Derivation Of The Energy Stored In A Capacitor

[Bashyboy]

Hello,

I am currently reading about the topic mentioned in the title of this thread. For the most part, I understand the derivation; however, at the end of the derivation, the author adds one little condition:

"Where is the energy in a parallel plate capacitor actually stored? Well, if we think about it, the only place it could be stored is in the electric field generated between the plates."

How is the energy not stored in the capacitors? Can't the charges, that comprise the charge of the plates of parallel capacitor, possesses the potential energy? I mean, after all, they the ones having a force applied over them through a distance (the distance between how far apart the parallel plates are)? I know the idea is useful, for it allows us to define the energy density of an electric field. Why and how is it stored in the electric field between the plates?

 

[mfb]

The charges have a potential energy, but where is this energy stored?
In the electric field between the plates.
This is similar to gravity: Gravitational potential energy is stored in the gravitational field.

 

[Bashyboy]

Well, why is gravitational potential energy stored in the gravitational field?

 

[Nugatory]

Well, why is gravitational potential energy stored in the gravitational field?

It's somewhat analogous to asking where the potential energy in a stretched string is. We have to do work, increasing the potential energy, to move the two ends of the spring further apart, but we can't easily associate the energy with either moving endpoint.

 

[phyzguy]

The Feynman Lectures on Physics, Vol 2, Ch 27 has an excellent discussion of electromagnetic field energy and momentum, and why we think the energy is located in the electric field region between the plates.

 

[cwilkins]

The way I think about it is you had to put some energy into forming that configuration of charges. The energy went into separating the charges and putting them in place, which means there will be some potential energy associated with holding those charges apart from each other. The result will be a net electric field which does have an energy density. In that sense the energy is stored in the resulting net electric field and not the charged particles themselves. The distinction is subtle, but the particles themselves are not storing the energy; it is the result of their charge and position, whose effects can only be manifested through a field. (I wasn't really sure how to word this, so I apologize if it isn't clear.)

 

[mfb]

Apart from those static setups, radiation shows that electromagnetic fields have energy.

The energy density of an electric field is $u=\frac{1}{2}\epsilon \left(\vec{E}\right)^2$.
A capacitor with area A and distance d (without dielectric) has a capacitance of $C=\frac{\epsilon A}{d}$.
Charged up to a voltage of V, the electric field is $\frac{V}{d}$ in a volume of $Ad$. This gives a total energy content (of the electric field) of $E=\frac{1}{2}\epsilon Ad \frac{V^2}{d^2}=\frac{1}{2} \frac{A}{d} V^2 = \frac{1}{2} CV^2$ - which is exactly the energy stored in the whole capacitor.

 

[Bashyboy]

Is this a proper way of thinking about why the potential energy is stored in the electric field?

So, if we have a parallel plate capacitor, initially uncharged, which is connected to a battery, the battery will use its chemical energy to push electrons to the plate of the capacitor that is connected to the negative terminal. For the very first electron, no work is required to push it, because the plates have no charge, there is no repulsion. However, once that first electron reaches the plate, the plate will be charged, and will repel an electron in the plate connected to the positive terminal. As more and more electrons are pushed towards the plate, the repulsion becomes stronger, and to push an electron requires more chemical energy. Once the capacitor is charged, it is then disconnected and connected to, say, a light-bulb. Because of the configuration on the charges, the electric field lines that originate from the positive plate do work on electrons in the negatively charged plate...

Hmm I seem to have come to a dead-end in my discussion. I was going to mention something out the electric-field lines having the potential to do work, and that's why the electric field stores the energy. But how can the electrons go from the negative side of the capacitor, through the wire to the light-bulb, and into the positive side of the capacitor, if electrons are directly attracted to positive charges? Won't the electrons go directly through the gap to the positive plate?

 

[mfb]

Won't the electrons go directly through the gap to the positive plate?

That gap is an isolator, it does not allow electrons to go that way.

 

[Bashyboy]

Oh, I see. So, my train of thinking, was it correct?

 

[mfb]

How the charging process looks like? Yes.

 

[Bashyboy]

Well, the electric field still has the potential to do work, due to the configuration, right? Meaning it would possesses potential energy?

 

[mfb]

I would not say "the electric field possesses potential energy" - it just has energy, which can be used to perform work, or other things.

 

[xAxis]

Imagine two identical capacitors, but only one is charged. Put some small charge in the gap of uncharged capacitor, noting happens, it just hangs there. Put the same charge in the charged capacitor, it will imidiately move towards one end. This will happen in every point of the gap, so we see that the energy is in the gap, because the field is there.