Ontologies in physics—4: Minor changes in the ontology of EM force-fields. Understanding potential energy.

OK. After posting the last entry in this series, a thought—actually, a problem—occurred to me right the next day. I’ve noted it via an update at the end of the last post; see if you didn’t check it out again [^]. This is not the only change though. The entirety of the last post has undergone substantively revised wording, though I left the positions untouched. …All my posts are always are transient—drafts subject to change without notice. (No, I don’t care for you. Really.)

But anyway, coming back to the problem with the ontology as presented in the past post: I also saw a way to get out of that problem right on the same day, just before going to sleep at around 9 PM or so. However, this issue being tricky, I decided to wait for a few days before giving my solution. Further, since the last post had already become so huge (5,000+ words), I decided to include the solution not there, but here, in this post. Let’s begin with it.


1. The ontological problem in regarding \vec{E} as physically existing:

1.1 The problem in essence:

The ontological problem which comes up with the description as given in the update to the last post is the following.

If we accept a physically existing field for the electrostatic phenomena, precisely of what quantity is its force field? Is it a field of the electric field vector \vec{E}_{1} = \dfrac{\vec{F}_{1T}}{q_T}, or is it of electric force itself \vec{F}_{1T}?

That’s the question to be settled.

1.2 The reasons why it is a problem:

If it’s a field of \vec{F}_{1T}, then we haven’t succeeded in isolating a field quantity that depends on q_1 alone, because \vec{F}_{1T} additionally depends also on q_T. On the other hand, if it’s a field of \vec{E}, then the physics of the interaction becomes quizzical. It is not clear precisely what happens physically so that a local field of \vec{E}_{1} comes to produce the effect of \vec{F}_{1T}?. It is not clear what physical process must occur at \vec{r}_T such that it corresponds to the required multiplication of the local \vec{E}_{1} by q_T.

Note, mathematically the difference is trivial. In fact, physicists (and even engineers) routinely do dimensional re-scaling of their equations. Dimensional scaling is just a handy trick that keeps the underlying physics the same. However, note, such a scaling procedure modifies the entirety of an equation; it affects all terms in the equation. In contrast, the key issue here pertains to only one term in the equation; it pertains to a difference of physics. The quantities \vec{E}_{1} and \vec{F}_{1T} are of different units, and this difference will come to exist as is in every dimensional re-scaling. It will remain intact in any unit system you choose, whether Gaussian, imperial, SI, or any other.

It is clear that experimentally, we measure only \vec{F}_{1T}, not \vec{E}_{1}. Therefore, if you assume a physical existence for \vec{E}_{1}, converting it to the observed \vec{F}_{1T} quantity requires a detailed model, a separate physical mechanism to be given.

1.3 What I figured about this issue:

There is simply no way to resolve this issue using the EM ontology as we have presented it thus far. The reason is that in classical EM, the charge magnitudes are arbitrary. This is the key reason why we need to factor out q_T in the first place.

The only way out of the tricky situation, I think, is to make reference to a physical fact discovered after the systematization of EM theory had already occurred, and to bring the import of this fact into our ontological description, thereby revising all the relevant parts of it. The experimentally discovered fact which provides the way out is this:

In nature, there exist only elementary charges (of electrons and protons), and their charge is the same in magnitude, just opposite in sign. The electronic charge e is a fundamental constant of the universe.

Therefore, in a most fundamental description, i.e., in ontology, we never have to worry about multiplication by arbitrary charge-magnitudes. We can say that the actually existing field is only of the force \vec{F}, and never of the electric vector \vec{E}. Each charge causes an exactly identical field to be developed in the background object, and any other charge at a distance which interacts with a given field also must possess the same universal magnitude of e (within the algebraic sign).

In short, Nature never does multiplications. We have to make this change to our ontological description.

Thus, in our revised ontology, \vec{E} is only a convenience of mathematics. This quantity is useful when we want to treat a system of a large number of charges as if it were a single charge. In classical EM, we in fact don’t even require q to be an integer multiple of e; through homogenization, we allow q to have any real value. Once again, this too is only a mathematical approximation, a convenience to adopt at a higher level of abstraction. But in terms of what actually exist out there, there are only the \vec{F} fields produced by an integer number of elementary charges. The ontology as such has only \vec{F} as a physically existing condition.

