# Potential energy vs field energy

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I am investigating the relation between potential energy and field energy. I think electric and magnetic fields might be simplest to consider.

Imagine a simple system of two small charged particles (not elementary particles) separated some distance apart.
- we define that the potential energy of this system is U0.
- we compute that the energy stored in the electric field is: E0
(note: we computed E0 by integrating the energy density formula for the electric field over the whole space. No need to consider our particles as point charges - say they contain uniformly distributed finite amount of charge in a small finite volume.)

We slowly separate the two charged particles some distance farther apart. To do this, we invested some work W.

What is now the potential energy U1 of the system? Is it: U1=U0+W?
What is now the field energy E1? Does it equal to E1=E0+W?
What is the change of the total energy of the system - does it equal to W?

My opinion: Yes, the U1=U0+W; yes, the E1=E0+W; and yes, the total energy change equals W. Therefore the change in potential energy and the change in field energy represent the same thing. We should consider either potential energy or the field energy when we compute energy balance. We cannot consider both energy changes as we would be doing double-counting error (in our example, we might compute that the energy difference is 2W instead of W).

Let's be stubborn and try the opposite anyway.... let's suppose that potential energy and field energy are two separate things. In this case we could claim:
U1=U0+x*W
E1=E0+(1-x)*W
(where x is a number between 0 and 1) Using these claims the energy will still balance (the energy change is x*W+(1-x)*W=W), but the question would be what is the factor x? Is it x=0.5? Why?

So, can we safely say that the change in the potential energy and the change in field energy represent one the same thing? Are there exceptions?

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Potential energy for certain configurations is often referenced to U=0 at infinite separation, but what is important us that you are always calculating the change in PE between two configurations, so any nonzero reference value will cancel.

Field energy is irrelevant to the PE; consider the field energy of a capacitor. The energy stored can be different because it depends on area and separation distance while holding V constant, and the PE is qV. That’s the work done, by one field on the charge in question.

If, as you say, the change in field energy is the same thing, then why does this matter? It would simply be a second way of doing the calculation.

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Posted (edited)
14 hours ago, Danijel Gorupec said:
I am investigating the relation between potential energy and field energy. I think electric and magnetic fields might be simplest to consider.

Imagine a simple system of two small charged particles (not elementary particles) separated some distance apart.
- we define that the potential energy of this system is U0.
- we compute that the energy stored in the electric field is: E0
(note: we computed E0 by integrating the energy density formula for the electric field over the whole space. No need to consider our particles as point charges - say they contain uniformly distributed finite amount of charge in a small finite volume.)

We slowly separate the two charged particles some distance farther apart. To do this, we invested some work W.

What is now the potential energy U1 of the system? Is it: U1=U0+W?
What is now the field energy E1? Does it equal to E1=E0+W?
What is the change of the total energy of the system - does it equal to W?

My opinion: Yes, the U1=U0+W; yes, the E1=E0+W; and yes, the total energy change equals W. Therefore the change in potential energy and the change in field energy represent the same thing. We should consider either potential energy or the field energy when we compute energy balance. We cannot consider both energy changes as we would be doing double-counting error (in our example, we might compute that the energy difference is 2W instead of W).

Let's be stubborn and try the opposite anyway.... let's suppose that potential energy and field energy are two separate things. In this case we could claim:
U1=U0+x*W
E1=E0+(1-x)*W
(where x is a number between 0 and 1) Using these claims the energy will still balance (the energy change is x*W+(1-x)*W=W), but the question would be what is the factor x? Is it x=0.5? Why?

So, can we safely say that the change in the potential energy and the change in field energy represent one the same thing? Are there exceptions?

Hi Danijel,

You are  right  -  this may underlie your difficulties in the Zeeman thread.

Separating energy into 'Field Energy' and 'Potential Energy' is the difficult way to go about things.

Choosing electric or magnetic fields makes it even more difficult since you have polarity to consider.

There is only one polarity in gravity, for instance.

For a full accounting, you also need to take consider what creates the field.
Very often we ignore this, as for instance when considering the PE of a (much) smaller body in relation to the Earth.
We attribute the PE to the small body, when in reality it belongs to the Earth -small body system.

By far the best introductory explanation of all this that I know appears in Sears and Zemansky  - University Physics, in the chapter entitled Work and Energy.

Sears introduces Work, energy, the so called Work - Energy Theorem, conservation of energy, (system) internal work and internal energy and external work and external energy etc.

