Where do Electric Forces Come From?

There’s a good chance that, at some point in your life, someone told you that nature has four fundamental forces: gravity, the strong nuclear force, the weak nuclear force, and the electromagnetic force.

This factoid is true, of course.

But what you probably weren’t told is that, at the scale of just about any natural thing that you are likely to think about, only one of those four forces has any relevance.  Gravity, for example, is so obscenely weak that one has to collect planet-sized balls of matter before its effect becomes noticeable.  At the other extreme, the strong nuclear force is so strong that it can never go unneutralized over distances larger than a few times the diameter of an atomic nucleus (\( \sim 10^{-15}\) meters); any larger object will essentially never notice its existence.  Finally, the weak nuclear force is extremely short-ranged, so that it too has effectively no influence over distances larger than \( \sim 10^{-15}\) meters.

That leaves the electromagnetic force, or, in other words, the Coulomb interaction.  This is the familiar law that says that like charges repel each and opposites attract.  This law alone dominates the interactions between essentially all objects larger than an atomic nucleus (\( 10^{-15}\) meters) and smaller than a planet (\( 10^{7}\) meters).  That’s more than twenty powers of ten.

But not only does the “four fundamental forces” meme give a false sense of egalitarianism between the forces, it is also highly misleading for another reason.  Namely, in physics forces are not considered to be “fundamental”.  They are, instead, byproducts of the objects that really are fundamental (to the best of our knowledge): fields.

Let me back up a bit.  To understand what a force is, one first has to accept the idea that empty space is not really empty.

Empty space, or vacuum, is the stage upon which the pageantry of nature plays out.  Just as the setup of the stage in a theater determines what kind of plays can be performed, so too do the properties of the vacuum determine what kind of natural laws we have.

Let me be a little less wishy-washy.  As we currently understand it, empty space is filled with a number of all-pervasive, interpenetrating fields.  To a physicist, these fields are mathematical objects: they are functions that take a particular value (or vector of values) at every point in space.  But for the daydreamer (which, of course, includes those same physicists), these fields can be visualized as something like a stretchy fabric, or a fluid.  To be concrete with the imagery, let’s say that a field is something like the surface of a pond.  When not perturbed, that surface is placid (as long as you don’t look too closely, say, at the molecular level).  But when something disturbs the pond, it creates a ripple that propagates stably across the surface.

In the modern view of physics, what we call “particles” are really just ripples across a field.  The word electron, for example, is what we use to refer to a ripple on the electron field.  The photon is just a ripple on the photon field (also called the electromagnetic field).  And so on.  For each of the elementary particles there is a corresponding field upon which that particle is a ripple.  It is these fields (and they alone) that define the properties of the universe: what kind of particles can exist in the universe, and how they interact with each other.

Just to belabor the point a little more: a particle like an electron is not any more “fundamental” or “elementary” than a wave lapping the shore of a beach.  It is the sea that is fundamental.  If you want to understand where waves come from and how they move, you must first understand the water.

(Aside: If you are encountering for the first time this idea of fields as the fundamental objects of the universe, I hope that it bothers you.  I hope that it makes you feel uncomfortable and a little incredulous.  That’s certainly the way I felt at first, and for me those feelings were the beginning of my ability to appreciate an idea that has come to feel deeply wondrous, and deeply useful.  If you do indeed feel unhappy with this idea, all I can suggest is time and perhaps the wonderful essay about fields (classical and quantum) by the great Freeman Dyson.)

With that pictorial overview, let’s return to the Coulomb interaction.  What I want to claim is that the Coulomb force, like any other force, is an emergent property of the field that mediates it (in this case, the electromagnetic field).  And so, if the force between two charged particles is to be properly understood, then it must be explained in terms of how the field behaves in the vicinity of those two charged particles.  From the resulting picture of a perturbed field one should be able to see, in a quantitative way, that Coulomb’s law emerges.

Let’s start with the basics.

Coulomb’s law, as it is usually written, says that the strength of the force \( F\) between two charges \( q_1\) and \( q_2\) separated by a distance \( r\) is

\( F = (1/4\pi\epsilon_0) q_1 q_2/r^2\).                                                      (1)

(You can think that the term at the beginning of the equation, \( 1/4 \pi \epsilon_0\), as just a constant conversion factor that turns the nebulous units of “charge” into something that has units of force.  But I’ll say a little more this factor in a bit.)