1.4 Another way to look at the problem and our solution:

Realize, this solution also takes care of another problem that I didn’t note but spotted later on.

The field is an attribute of the background object, even if it is caused by, or having effects on, the EC objects. Therefore, ideally, a physically existing field should show an equal abstractness from both the charges.

However, the electric vector quantity \vec{E}_1 comes with the magnitude of the first charge built into it. That’s why \vec{E}_1 is not “symmetrical”: since the multiplication by q_1 is built into it, we are then left looking for the physics of multiplication by q_2 at the other end.

Adopting a \vec{F} field as the actual existent makes the description at both “ends” of the field symmetrical: the same process can govern the production of a field by a charge, and the production of a force on a charge.

1.5 We know nothing about how a charge causes a force-field or vice versa:

But how exactly does a charge produce a force-field in its vicinity? Ontologically, there is no evidence to make this an issue of any significance. In fact, any answer you try to think of will end up replacing a more general phenomenon by a more specific model.

For instance, you may put forth a model which says that a charge acts like a press-fitted ball inside a continuum. However, this “model” doesn’t have much value, because the background object is basically not mechanical in nature (it is not an NM Object), and so, its interactions with EC objects also can’t mechanical in nature. In fact, a purely mechanically inspired explanation carries a definite risk that we may end up taking it too literally and forget the actual phenomena.


2. The points about the history of EM fields which I learnt only this week:

After I had thus figured out what should be the proper EM ontology in this \vec{E} vs. \vec{F} issue, I made a few rapid references to books and papers on the history of EM. I came to know a few very valuable points for the first time in my life. (My earlier readings of history were mostly focused on developments in continuum mechanics, variational calculus, and to some extent also QM, but not on EM.) Here are the main points I learnt this week:

2.1 Maxwell had a dielectric aether:

I already knew that Maxwell had a mechanical aether; that he ascribed NM-object properties to his aether. But I didn’t know that he in fact also had dielectric properties ascribed to the aether—which was rather unexpected to me.

2.2 Lorentz beat me to it by 90+ years:

All along I had thought that the kind of “aether” which I had in mind was a new development.

I had thought of a non-mechanical aether way back, probably while in UAB (following a remark by Ayn Rand in her seminar in the ITOE book, second edition). Soon later, I had also used it in my first attempts in resolving the QM wave-particle duality, which started from the early 1990s. Much later on, I had also used it during my PhD-time research on QM. In the Abstract of my PhD thesis, I had said:

“The present view of aether differs from all the previous views in the sense that the author takes luminiferous aether to be a physically existing but non-material substance.”

It was only in this week that I came to know that H. A. Lorentz (the 1902 Nobel laureate) had already figured it out right, some 90 years earlier. It’s just that people never highlighted the difference of his view of aether from Maxwell’s in any discussion on aether, so I couldn’t even suspect that I was not the first.

2.3 The only (minor) difference in Lorentz’s idea of the aether, and our background object:

The only difference between Lorentz’s ontological ideas and mine seems to be that implicitly, by always visualizing a singularity for the electrostatic field (albeit only as an approximation for the unknown physics of a charge of a finite size), I naturally came to imagine that no field would be present at the location of the charged object itself. Even if it’s not singular, ontologically, it’s in an entirely different category from the aether.

In contrast, Lorentz thought that what we call the EC Object had some structural detail.

Initially, he sought a dynamical description, inspired from mechanical models, even though he was clear that his aether was not mechanical in nature. But he did think of a charge has having a structure; he thought in terms like changes in shape that are undergone by a charge when it is in motion. His motivation was to supply a mechanism for his relativitistic contractions.