The Work - Energy Theorem is the basis for the correct version of the equations you are trying out here.

Here are two scans from the chapter.

If you find these interesting there are 13 in all and a PM letting me have an Email that can receive jpegs would get you the rest.

Once you have this chapter under your belt we can make more sense in your Zeeman/magnetic thread.

Edited by studiot

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@studiot Thanks. I obtained the book (obviously a different edition than yours). Looks fine, will try to read some interesting chapters in my spare time.

I am considering electric and magnetic fields because I know them better. The gravitational field might be simpler from the potential energy point of view, but I wouldn't know how to handle its filed energy. Furthermore, the ultimate problem I am considering (what does it mean when we say that electron shifts its energy in magnetic field, like in the Zeeman effect) is related to electric and magnetic fields.

@swansont Great that you introduced the capacitor example - it is simple enough that I can make some calculations (see below). But first to answer your 'philosophical' question...

18 hours ago, swansont said:

If, as you say, the change in field energy is the same thing, then why does this matter? It would simply be a second way of doing the calculation.

Yes, these are two ways of doing calculations. But, imo, not equivalent... The 'field energy' is more precise. Specifically, it precisely defines how the energy is distributed in space (energy density distribution). The 'potential energy' obscures this information.

The energy distribution is important for the local energy conservation idea. Imo, if you cannot tell the energy distribution, then the idea of local energy conservation becomes moot. As I take the local energy conservation as a strong requirement, so I think that, at least in principle, we should be able to describe each form of energy by its energy density distribution... It is for this reason that I expect that each forms of potential energy is describable by field energy (or at least by some other way that does not obscure the energy distribution information).

As you know from my other thread, I found one example (spin magnetic moment in external magnetic field) where I cannot relate the potential energy change to the field energy change and this itches me.

Now the computations... they are striking... We consider a charged plate capacitor. During the experiment, the plates are moved from the starting distance d1 to the ending distance d2. We compute the potential energy change (first line of equations) and field energy change (second line of equations).

CASE 1 - constant charge. Everything is clear here and everything fits nicely. The PE change equals the field energy change.

CASE 2 - constant voltage. This is interesting. When we look at the system from the potential energy viewpoint, the total energy of the world does not fit! But when we look from the field energy viewpoint, everything fits nicely. Note that in the total energy of the world must include the energy in the idealistic battery that is used to ensure the constant voltage on the capacitor.

Note that in the constant-V case the field energy decreases as we are separating the plates. However the energy that is 'pumped' back into the battery is twice the work done... The potential energy viewpoint would give balanced energy only if you avoid looking into the energy change of the battery. Once you peek inside your battery,  your energy balance falls apart.

So, if I didn't do any major mistake in my reasoning, it seems that the field energy is more precise and more fundamental than the potential energy. If one really, really wants to give a precise answer to the question 'and where is the energy stored' he/she should give the answer from the field energy viewpoint and should avoid the potential energy viewpoint. What do you think?

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4 hours ago, Danijel Gorupec said:

@swansont Great that you introduced the capacitor example - it is simple enough that I can make some calculations (see below). But first to answer your 'philosophical' question...

Yes, these are two ways of doing calculations. But, imo, not equivalent... The 'field energy' is more precise.

If they give the same answer, they are equivalent and equally precise.

4 hours ago, Danijel Gorupec said:

Specifically, it precisely defines how the energy is distributed in space (energy density distribution). The 'potential energy' obscures this information.

Why does that matter, unless that’s the specific question being asked?

4 hours ago, Danijel Gorupec said:

The energy distribution is important for the local energy conservation idea. Imo, if you cannot tell the energy distribution, then the idea of local energy conservation becomes moot. As I take the local energy conservation as a strong requirement, so I think that, at least in principle, we should be able to describe each form of energy by its energy density distribution... It is for this reason that I expect that each forms of potential energy is describable by field energy (or at least by some other way that does not obscure the energy distribution information).

Energy conservation means the energy does not change, not that you know where it is located.

I can have a rotating object and know the rotational KE without knowing the mass distribution.

4 hours ago, Danijel Gorupec said:

As you know from my other thread, I found one example (spin magnetic moment in external magnetic field) where I cannot relate the potential energy change to the field energy change and this itches me.

Because it doesn’t matter, perhaps. There are a number of problems in physics where there are multiple approaches to a solution. You use the one that works.

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