We've established that charges attract and repel each other even if they're separated by a distance, but we haven't said anything about how this happens. It's not too hard to understand how my feet can exert a force on the ground when I walk, but how can one object exert a force on another when they aren't in contact?

The honest answer is that we don't know the exact mechanism. This is very common in science: while we would like to be able to explain precisely how and why something occurs, the best we can do is to construct a model which predicts the behavior of the world. Whether this model is true or not is hard to determine and in some sense irrelevant: it's more important to ask whether a model is useful, and makes accurate predictions.

The model we will use to explain the electric force is called the electric field model. In this model, every charged object creates an electric field which permeates the entire universe, combining with the electric fields of all the other charges in the universe. And when a charged object is in the presence of an electric field created by other charges, it feels a force. (Charges don't react to their own electric fields.) In terms we've discussed before, sources create electric fields, and targets experience electric fields.

We define the electric field at a particular point in space by how a target charge would react if it were placed there. Specifically, if a charge \(q_T\) is placed at a spot with an electric field \(\vec E\), then it will feel a force

$$\vec F=q_T\vec E$$

If the charge is positive (\(q_T>0\)), then the force and the electric field point in the same direction; if the charge is negative, then the two point in opposite directions. Thus, if we want to know the direction the electric field will point in any situation, we need only ask WWPD: that is, What Would a Proton Do? It is important to remember that the electric field exists everywhere in space at all times, whether or not a charge is there to experience it.

The figure shows three charges in a uniform electric field (i.e. the field is the same everywhere). The charge on the left feels a force to the left of 5 mN.
  1. What is the magnitude of the electric field?
    The particle in the diagram has a charge qT = 5 µC feels an electric force of 5 mN upward. This means that the electric field at its location must be
    $$\vec E=\frac{\vec F}{q_T} =\frac{5\ten{-3}\u{N}\leftarrow}{5\ten{-6}\u{C}} =\boxed{1000\u{N/C}\leftarrow}$$
    which points to the left. Notice that the units of electric field are Newtons/Coulomb (which for some reason hasn't been named after anyone.)

  2. What is the force on the 1 µC charge?
    The 1 µC charge feels a force of
    $$\vec F=q_T\vec E=(1\ten{-6}\u{C})(1000\u{N/C}\leftarrow)=\boxed{0.001\u{N}\leftarrow}$$
    The force is proportional to the target charge's magnitude, so a charge of one-fifth the magnitude naturally feels one-fifth the force.

  3. What is the force on the −1 µC charge?
    The −1 µC charge feels a force to the right instead:
    $$\begin{eqnarray} \vec F&=&q_T\vec E=(-1\ten{-6}\u{C})(1000\u{N/C}\leftarrow)\\ &=&-0.001\u{N}\leftarrow\\ &=& \boxed{0.001\u{N}\rightarrow} \end{eqnarray}$$

It is good to remember that

positive charges are pushed with the field, while negative charges are pushed against it.

Notice that we don't need to know anything about the source charges: as long as we know what the electric field is, we don't need to know what created it. (And in fact there are circumstances where an electric field may be created by something other than a source charge, which we will discuss later.)