Now, Lorentz did advocate the position that the electron is to be regarded as the source of all electromagnetic phenomena. However, to him, a charged body still was a composite object, with both charges (elementary ones) and aether present within its volume—as happens in ions. (He had explicitly used ions in his initial thinking.) This is the reason why the \vec{E} field would be present “inside” his “charge” too—like gravitational field is present inside the earth too, not just outside.

Apparently, Lorentz didn’t notice our above-mentioned difficulty (regarding the physics of multiplication). However, such differences between Lorentz’ view and ours amount to nothing as far as the basic ontological scheme is concerned.

2.4 Lorentz’ work is grossly under-rated:

Just think: How much independence of thought and conceptual skills Lorentz must have required in originally (and correctly) isolating the aether from the charges—and then, even using them to simplify Maxwell’s (very) complicated theory.

Note, other people were still working with only a mechanical aether, even people of the stature of Maxwell, Stokes (who formulated the aether-wind idea), Helmholtz, Hertz, FitzGerald, and others—all first-rate thinkers.

Further, some other brilliant people like Poincare had already begun becoming fully skpetical of the entire idea of the aether.

Situated in this intellectual climate, Lorentz still managed to reach an essentially right ontological position to be had, and then held fast to it, saying that

the universe was basically electromagnetic in nature, not mechanical; but there still had to an aether, although it was of an electrical nature, not mechanical.

All in all, a unique position in every respect.

2.5 My “Maxwell’s equations” always were in Lorentz’s/Heaviside’s form:

The second fact I learnt was that it is Lorentz’s reformulation of the electrodynamical equations which are now taught as Maxwell’s equations. This fact is important.

The basic form in which these equations come itself has been guided (“informed”) by Lorentz’ ontology. Therefore, if you just “stare at” these equations long enough, the right ontological structure implied by them begins to become clear to you. Such a thing would have been plain impossible using the form of Maxwell’s original equations.

So, in a way, it is not at all surprising that starting with Maxwell’s equations as recast in Lorentz’s mould, I should have been able to slowly but correctly trace the logic back, and thereby arrive at the same ontology. The form of the equations itself makes the task so easy for us today.

The only thing you require here is to learn to ignore teachers like Feynman on points like the aether or the proper origins of any mathematical concepts. It also helps that other teachers like Resnick and Halliday cover the same material very carefully, in a manner that develops real understanding.

Addendum on 2019.09.24 17:53 IST: Added Heaviside’s name. The form of the equations as taught today might be following a notation which is based on what was put forth by Heaviside. However, I also know that Lorentz was very clear that the aether had to be electrical i.e. non-mechanical in nature, but I am not sure where Heaviside stood with respect to aether (though he might have suggested or worked on the contractions). … Would need to dig through the history to be able to identify exactly who contributed what, and priority. Don’t have the time to go into it all right now.


3. Summarizing the changes to our EM ontology:

For obvious reasons, what we have so far called the background object (B^{0}) shall henceforth be called Lorentz’s Aether in our ontology.

Also, in view of the difficulty with \vec{E}, we now declare that our entire EM ontology has been changed in such a way that any EC Object will be regarded, in any fundamental ontological discussion (unless otherwise stated), as being only an elementary charge, having the value such that -|q| = e.

We now return to the planned topics. However, it’s obvious that we won’t be able to finish them all right this time around.


4. The potential energy as an internal energy of a thermodynamic (and mechanical) system:

We begin by regarding two or more EC objects as forming a thermodynamic system. We first want to make sure that we understand the potential energy of a system.

The archetype of thermodynamic systems is the usual piston-cylinder arrangement which exchanges heat and work with the rest of the universe, i.e., with the (given system’s) environment.

4.1 Two transient forms of energy that appear only on system boundaries: heat and work:

Heat and work are the two transient forms of energy that come to appear on the system boundary—and at no other point in space. Their quantities are nonzero only for the time when an actual exchange of energy (of their respective form) is actually ongoing.

Heat cannot be stored in a volume (say, the volume of a system) because heat is not a quantity defined in reference to a volume. It is defined only at the boundary surfaces of a system. We call heat that energy which passes a system boundary because of a single reason: the temperature gradient as it exists at the boundary surface. (Temperature gradients purely internal to the system or purely external in the environment don’t count—only that at the boundary does. If concerned with internal temperature gradients, you have to divide the system into as many sub-systems as you wish, and then account for them but only at the boundaries, again!)

