0. Before we begin:
I wish I could have undertaken some fresh re-readings on the history of science before setting out to write this series of posts. I have most of the material, and over the years, I have gone through a lot of books on history / conceptual aspects of physics and maths. (Off-hand: Kline, Kolmogorov (ed.), Dugas, Truesdell, Hesse, Coopersmith, Encycl. Brittanica, and many, many others, not to mention the Wiki). Further, I’ve just came across a new source: “Energy, Force and Matter,” by Harman. Wish I could’ve gone through at least this one. But I just don’t have the time. … I would like to finish this series as fast as I can, so that I can then go over to my new theorization regarding QM. So, overall, what I write here turns out to be in a rather abstract way, and without specific references to the original writings. I rely too much on my memory and the view of these matters I developed. There’s a risk here that in writing without fresh referencing, I might be mixing my own views with the original ideations. All in all, not at all to my liking. But I have to make do with it for now, purely out of a lack of time.
A great resource for this post (and the last) is the online notes: “A visual tour of classical electrodynamics,” written for an undergraduate course at MIT [^]. Apart from the books, here is a relevant paper: Ernan McMullin, (2002) “The origins of the field concept in physics,” Phys. perspect, vol. 4, pp. 13–39 [(PDF) ^]. Also see a brief note at Prof. Philip C. E. Stamp’s Web pages: “The concept of the field in physics.” [(PDF) ^]. Both these resources were mentioned in a tweet I made on 04 September 2019.
1. The background object and spaces in our EM ontology:
We mentioned the last time that in our EM ontology, there are only two types of objects: (i) EC objects, and (ii) the background object.
In our EM ontology, the background object is a physically existing but a non-NM kind of an object.
Our view that the background object is not an NM kind of an object makes it sharply different from Faraday and Maxwell’s view, but more on it later in this series. [As an aside, even my c. 2005 paper on QM (covering my old, PhD-time approach) had mentioned the “aether” as a physically existing but non-material object. I guess this idea has been with me for a very long time.]
When you point your finger to some place in between two EC objects, i.e., if you point out the “empty space” in between any two “classical” objects, what you are actually pointing to an invisible background object-–not space—which is present at that place. We can justify this position, though its justification will progress slowly over this and the next two posts.
So, what exists in between two EC objects is not the “empty space”—not even in the physical sense of the term “space”.
Physically, the concept of space denotes the fact that physical objects (both the EC objects *and* the background object) have spatial attributes like extension, location, and also other spatial attributes (like the topological ones). The concept also includes the physically existing relations between all these spatial attributes. Mathematically, the concept of space denotes a quantitative system of measuring the sizes (or magnitudes) of these spatial attributes with which objects actually exist. [For a very detailed, in fact very long-windingly written series of posts on the philosophical ideas behind the concept of space, see my earlier posts here [^].]
Notice the logical flow: What ultimately exist are objects—that is the most fundamental fact. In fact, it is the primary (not further analyzable) fact assumed by all physics (and also by all philosophy). So, objects exist. Every object exists with all the attributes that it has; each object has a certain identity. Each object possesses every one of its attributes in some specific quantity or size. There can be no size-less attribute. As an example, a pen has the spatial attributes of length, diameter, shape, etc., and each pen exists with a specific quantity or measure of length etc. Thus, sizes do have a physical existence; sizes do exist in the concrete physical reality out there. However, sizes don’t exist as apart from the objects whose sizes (in different respects) they are.
Mathematics then comes into picture. Mathematics is a science that develops the methods using which physical sizes of comparable objects (i.e. objects having the same attributes but in different measures) can be quantitatively related to each other. Mathematical concepts refer to mathematical objects, not physical. Mathematical objects are a result of objectifying the methods devised by us for measuring the existing sizes of the physical objects.
Coming back to the concept of space: There can be many mathematical spaces (systems of measuring physically existing spatial attributes of objects), but they all refer to the same physical space. The physical space is the sum total of all the spatial attributes of all the physically existing objects, with all the interrelations between them.
The referents of the concept of the physical space are perceived directly.
A mathematical space does not physically exist. What you directly see, the sense that there is some solidity or volume-ness to the physical world, is the basis of the concept of the physical space, not of a mathematical space. A mathematical space is, most prominently, an abstract quantitative system of measuring extensions and locations of physical objects. It is a mathematical object abstracted from the concept of the physical space.
The background object is a physical substance, but having a non-material nature. In particular, it is not an NM object.
The physical space is just one way to characterize the background object because all parts of the latter do possess spatial attributes (like extension, location, etc.). However, therefore identifying the background object with the physical space (or worse, with a mathematical space) is a very basic error.
