Question: My physics teacher told me the statement "The energy of a capacitor is stored in its electric field".

Now this confuses me a bit. I understand the energy of a capacitor as a result of the work done in charging it, doing work against the fields created by the charges added, and that the energy density of a capacitor depends on the field inside it. But how exactly do I interpret energy being stored "within" a field?

I saw that there have been similar questions asked here, but I wasn't able to understand the answers clearly, or found them unsatisfactory.

Answer: 
**What "is" an electric field?**

One way to look at the world is to see it as a cellular automaton; something resembling [Conway's game of life](https://playgameoflife.com/). Of course it's not so simple — the playing field looks different from observer to observer, everything is fuzzy etc. — but as a mental *model* it's quite productive.
Each little volume has a set of properties which interact with other properties and propagate with each "tick" (only) to neighboring cells, thusly establishing the speed of light.

Such properties are essentially a set of "influences" on various kinds of "particles"1: Charges are attracted, repelled or deflected. Matter is attracted. Quarks are glued together. These "influences" are what we call "fields" in a macroscopic view. **Fields are properties of points in space time** that describe how this point in space time interacts with "particles".

**How can a field store energy?**

One of these is the electrostatic field, for example between capacitor plates. It can accelerate charges, which means that it gives them kinetic energy. ***The field can do work on matter*; that is why we say it "contains" that energy.** Like all energy it is conserved: Charges flowing along the field will weaken it until its energy is gone. "How" exactly it stores the energy is a somewhat meaningless question: "How" does a missile store its kinetic energy? Fields and their energy are properties of spacetime volumes; that is all we can say.

While "fields" (i.e., spacetime properties) are "concepts" they do have a certain reality (beyond the fact that we can measure "their" effect, which is somewhat circular): Their energy actually makes the space heavy, although that wording is a bit awkward: Energy *is* mass, so it doesn't *make* anything heavy, it *is* heavy. If you want a bit of pop science, a magnetic field strong enough could conceivably be "heavy" enough to [produce a black hole](https://www.wtamu.edu/%7Ecbaird/sq/2014/08/21/what-is-the-strongest-magnetic-field-possible-is-there-a-limit/), unless something else happens first, like spontaneous pair production etc. There is no principle reason why this should not be possible with other fields as well.

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1 I'm putting "particles" in quotes because what we call "particles" is, at a closer look, a volume with special set of properties that just happen to perpetuate themselves, sometimes infinitely, sometimes not.
The stress-energy tensor
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At every point in space there is a density: Build a sphere (or any shape) of gravity sensors and measure the net acceleration into the sphere (technically the surface integral of F/m\*dA). For normal matter [greens theorem](https://en.wikipedia.org/wiki/Green%27s_theorem) means you get the total mass inside regardless of external masses.

Note how I said "normal matter"? The actual measurement is *mass + 3 volume pressure*. For normal matter the volume-pressure (which has units of energy) is far smaller than the mass. But for fields and light this is not the case.

What we actually have at each point is a [stress-energy tensor](https://en.wikipedia.org/wiki/Electromagnetic_stress%E2%80%93energy_tensor) which enters directly [into general-relativity](https://en.wikipedia.org/wiki/Einstein_field_equations). This tensor bundles up density, momentum flux (i.e. a moving river) and stress (tension/compression/shear) all into one 4x4 symmetric matrix. A gas of photons has is under hydrostatic pressure equal to 1/3 of it's (energy) density (denoted as *w*=+1/3). An electric field has *w*=-1 in one direction like an extremally lightweight string under tension. This tension pulls the plates together in a capacitor.

The entire stress-energy tensor [Lorentz transforms](https://en.wikipedia.org/wiki/Lorentz_transformation) and so can be deduced by modifying our sphere experiment to measure the gravitational accelerations of objects moving at different speeds in different directions.

This all sounds abstract, but for a [charged black hole](https://en.wikipedia.org/wiki/Reissner%E2%80%93Nordstr%C3%B6m_metric) all the energy and stress in the electric field around the hole modifies the spacetime curvature: particles *even if uncharged* will orbit differently.