To repeat, heat cannot be stored. If you want to say “heat energy stored in a system,” just gulp the word “heat” and utter the word “thermal” in its place, and you will be perfectly OK—it will keep professors like me happy. (Thermal energy is due to a non-zero kelvin temperature and can be stored; heat energy is due to a difference in temperatures and cannot be stored.)

Work cannot be stored in a volume (say, the volume of a system) because work is not a quantity defined in reference to a volume—only for the boundary surface, and only during the time when the boundary is physically undergoing a displacement.

Heat doesn’t require a displacement of boundary, but work does. If a system volume is forced to remain constant, then the system cannot exchange work with its surroundings.

To repeat, work cannot be stored. If you want to say “work stored in a system,” just say “increase in the internal energy of a system due to work done on it,” and you will be OK. However, since most professors themselves say things like “work done on a system gets stored in it,” guess you are only technically wrong when you say things like “work stored in a system.” It’s an informal expression that does not explicitly state all the essentials by adding “as its internal energy”. Professors too use the curtailed expressions. That’s why, even if you are technically wrong, you too shall come to pass your viva. [Your professors did the same.]

Sign convention: Here, it’s useful to keep in mind the steam engine and what we do with it. Heat supplied to the system (by its environment) is algebraically positive, and the work done by the system (on its environment) is algebraically positive.

So, the positive quantities are: heat supplied to, and work done by, the system.

4.2 Internal energy:

The total energy stored inside a system is called its internal energy.

4.3 The first law of thermodynamics:

A supply of heat to the system increases its internal energy. A supply of work by the system (because the system boundary moves with the motion of the piston due to the expansion of gas) reduces the internal energy.

The first law of thermodynamics says that the preceding description is complete. For instance, aliens don’t get to change the total energy content of a system. Accordingly, the first law of thermodynamics can be stated as:

\text{d}U = \delta Q - \delta W.

Notice that, strictly speaking, the first law does not make a statement about the quantity of energy stored in the system. It makes a statement only about changes in the stored quantity.

The \delta in front of Q and W denotes an inexact differential—an infinitesimally small quantity whose value depends on the path taken in a state-space like the PVT diagram. The \text{d} in front of U denotes an exact differential. The internal energy of a system is a state variable; it doesn’t depend on the particulars of how the energy changes. Yes, here, the difference in two path-dependent infinitesimals is a path-independent infinitesimal.

In the basic EM theory, we don’t care for thermal considerations, and so for our purposes here [why did I type so much?], the first law becomes:

\text{d}U = - \delta W.

Effectively, in EM theory, there are no temperature gradients at the boundary between an EM system and its environment. Everything in the EM theory stays at the same temperature. There are some important points regarding what is regarded as the boundary of a basic EM system. We will come to it in a short while.

4.4 Work done on a system, internal energy, and potential energy:

The work done by the system is positive, i.e., +\delta W. Hence, work done on the system is negative, say, -\delta W.

When work is done on a system, there is a corresponding increase in its internal energy. This increase in U makes the system acquire a certain potentiality—the potentiality to produce some amount of work during its interaction with its environment.

If heat is supplied to a system, there again is a corresponding increase in its internal energy. This increase in U too can make the system acquire a certain level of potentiality to produce some amount of work. In EM discussions, we neglect heat, but remember, U can increase via heat supply at the boundary too.

Thus, the actuality of some energy already supplied to a system imparts to it the potentiality that it can later on produce some amount of work. As Aristotle said (and Dr. Binswanger once pointed out in one of his lectures), actuality precedes potentiality.

The potential energy of a system is that part in the increases in its internal energy which it can later on convert into work.

The potential energy, thus, is a form of internal energy. However, if a system cannot produce work, it may still have a reservoir of an internal energy, but you can’t call this stored energy a potential energy.