Just the fact that the background object does possess spatial attributes, does not make it the same as the physical space. … Would you call an NM-object (like a ball or a table) “space” just because it has spatial attributes? … Realize, the background object, though it is invisible (in fact it is as good as non-existent for the passage of NM-object through it), is physically an active existent at all times. Forces are being sustained in it at all times, which it makes it active in the same sense a foundation stone of a building is: no overall motion but still transmission of forces through it.
Neglecting gravity, the background object does not interact with the NM-objects. It is for this reason that no inertia or mass can be ascribed to it—one of the easy reasons why it can’t be regarded as an NM-object. However, as we shall see later, the background object can come to possess something like a state of a stress within itself. It is worth noting here that even in the NM-ontology (i.e., in solid- and fluid-mechanics), the equations governing stress/strain fields do not have mass appearing in them.
2. A few words on the difference between the 19th century view of space and ours:
The 19th century physicists (like Maxwell) took space in the same sense as Newton did—in the NM-ontological sense of the term. Thus space, to them, is a directly given (perceptible) absolute (having an independent physical existent apart from any NM-objects). They would regard objects as filling or occupying some part (some regions) of this *already* given absolute space. In our opinion, this view is somewhat Platonic in nature. They received this view from Newton.
In contrast, we take the view that only physical objects are the directly given in perception. (Perception here also includes observations of EC objects made in controlled EM experiments.) In our view, objects are not to be seen as filling space. Rather, they are primarily to be seen as possessing spatial attributes (of extension and location). Of course, we do directly perceive something spatial—the solidity or volume-ness of objects. We also directly perceive the particulars of their configurations such as their locations. But that’s all. We don’t perceive space as such. We conceive of the idea of the physical space (based on the perception of spatial attributes).
It is only at this point in development that we are able to “invert” the relation and say that objects can now be seen as occupying regions in space. But, in our view, this statement involves an application of a concept, an inversion of sorts, and such an inversion becomes possible only if you start from an abstract level. Perceptual level does not permit such inversions. This point is somewhat similar to saying that what we directly perceive are blue objects (of different hues). We do directly see that they possess blueness. But the concept “blue” is only an abstraction; it is a product of concept-formation. The blue is not out there; only blueness is, and that too, only as characteristic of actual objects.
To Newton, Faraday and Maxwell, if all physical objects were to be annihilated, then the absolute space would still be left. To us, this is an impossibility. There cannot be spatial attributes without there being objects that have spatial attributes, and there cannot be space without such objects or without our grasping a certain extended-ness or volume-ness that they have and making a concept out of it.
3. Causality and interactions:
Ontologically, in any interaction between any two objects of any kind, both the objects participating in the interaction must be seen, simultaneously, as being agents of causal actions.
A word about causality. We follow the Objectivist view of causality, put forth by Ayn Rand [^]. Thus, at the most basic level, the idea of causality has nothing to do with an orderly progression in time. Properly speaking, the nature (identity) of the objects that act is the cause, and the nature of the actions they show or undergo are the effects.
It must be understood that the concept of causality is in principle applicable to single objects as well as to interactions between two or more objects.
When certain actions of a given object leads to certain other actions by some other object, we say that they are interacting. Suppose one billiard ball hits a second ball that was initially stationary. We often loosely say that the motion of the first ball is the cause and the motion of the second ball is the effect. Actually, the respective natures of the balls themselves are causes—both of them; and the natures of their individual actions are the effects—both theirs. At the instant when the two balls are in contact, both their natures taken together are such that they determine both their subsequent actions (here, motions). Thus, there are two causes and two effects.
If in order to determine an effect (an action by some object) you have to consider the nature of some other object(s) too, then we say that they all are interacting with each other. Thus, in collision of two balls, the ball that is initially moving (the one that hits the other) is not the only cause. Both the objects are causes (and they both hit each other). They both produce effects, even if you typically focus on only the second ball for characterizing the effect (of that interaction).
To repeat, it is wrong to characterize the idea of causality in reference to an orderly progression in time. When two NM-objects remain in static equilibrium for a long period of time, they still are obeying (and exhibiting) causality even if nothing about their dynamical states is ever undergoing any changes with the passage of time. Both are interacting at all times, both are causes (they exert forces like weight and reactions), and both their stationarity (“motionless-ness”) is an effect.
4. The electric field () as a mathematical generalization from Coulomb’s law of force:
The Coulombic force arises between two EC objects due to their charges.
Consider an EC object of a charge fixed at a position . Consider a “test charge” of magnitude at some arbitrarily chosen but fixed position .
The idea behind calling the second charge a “test” charge is that it can be placed at any number of locations. However, the position of the “given” charge always stays fixed.
