Why do we say that electric potential energy is stored in the electric field?

I have been learning electrostatics and came across capacitors. I don't really get why do we say energy is stored in electric field rather than in the charges upon which we or the battery does work. I hope you could provide me the intuition. I don't understand the other answers on the same question.

• Hi, welcome to Physics SE. The key insight is this.
– J.G.
Commented Dec 3, 2023 at 16:40
• Intuition? How could two charges interact directly when they are not at the same place? Can there be anything between them when they are far apart? Commented Dec 3, 2023 at 17:10
• Related (if not a duplicate). Can you explain what it is in the answers you've seen that you don't understand?
– Puk
Commented Dec 3, 2023 at 18:39

It can be difficult to see why the electric field has to store energy when studying electrostatics alone. Electrodynamics provides the real motivation. As David Griffiths puts in his text Introduction to Electrodynamics,

When a charge undergoes acceleration, a portion of the field “detaches” itself, in a sense, and travels off at the speed of light, carrying with it energy, momentum, and angular momentum. We call this electromagnetic radiation. Its existence invites (if not compels) us to regard the fields as independent dynamical entities in their own right, every bit as “real” as atoms or baseballs.

So electromagnetic radiation (or light) is made from the electric and magnetic fields alone, without charge. Since light carries energy, the fields must store that energy.

• Your last sentence is interesting. Since light is emitted when an electron relaxes in an atom, it would not be wrong to explain this by the weakening of the electric field of the bound electron. This would give us a better understanding of how an atom holds together without being blown apart by the forces between the electrons. Of course, the unit charge of a free electron - as measured - is constant. But IM Atom has probably still been able to convince us of the constancy of the charge. Commented Dec 5, 2023 at 5:04

Here is a simple argument that I find suggestive ...

The capacitance of an 'ideal' vacuum-spaced parallel plate capacitor (one for which the plate dimensions are much greater than the plate separation, $$d$$), is $$C=\frac{\epsilon_0A}d$$ The energy $$U$$ stored in the capacitor is

$$U=\tfrac 12C\ (\Delta V)^2=\tfrac 12\frac{\epsilon_0A}d\ (\Delta V)^2=\tfrac 12\frac{\epsilon_0A}d\ (Ed)^2=\tfrac 12 \epsilon_0Ad\ E^2$$ We see that if we express $$U$$ in terms of the magnitude, $$E$$, of the electric field strength in the space between the plates, then $$U$$ is proportional to the volume, $$Ad$$, of that space. To me, this suggests energy storage within that space; for a given $$E$$, the more field-filled space you have, the more energy you are storing.

I don't really get why do we say energy is stored in electric field rather than in the charges upon which we or the battery does work.

It is stored in both. Electrostatic potential energy, just like other forms of potential energy (gravitational, elastic) is a system property. The charges themselves don't store potential energy nor does the field alone store potential energy. It is the system (combination) of positive and negative charges and the electric field that store electrostatic potential energy.

That stored electrostatic potential energy of the capacitor comes from the work done, by say a battery, to move electrons from one plate (making that plate net positively charged) to the other plate (making the other plate net negatively charged) with an electric field between the plates.

Hope this helps.

• I always thought the stored energy is the volume integral of $(1/2)({\bf E \cdot D + B \cdot H})$. No mention of charge there. Commented Dec 4, 2023 at 12:23
• @AndrewSteane You are probably right. I was taught it was stored in a configuration of charges Commented Dec 4, 2023 at 13:05
• Since field and source of field are intimately linked it is not wrong to talk about the configuration of charges, but if push came to shove and we had to say "where exactly is the energy?" then one way to frame the question is in terms of its gravitational effects. This has never been experimentally tested but I think the idea is that the gravitational source term is indeed the field energy (and stress) as one would guess. At least that's the standard formulation. Commented Dec 4, 2023 at 14:35
• @AndrewSteane Good points. Would it be better if I said the electrostatic potential energy is stored in a volume containing a configuration of charges? Commented Dec 4, 2023 at 21:54
• I leave that to you! Readers will probably see these comments if they are interested. Commented Dec 4, 2023 at 23:01

If you wiggle charges in the right way, they produce electromagnetic waves. These waves can extend for great lengths. For example, the Voyager space probe is a spacecraft that is headed for deep space. It is currently past the outer edge of the solar system at a respectable 11 billion kilometers and it is still transmitting radio waves. These waves carry energy, equal to $$\mathcal E(x)=\epsilon_0E^2(x)$$. You can easily imagine these waves in isolation, since these waves have little to do any more with the charges that originally created them.

You can see that these waves carry energy by the fact that after this huge amount of travel, they can still cause charges to oscillate in the receiving antenna here on earth.

• I mean, using the appropriate metric, the distance those waves have from their source charge is ... 0.
– Yakk
Commented Dec 4, 2023 at 18:28
• @Yakk, Yes, electromagnetic waves move on null geodesics, approximately, which have zero length. But the 11 billion km cited is in the reference frame of the Sun or probe or similar. More precisely, it would use the post-Newtonian approximation to gravity in our Solar System. One might also speak of affine parameters. Anyway, the answer is correct that the source and wave are widely separated regarding any ongoing causal influence Commented Dec 6, 2023 at 1:09

There are two things to note here:

1. Electrons are free to move wherever they like to be within any electrical conductor. The steady state is that they do not feel the urge to move at all (ignoring thermal movement). When you have a charged metal sphere, there is no energy available within that metal sphere.

2. If you take an electron from a charged metal sphere, it is going to feel a force near the sphere's surface. And when the electron moves along the direction of that force, it is going to gain/loose energy accordingly. And if this accelerating force is strong enough (high enough field strength), it is going to gain enough energy to knock off another electron from an air molecule before it hits that molecule, allowing a cascade of electrons and ions to form that we call a spark or lightning bolt. This accelerating force does not exist within a conductor, it exists within the isolators around it. We call it the electric field. And it is the source of the energy that the electrons gain in a spark.

We basically use wires to remove any potential difference between two points (unless we talk high frequency AC). The energy, however, moves through the electromagnetic field around those wires. That is where charges are accelerated.