Presentation: electric forces and fields

Look at this van de Graff generator exerting attractive and repulsive forces on these soap bubbles. Just look at it. (Video link: "Static Electricity and Bubbles!")

While we will quantify how charges exert electric forces on other charges, consider the philosophical problem of why charges are able to exert electric forces on other charges, without touching, across empty space. This is the action-at-a-distance problem. We are going to "solve" this problem in this presentation--well, sort of--and much later will only be able to completely "solve" the action-at-a-distance problem near the end of this semester by introducing the concept of quantum electrodynamics.

In this presentation we will discuss two models of how charges exert electric forces on other charges.

First, a "direct" model of electric forces.

By convention we label the two charges that exert forces on each other as "source" and "test" charges, where the source charge q₁ is said to be exerting a force on the test charge q₂. This force is attractive if the source charge and test charge have opposite signs, and repulsive if the source charge and test charge have the same sign.

Coulomb's law quantifies the magnitude of the force that a source charge exerts on a test charge, separated by a distance r. Note that the signs of the charges do not matter when calculating the magnitude of the force, and that the constant k handles unit cancellation and conversion to express the force with the proper units of newtons.

Second, an "indirect" or two-step model of electric forces.

Here the convention is to label the source charge and test charge as Q and q, respectively. Instead of the source charge Q directly exerting a force on the test charge q, in this two-step model, the source charge Q is said to create an electric field everywhere around it, and it is this electric field that exerts a force on a test charge q.

Let's focus on the first step of this two-step model. The source charge Q creates an electric field everywhere around it. The magnitude of this electric field can be calculated for a location at a distance r from the source charge, and has units of newtons per coulomb (N/C). The direction of this electric field depends on the sign of the source charge Q: if the source charge is positive, the electric field vectors point outwards from it; if the source charge is negative, the electric field vectors point in towards it.

An electric field is depicted as filling in all space surrounding a source charge, and is drawn as field lines that point outwards from a positive source charge, and point in towards a negative source charge.

Now what? If there is another charge anywhere in the presence of an electric field, this electric field will exert a force on the test charge q. Note that this is a vector equation, where the force F (magnitude and direction) exerted on the test charge q is the test charge q multiplied by the electric field E (magnitude and direction) at the test charge's location. Since the test charge q could be positive or negative, then the electric force exerted on it could have the same or opposite direction as the electric field lines.

Here, from before, we show the electric fields filling in all space surrounding a positive or a negative source charge. Putting a positive test charge in the presence of these electric fields will cause an electric force to be exerted along a field line, while putting a negative test charge in the presence of these electric fields will cause an electric force to be exerted in the opposite direction of a field line. (In any case, as a check the direction of the force on any source charge q should be attractive or repulsive depending on whether it has the opposite or same sign as the source charge Q.)

So if using Coulomb's law in the direct model of calculates the electric force a source charge exerts on a test charge, why do we need a two-step model of a source charge generating a field, and this field exerting an electric force on a test charge?

Here in this excerpt from Star Wars Episode IV: The Empire Strikes Back (Lucasfilm/Twentieth Century Fox, 1980), Luke Skywalker is trying to grab his lightsaber, which is just beyond is reach. He could use the "force" to pull his lightsaber, but this brings up the philosophical problem of how a force of one object on another can be exerted across empty space.

Instead, let's reinterpret this in terms of the two-step field model. Luke goes into a trance, and let's say that he is generating some sort of field around himself (as Obi-Wan Kenobi explains: "...an energy field created by all living things. It surrounds us and penetrates us; it binds the galaxy together"). This field spreads out from and surrounds Luke, and it is this field that exerts a force on the lightsaber, pulling it out of the snowbank. Mayhem ensues.

So now we've "solved" the action-at-a-distance problem here by discussing a field as occupying the empty space surrounding and between the source object (Luke) and the test object (the lightsaber). (Perhaps, then, Luke is really using the "Field," and not the "Force" in the Star Wars universe.) And we put "solved" in quotes because this really recasts the philosophical action-at-a-distance problem as a philosophical question of what exactly is a field (and is it really generated by and surrounds a source object)? In a subsequent presentation near the end of this semester we will explain the physical nature of what a field is, but for now we will merely take an electric field as a given in our two-step model of how charges exert electric forces on other charges.

In this discussion we only considered the force exerted by a single source charge on a test charge, or the field generated by a single source charge. Later in class we will go through more complex situations where there may be multiple source charges, each exerting forces on the same test charge (thus requiring us to determine the net force on the test charge); or multiple source charges, each creating an electric field at the same location in space (thus requiring us to determine the net electric field at that location). So be sure to brush up on one-dimensional vector addition, as discussed last semester.

So remember, may the field be with you...