The force exerted by on the test charge is given by Coulomb’s law:
The equation of the law tells us that the force is a function of both and . If we could somehow take out the effects of these two factors (both being related to the test charge), then the field will come to depend on the first charge alone. In short, we want to isolate the action of the first (given) charge from that of the test charge.
If the position of the test charge is different, we will obtain another force vector acting at that position. We can imagine having the test charge placed at different locations, successively, in different system configurations. Each pair is unique. If we collect together all these (infinity of) unique force vectors, they form a vector field—a function that gives a vector once you plug in some specific coordinates. Since the idea of the field includes all possible force vectors for all possible positions of the test charge, we don’t have to separately state a specific position—no matter what be the position, it’s already there somewhere in that infinite collection. So, by giving the entire field, we make it redundant to have to specify a single specific position for the test charge. We remove the necessity for specifying any specific position.
To factor out the effect of the magnitude of the test charge, we now propose a new vector quantity called the electric vector, denoted as . It is defined as:
What we have effectively done here, by dividing the force at by , is to suppose that always remains a unit charge. Since neither position nor charge-magnitude of the test-charge have to be specified, we have dropped the subscript.
Whereas the electrostatic force field depends on the magnitude of the test charge, the electric vector field does not. Thus, we have found an object that captures the effect of the first charge alone. Hence the subscript to still remains.
Actually, the second charge () is physically still there, but its role has been pushed back into an implicit background, using our procedure—via generalization to all space, and normalization to unit charge, respectively.
A field is actually nothing but a simple function of spatial coordinates. You plug in any specific position into the field-function, and you get the electric field vector () that would be produced at that point—if a unit test charge were to be actually present there. Once you know the electric field vector present at a point, you can always find the Coulomb force which would be exerted by the first charge (which generates the field) on any arbitrary second charge, if it were to be actually situated at that point: you just multiply the electric field vector at that point by the magnitude of the second charge.
This is the meaning which people (the continental mathematical physicists) had in mind when they first put forth the mathematical idea of such functions. These functions later came to be regarded as fields.
5. Some comments on the MIT notes:
Since I gave a reference to the notes at MIT, a comment is in order. The MIT note defines the electric field via a limit: . Taking this limit is not at all necessary. In fact I fail to see even its relevance. For others’ comments, see, for instance, here: [^]. The answer given at a JEE-preparation site also is somewhat misleading [^].
The correct way to think about it is to think of a static situation, always. A static situation can be had either (i) by considering just one instant of time in the motion of a movable configuration of charges (EC Objects), or (ii) by introducing some imaginary support forces which keep all the charges fixed at their respective positions at all times.
We differ from the MIT notes in one more respect. They first define the electric field as a limit of the force at a point on a test charge, but without any reference to the other source of that force. Its only after thus defining the electric field that they come to relating it with the Coulomb force exerted by the first charge. Thus, their definition is, strictly speaking, half-arbitrary: it misses one of two crucial objects that are present in the empirical observations.
We regard the idea of an arbitrary field as existing at a higher level abstraction, but insist on noting that no matter how arbitrary an electric field (its pattern or distribution) might get, it still cannot come into existence without some or the EC object(s) producing it. That’s our viewpoint. We emphasize the role of the field-producing charge.
Indeed, when it comes to QM ontology, we do away completely with the idea of arbitrary fields and even arbitrary continuum charge distributions (which they demand). We *restrict* the generation of all permissible electric fields only to point charges because elementary charges are point-particles.
6. An ontological breakthrough: The entire electric vector field seen as existing physically
Coulomb’s law states a relation for what happens at two specific points in space. The law is completely silent on what happens at any other points of space. In contrast, the electric field is mathematically defined for all points of space.
As mentioned in the last post, the continental physicists did work with the mathematical notion of fields for a long time (I off-hand suppose, for at least 70 years) but without thereby necessarily implying its physical existence.
It should not be too difficult to understand their perspective. A field would be just a mathematical device for them; using this mathematical object meshed well with their energetics program, that’s all. Their program primarily relied on the idea of energies, and so would require the potential energy, but not forces. Strictly speaking, having a field of potential energy was not necessary for the formulation of laws of physics in their program. However, the field idea would be convenient when it came to applying their formulation (essentially based on the energy concept, even if the concept was yet to be isolated as such—they simply used the mathematical functions that we today regard as defining the quantities of energy). Specifying continuous boundary conditions required that a field be supposed for the potential energy. Hence their development of the mathematical functions of spatially spread potential energy. They would see forces as gradients of potential function; forces were secondary or derived quantity for them. Thus, all in all, the field was primarily a mathematical idea for them—just a device of calculations.