To see how the Coulomb force arises from the properties of a field, it makes sense to first talk about the electric field created by a single charge.  For this we can invoke the second piece of canonical knowledge related to point charges, which says that the strength \( E\) of the electric field around a point charge \( q_1\) follows

\( E = q_1/r^2\).                                                      (2)

This electric field points radially outward (for positive charges) or inward (or for negative charges) from the position of the charge.

 

At first sight, this equation looks essentially identical to the previous one: the only difference is that there is only one \( q\) in it instead of two. In fact, high school classes tend to explain the idea of electric field in a way that makes you question why it needs to exist as a separate concept from force.  I think my high school physics class, like many others, literally defined an electric field as “the force that would be felt by a unit charge if it were placed at a particular location.”)  In this sense the concept of electric field often sounds subsidiary to the concept of force.  But if you’re prepared instead to think about the field as a truly fundamental object, then this second equation becomes quite interesting.

In our analogy, the electromagnetic field is something like a fluid that fills all of space.  This fluid exists at every point in space and at every moment in time, but at moments and locations where there are no charges around you can imagine that it is stationary and calm.  In the presence of electric charges, however, the fluid starts to move.  What we normally call the electric field strength, or (perhaps confusingly) just the electric field, can be imagined as the local velocity of the fluid at a particular point.

What emerges from equation (2), then, is a picture of how the fluid is moving in the vicinity of a point charge.  For a positive charge, for example, it moves radially outward, in such a way that the speed of the fluid falls of as the square of the distance from the center.  As it happens, this inverse square law is special: it guarantees that the total amount of fluid flowing across any closed surface containing the point charge is the same, regardless of the shape or size of the surface.  What’s more, the fluid flow rate is directly proportional to the charge \( q_1\).

The simple way to see this is by drawing a sphere of radius \( R\) around the point charge.  The flow rate of water across the surface is equal to the product of the “velocity” \( E\) times the area of the sphere.  Since the surface area of a sphere is proportional to \( R^2\), and the velocity is inversely proportional to \( R^2\), the flow rate of water is the same for any sized-sphere and is proportional to the charge inside.  This is basically just a restatement of Gauss’s law.

All this is to say that, in our fluid analogy, the (positive) point charge is something like a spout of water or a hose: fluid comes flying out of it in all directions.  This fluid is fast moving at the source, and slower as it spreads out. It is not created or destroyed anywhere except at the point charge itself.  Conversely, a negative point charge is something like the opposite of a hose: a suction source or drain that pulls water into itself.  In both cases, the total flow rate of water (either sprayed out or sucked in) is quantified by what we call the charge, \( q\).

 

This analogy might strike you as perhaps a little too precious, but it turns out to give an intuition that works at a surprisingly quantitative level.  In particular, the analogy gets quite good when you ask the question “how much energy is there in the field?”

When fluid is in motion, it has kinetic energy.  You may remember from high school physics that an object with mass \( m\) and velocity \( \vec{v}\) has kinetic energy \( m|\vec{v}|^2/2\).  For a fluid, where different locations have different velocities, you can generalize this formula by integrating over different locations:

\( (\text{kinetic energy of fluid}) = \int \frac{1}{2} \rho |\vec{v}|^2 dV\).         (3)

Here, the symbol \(\int … dV\) means an integral over all different parts of space, and \( \rho\) is the density of the fluid.

As it turns out, the expression for the energy stored in an electric field looks almost exactly the same:

\( (\text{energy stored in an electric field}) = \int \frac{1}{2} \epsilon_0 |\vec{E}|^2 dV\).  (4)

By comparing the last equations, you can see that the value of the electric field, \(\vec{E}\), really is playing the same role as the fluid velocity.  That constant \( \epsilon_0\) in equation (4) is a special constant called the permittivity of free space.  In our analogy, you can think of it as something related to the “density of fluid” in the electromagnetic field.

You can also notice that the quantity \( \frac{1}{2} \epsilon_0 |\vec{E}|^2\) is an energy density.  It tells you how much energy is stored in the electric field at a particular location.  For fluids, energy density is closely related to the concept of pressure: a fluid under high pressure has a lot of energy stored in it.  (You can check, if you want, that energy per unit volume and pressure have the same physical units.)  We can therefore think of the quantity \( \frac{1}{2} \epsilon_0 |\vec{E}|^2\) as something like a pressure that builds up in the electromagnetic “fluid”.

