assume that we are interested in a particular instant in time and examine what is happening for that situation (like taking a single frame from a movie or freezing time).

It turns out that this assumption is fairly useful. In the case of charges fixed in place, this often true for metals and insulators that are experiencing a fixed external electric field. The electrons redistribute, but do not continue moving because they have either moved as far as they can (in the case of conductors (metals)) or are strongly bound to their nuclei (in the case of insulators). This assumption also works if we rub some excess charge off onto an insulator as the excess charge stays fairly localized to the place where it was rubbed off. If we are interested in the distribution at a given time, that is an even stronger assumption as we are taking a snapshot of the distribution at a specific instant in time, so the electrons are fixed in the snapshot.

To illustrate the idea of superposition of electric field and potential, we will use the example of a fixed electric dipole, which is a special term for when you have a positive and negative charge next to each other (shown to the right). An electric dipole is a common example in physics because it is a fairly good model of an atom in an insulator - where the positive charge represents the positive nucleus and the negative charge represents the negative electrons.

As you have learned, the electric field from a single positive charge at any given point will point away from the charge, and the electric field at any given point from a negative charge will point toward the point charge. So what happens to the electric field when you have a positive charge next to a negative charge? The field at any point in space around the two charges will be given by a net electric field, which is the vector addition of the electric field from the positive charge and the electric field from the negative charge.

$$\vec{E}_{net}=\vec{E}_{+}+\vec{E}_{-}$$

The figure on the right shows how the electric fields from the positive and negative charges combine at a single observation point to give the net electric field at that location. Remember electric field is a vector, so here the vertical components cancel out and the horizontal components add together. (This should make sense intuitively - if you add something that points up and to the right to something that points down and to the right you should be left with something that just points to the right.)

If you repeat this process for many observation points all around the dipole, you will end up with the electric field shown in the next figure to the right. Close to the individual charges, the electric field looks much like it would for a single point charge (the electric field points away from the positive charge and toward the negative charges); however in the middle of the two charges the field bends to point from the positive charge to the negative charge.

If you have more than two charges, the net electric field is simply found by adding up the individual fields from each point charge.

$$\vec{E}_{net}=\vec{E}_{1}+\vec{E}_{2}+\vec{E}_3+...$$ Just remember that you must add the vectors together and not simply the magnitude (in other words, the direction matters!). Superposition might seem like a new idea (adding of vector fields), but it is the same process that you do when you add together all the forces acting on an object to find the net force. In fact, the net force is the superposition of all the forces acting on an object. So the difference here is that the forces are often defined at an object or collection of objects (i.e., where the force is applied to a system) while the electric field for any charged object permeates all of space (i.e., can be found at any observation location around the charge/charges).

Because electric potential is a scalar, it means adding together the electric potentials can be quite a bit simpler than adding electric fields (you don't have to consider direction); however, you must first check that the reference point for each of the individual potentials is the same. This is the same idea that we used with