How can energy be stored in a field?

Question

Textbooks say that in a capacitor or inductor, energy is stored in a electric or magnetic field. How can energy be stored in a field? Mathematically it can be proved but I am not able to feel what it means physically.

Answer 1

Energy stored in fields = the total energy required to assemble the fields

It takes energy to bring the charges to specific positions to assemble the field, and when you let everything go, the charges will just fly apart. The energy you stored in the field becomes the kinetic energy of the charges once you let them go.

Answer 2

A field is a difficult concept to grasp and then assigning energy to a field makes it more difficult.

In some situations it is easier to consider that the energy is stored elsewhere than in a field because you can visualise one approach much easier than another.

You have a system of two masses and you separate the two masses and the net result is a gain of gravitational potential energy.
The question is "Where is that energy stored?"
The answer is that it is stored in the system of two masses.
When the question becomes "Where in the system of the two masses is the energy stored?" the question is more difficult to answer.
I think what you are asking is what is the rational for saying that it is stored in the field.

I think that the best answer that I can give is that as part of a theory it works.
From the idea that a field is a store of energy lots of predictions can be made which are shown to be correct when experiments are performed.

Perhaps you can look at the idea that a field has energy is a step forward as a theory?

You will know that the electric potential energy $U$ stored in a capacitor is given by the equation $U = \frac 12CV^2$.
The origin of this potential energy is in some to assemble positive and negative charges on two electrically isolated conducting plates.
This description is probably very clear because you know that to separate positive and negative charges work has to be done and that work increases the electric potential energy of the system of charges.
You know the electric potential energy is there because if you connect a resistor between the two plates a current will flow and heat will be dissipated in the resistor.
There is a change from electric potential energy to heat.

However there is another way of describing this situation using the idea that the electric potential energy is stored in the electric field which was produced when the capacitor was charged.
For an ideal parallel plate capacitor the capacitance $C = \dfrac{\epsilon_{\rm o} A}{d}$ where $d$ is the separation of the plates and $A$ is the area of the plates.
If there is a potential difference between the plates, $V$, then the uniform electric field between the plates $E = \dfrac V d$.
Substituting for $V$ in the energy stored equation gives $U = \frac 12 \epsilon_{\rm o} E^2 \cdot Ad$.
Now $Ad$ is the volume of the space between the plates and so $\frac 12 \epsilon_{\rm o} E^2$ is the energy stored in the space per unit volume and what is in that space?
An electric field.

This is a result that can be derived from Maxwell’s equations.

Now in a sense this is where you have to leave the easy to imagine picture of separating charges this producing a store of electric potential energy and perhaps start believing that “energy is stored in a field" is a good idea.

You have learnt that an electromagnet wave comprises an electric field and a magnetic field oscillating mutually at right angles to one another.
Being a wave it carries energy and so an electromagnetic wave must have energy associated with it.
Where is that energy stored?
Well it must be part of the wave and from Maxwell’s equations you can show that it is $\frac 12 \epsilon_{\rm o} E^2$ per unit volume for the electric field part and the same amount again for the magnetic field part.
So the energy propagated by the wave is “stored” in the electric and magnetic fields.
Is this proof enough that an electric field is a store of energy?

To create the electromagnetic wave work had to be done and then that energy is transferred in the electric and magnetic field of the wave.
It is a difficult idea to grasp in that there seems to be nothing there and yet that nothingness (the electric and magnetic fields) has energy.

Answer 3

Here's a very nice argument for energy storage in fields, which I learned from a textbook by Chabay and Sherwood. I'll phrase it for the gravitational field, but you could use electric charges just as well.

Imagine that you have two masses that are separated from each other and at rest. Due to gravitational attraction, they'll start to fall together, so after a while your system will have some positive, nonzero kinetic energy. But the gravitational potential energy becomes more negative as the two masses approach each other, so we can say that total energy is conserved.

Now let's take this system and partition it, accounting for every part separately. There are two masses, so split the system into two parts, right? The first mass starts at rest and ends up moving, so its kinetic energy is increased due to work performed by its surroundings. The second mass also starts at rest and ends up moving, so its kinetic energy is also due to work performed by its surroundings.

This should bother you a little bit. Here is a universe that has two things in it. If I consider the entire universe, both things together, energy is conserved; however if I consider all of my things separately, energy is not conserved. Isn't this bookkeeping inconsistency troubling?

The way to recover conservation of energy here is to recognize that my partition is missing a member of my universe: the gravitational field through which the two masses interact. If there's energy stored in this field, and that energy is reduced as the masses speed up and approach each other, than I can reconstruct my entire system and preserve conservation of energy: the system is (mass + mass + field).

In advanced electrodynamics you go further, actually computing where the energy is stored for a given electromagnetic field configuration. But the idea is just the same.

Answer 4

How can energy be stored in a field?

Well, fields are not some "fairy-tale" fictions just used to compute forces.

They are real; they have momentum, stress, energy; they interact with matter, charges; exchange energy and momentum with them.

Energy conservation is a local process which evidently implies electromagnetic field between two interacting charges must mediate the energy and momentum exchange between the charges and hence must have energy density and momentum.