Having made a correspondence between electric field and pressure, the final step toward understanding Coulomb’s law is relatively straightforward.  When two spouts of water are brought together, pressure builds up between them, and they are pushed apart.  Similarly, when two electric charges are brought near each other, pressure builds up in the electric field between them.  This pressure ends up pushing the two charges apart from each other, in the same way that two fire hoses would be pushed away from each other if you fired them toward each other.

With equations for the “field pressure” in-hand, you can even calculate the exact mathematical form of the repulsive force between the two “hoses”.  If you want the technical details: you can calculate the pressure at the midplane between the two, and then integrate the pressure over the midplane area.  (This is the same procedure that you would follow if you wanted to know the force of a fire hose spraying against a wall.  Of course, there are other approaches for doing the calculation.)  What comes out of this procedure is exactly what I promised from the beginning:

\( F = (1/4\pi\epsilon_0) q_1 q_2/r^2\).

If you want a conceptual picture for the attractive force between opposite charges, you can approach it in a similar way.  In particular, when two opposite charges are brought near each other, this is like bringing a hose that emits water close to a strong suction hose.  One of the two hoses is furiously emitting fluid, while the other is happily sucking it in, and consequently the pressure between the two of them becomes relatively small.  As a result, the two “hoses” end up being pushed together by the larger pressure of the fluid outside.

At this point, we have a conceptual explanation of where electric forces come from.  But to close this post, it is perhaps worth making a remark about simplicity in physics.  It may strike you that the story I have told here is not simple.  I started with a very simple equation – Coulomb’s law, which is usually introduced as the simplest quantitative starting point for thinking about electric charges – and I gave it a complicated origin story.  This story required me to invoke nebulous, space-filling force fields; to make questionable, convoluted analogies; and to compute multi-dimensional integrals over vector-valued functions.  This story also never explained why a point charge behaves like a “source” or “sink” of “fluid”; it just does.  Or, at least, it needs to, if the story is to hold together.

You may reasonably feel, then, that the picture I painted is essentially worthless.  It is much easier to simply remember equation (1) than to remember how to describe the way that pressure builds up in a space-filling, fluid-like field.  And it requires essentially the same number of arbitrary assumptions.

If you feel this way, then probably all I can offer is an apology for wasting your time.  But for a physicist, the construction of such “origin stories” is perhaps the very most important part of the profession.  It is absolutely integral to physics that its developers never be satisfied with any level of description of reality.  To every law or equation or theorem, we must always ask “yes, but why is it that way?”  This impertinent questioning, where it succeeds, ultimately always turns one question into another question.  But along the way it can rewrite very fundamentally the way we perceive nature.  And, when those revisions succeed, they pave the way for significant new insights and discoveries while recapitulating all the results that came before.  (For the record, the classical and quantum theories of fields are probably the most successful scientific theories that mankind has yet produced.)

You can also view the question of simplicity another way.  In telling this story, I have not been particularly simple, but nature has been very simple indeed.  It has provided an extremely succinct mathematical law and allowed it to govern the universe over more than 20 orders of magnitude in scale.  Perhaps the greatest proof of Nature’s simplicity is not that I can write Coulomb’s law in a single line, or that I can give it a particular origin story, but rather that I can think about it in many different ways and derive it through many different avenues, and all of those avenues turn out to be equivalent.

I’ll leave you with the words of Richard Feynman, who expressed this same sentiment very nicely in his Nobel lecture:

The fact that electrodynamics can be written in so many ways … was something I knew, but I have never understood. It always seems odd to me that the fundamental laws of physics, when discovered, can appear in so many different forms that are not apparently identical at first, but, with a little mathematical fiddling you can show the relationship. … I don’t know why this is – it remains a mystery, but it was something I learned from experience. There is always another way to say the same thing that doesn’t look at all like the way you said it before. I don’t know what the reason for this is. I think it is somehow a representation of the simplicity of nature. A thing like the inverse square law is just right to be represented by the solution of Poisson’s equation, which, therefore, is a very different way to say the same thing that doesn’t look at all like the way you said it before. I don’t know what it means, that nature chooses these curious forms, but maybe that is a way of defining simplicity. Perhaps a thing is simple if you can describe it fully in several different ways without immediately knowing that you are describing the same thing.