# Making sense of a vector field

#### Prerequisites

## Fields for non-touching forces

The general idea of a field is simply a function — an assigning of some kind of value to every point in space. The particular use we make of this idea in physics is a little more specific: we use it to talk effectively about non-touching (action-at-a-distance) forces such as gravity, electricity, and magnetism. In particular, we use it to separate the object that feels the force we are considering from the object(s) that are causing that force.

In all of the action-at-a-distance forces in physics, the objects causing and feeling that force have some property — a parameter — that tells us how strong the force is both that it feels and that it creates. For gravity, that property is *mass*; for electricity the property is *charge*; and for magnetism it's *magnetic moment*.*

In all of these cases, the actual force felt is proportional to the product of the two objects involved in creating and feeling the force. (In magnetism it's a little more complicated since magnetic moment is a vector.) So in the case of gravity the force between objects 1 and 2 is proportional to $m_1m_2$. In the case of electricity, the force between objects 1 and 2 is proportional to $q_1q_2$. In these two case (we'll treat magnetism separately later) interesting things happen as a result of the situations in which we most commonly consider these forces.

## The gravitational field

The case of gravity is the simplest since gravity is so weak. In most examples, we only have to consider sources of planetary mass or greater. For motions near the surface of the earth, the only source of gravity that matters is the earth. The parameter that couples to gravity is the mass. Therefore, for every object, the gravitational force that it exerts is proportional to its mass. If we divide that out, like this:

$$\overrightarrow{g} = \frac{\overrightarrow{W}_{E\rightarrow A}}{m_A}$$

$W_{E\rightarrow A}$ is the gravitational force of the earth (E) pulling on object A. If we divide by the mass of A, we get a result that is independent of A — it only depends on the earth and where we are; at least for direction and the distance above the center of the earth. The vector $\overrightarrow{g}$ is called ** the gravitational field**. It clearly has units Newtons/kg. (Because we use the same mass in Newton's second law this has the strange result that this is also an acceleration; but in all cases except free fall, it's best to think about the gravitational field as a force/mass.)

The value of introducing this in the case of gravity is that we have separated what depends on the source — the earth — and what depends on the object that is feeling the force from the earth.

## The electric field

Introducing the field is much more important in the case of electricity (at least for a biologist, if not for a rocket scientist). The electric force is many, many orders of magnitude stronger than gravity. Furthermore, all matter is made up of electric charges. As a result, in general we have to consider a LOT of sources when we are thinking about an electric force on an object. Each of the source charges may be at different distances so we have lots of different Coulomb's laws to calculate. Furthermore, each force is a vector and they have to be added up in components so their directions are correctly taken into account.

Worse, often we don't know where those charges are. By separating the electric force on an object into its charge and an electric field we can MEASURE the electric field using a test charge — map it out — and then be able to understand what other charges will do in that situation.

So here's the idea: We have a region of space that's influenced by a number of charges. We go into that region with a test charge small enough not to affect the source charges, and measure the force on that charge at a given point. We then calculate the electric field the charge feels at that point,

$$\overrightarrow{E} = \frac{\overrightarrow{F}^{elec}_A}{q_A}$$

where the force $\overrightarrow{F}^{elec}_A$ in this case is the electric force felt by A, caused, perhaps, by many (unspecified) source charges.

As in the case of gravity, by dividing out by the charge of A, we wind up with something that only depends on the sources, not on the charge we are looking at. (Of course to find the force rather than the field, we have to multiply back by the charge of the object we are considering.) Using various ways to represent the vectors of the electric field, we can get a good idea of what any charge will feel as a result of the source charges — without actually having to know where the individual source charges are.

## Just a calculational tool? Or...?

I like to describe the measurement of an electric field as an expression of "the cheshire cat" principle. In Lewis Carroll's "Alice in Wonderland," Alice encountered a cat who vanished, leaving only its grin. We measure a force with a test charge and then "vanish" the test charge, leaving only the electric field it measured — the grin.

Now it seems like this whole "action-at-a-distance field" concept is just a useful calculational device. The interesting thing is that as we get more sophisticated we learn that the field can carry both energy and momentum! It seems that this "calculational device" has real existence. In fact, we shall find out that light, which we know well can carry energy, since all life on earth depends on it, is just made up of electric and magnetic fields!

* There are corrections to this — coupling the fields to the velocity as well as the charge of the object. These are important in the case of electricity and magnetism and make generators and motors possible — hence our entire electrical economy.

Joe Redish 2/12/12

Last Modified: May 24, 2019