4.5 Example mechanisms to store energy:

But what is it which allows a system to at all store energy? What precisely happens physically when the internal energy of a system increases? Here, we have to look at the structural details of the system, its physical internals.

In a piston-gas system, there is a gas that gets compressed when you do work on the system. The gas consists of molecules; pressure is the momentum they impart during their collision with the cylinder and piston walls. So, when you compress a gas, something about its internal configurational state changes. The arrangement of the NM Objects contained in the system changes. The increase in the internal energy comes about solely because of the changes in the configuration of the NM Objects constituting the gas. So, the internal energy here is the energy of a configuration internal to a system.

You might as well have a mechanical spring in place of the gas in a piston-cylinder arrangement. Same thing. The “changes in the configurational states of the spring,” etc.

You might as well forget to have an actual piston-cylinder arrangement around it. You can have just the spring. Same thing. The spring can still store internal energy via its deformation. There is no cylinder, no piston, but only forces at its two ends, arising out of a direct contact with some NM objects in the environment. In short, the piston and the cylinder can be imaginary; the spring alone still qualifies to be called a thermodynamic system.

You might as well have just two NM Objects interacting with gravity. Same thing. Fix the system boundaries at the NM Objects (say at the centers of mass of the earth and the moon), but let the boundary be movable. Then, this arrangement sure forms a thermodynamics system. This system too is able to have (“store”) internal energy. It too can produce work at its moving “boundaries”.

4.6 The thermodynamic boundary for the earth–moon system:

Notice the fact that earth’s gravity affects not only the moon but also a satellite when it’s on the dark side of the moon. The logical system “boundary” for the earth-moon system still remains fixed at their respective centers of mass. So, the logical “boundary” of a thermodynamic system—the place where work-exchange occurs—is not necessarily a surface (let alone an enclosing one); it can also very well be a set of isolated point-positions in the physical space.


5. The electrostatic potential energy (\Delta U):

5.1 Electrostatic interactions as a mechanism to store energy:

Instead of two gravitating objects, you might as well have just two EC Objects interacting electrostatically.

Fix the movable system boundaries at the EC Objects (the elementary charges). This system too has an internal energy associated with the configuration—the relative positions—of the two charges, so long as the charges are able to occupy different positions. That’s what the electrostatic internal energy is. This system too can produce work at its moving “boundaries.”

The “boundary” of the EC system too is identified with point-locations, and not with an enclosing surface.

5.2 The electrostatic potential energy is \Delta U:

In basic EM theory, U doesn’t increase because of a heat supply—we ignore heat. Hence, work is the only mode available to cause increases or decreases in the internal energy. It so turns out that at the most fundamental level, any EM system is frictionless. So, any EM system should be able to produce just as much work as was put into it. This presumption is correct.

All the work done in moving EC objects indeed forms an increase in the internal energy of the EM system comprised of them, and this entire increase in the internal energy is available for producing work at any later time. We call it the electrostatic potential energy.

The electrostatic potential energy is the increase in the internal energy of an EM system due to the work done on it in shifting the EC objects (point-particles of elementary charges) contained in it.

5.3 The electrostatic potential energy is not the same as the internal energy U:

Note very carefully. It is the increase in U due to shifting of EC Objects which is being called the electrostatic potential energy, not the total energy content (i.e. the internal energy) denoted as U.

This distinction is important, and we will have a definite occasion to use it a bit later.


Enough for today. Got tired typing and re-arranging material. (Still, 4,100+ words already!) I will cover the rest of the planned portion in the next post, due soon. As usual, minor changes may be effected to this post (including even misleading formulations), but any significant change (especially that in positions/arguments) will be prominently highlighted.

Bye for now, and take care… See you the next time (if still left interested in this topic).


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(Marathi) “maavaLatyaa dinakaraa…”
Lyrics: B. R. Tambe
Music: Hridaynath Mangeshkar
Singer: Lata Mangeshkar


History:
Originally published: 2019.09.21 18:39 IST
Added a note about Heaviside in the section 2.5:  2019.09.24 17:53 IST