It was Faraday who vigorously advocated the idea that the force-field is not just a mathematical idea but also physically exists in the real world out there. He characterized it in terms lines of force. He believed that the space was not empty but filled with a fluid (a mechanical or NM-object like air, water, oil, etc.). The lines of force were imagined by him to be tubes formed by fluid flow. Maxwell then mathematically refined the idea.
It may be perhaps be noted here that the pattern of the magnetic field which is observed when you sprinkle some iron filings on a magnet, does not actually form enough of an evidence to prove the existence of fields. It merely suggests and supports the idea of a field. But strictly speaking, you can always argue that a field does not exist, only point-wise forces (action at a distance) do—may be via an invisible string that runs between the two charges. In the context of iron-filings and a magnet, you can argue that magnetic forces are present only at the points where the iron-filings are—not in the empty spaces in between them. The picture of the field pattern produced by the iron-filings, by itself, is thus not sufficient. The field idea, thus, has to be introduced only with a more general thought; it can be introduced only as a postulate, to ensure consistency in theory. We will touch on this issue later. For the time being, we will simply assume that the continuum field does indeed physically exist.
7. Our EM ontology: The electric force field as an attribute of the background object:
Let’s now note a subtle ontological point.
As the position of a given charge (the “first charge” or ) itself changes, the entire force-field shifts in space too. This is a direct consequence of Coulomb’s law. (The preceding sentence says “changes” and “shifts”. However, note that we don’t mean an actual motion here. We are merely describing the differences which are present between two fields when they are actually produced in two different system configurations. Within each system description, everything still remains static. Taking a difference between two system descriptions does not always have to involve a continuous motion connecting them. That is what we mean here. Thus, the “shifts” here are of the variational calculus kind.)
The fact that the field generated by an EC object “moves” with its position further reinforces the idea that the electric force field is a direct result causally produced by that object. However, an important point to note here is that this fact still does not make the field an attribute of that particular EC object.
Mathematically, the field due to a point-charge is a function that is defined at all points *other* than its own position. Physically, therefore, the field must exist only at those spatial locations where the field-generating EC object itself is not present.
In the EM ontology, EC objects and the background object are the only two categories of object. If the field is not even present at the location of a point-charge, it cannot be an attribute of that charge. Therefore, it must be an attribute of the background object. Matter cannot act where it is not.
Thus, there is a difference in the hierarchical levels of EC objects and electric fields. EC objects are primary existents. In contrast, electric charge and mass exist only as attributes of EC objects. Similarly, the background object is a primary existent. But the electric fields are secondary—they exist only as attributes of the background object.
In calculations, we often get so much used to associating the electric field generated by a point-charge with that charge itself, that we tend to forget that the generated field is not an attribute of that charge itself. Though produced by a charge, the associated field is actually an attribute of the *other* object—the background object.
The electrostatic (and in general, all electromagnetic) phenomena can be explained in terms of the direct contact forces which arise between an EC object and the background object, or between two control volumes (of arbitrarily small size) within the background object.
8. An electric vector field as an effect produced by an EC object acting in isolation of others:
Coulomb’s law has exactly two electrostatic forces. The force exerted by any one charge acts at only one point: at the distant location of the other charge. (The support forces are taken as mechanical in nature here, not electrostatic.) Coulomb’s law thus speaks in terms of a pair of forces. It is physically impossible to have a situation in which only one of the two charges is active (exerts a force on the other).
In abstracting the idea of the electric vector field, we had to factor out the effects due to the test charge. As a result, attributes of test charge are not present in the equation for the electric field. Interpreted physically, a single charge can be taken to generate the field associated with it; a second charge (say a test-charge) is not at all necessary.
In other words, a complete electric vector field (spread over the entire infinite space) can come into being with the existence of just one charge. In contrast, Coulomb’s law requires and simultaneously relates two different charges. Ontologically, this is a significant difference.
The electric field due to several charges is simply an algebraic sum of the fields produced by each charge acting separately.
Imagine an infinitely large universe that has nothing but just a single charge. (You can’t determine its location.)
If the electric force field exists physically, and if such a field can be produced by each single charge acting singly, then actual forces will come to exist everywhere even in this universe. The field will be spherically symmetric. Faraday’s lines of forces will be straight lines that emanate from the point-charge—they will look like symmetrically distributed “spokes.”
When a second charge is added at a finite distance from the first, then the field condition actually existing in the universe is obtained simply by a linear superposition of the two fields. The effective lines of force will look distorted from their initial symmetrical shape. A distortion of lines of force is implied by (and implies) the existence of the second charge.
9. A preview of the things to come:
The description in this post was mostly in terms of forces and quantities derived from them. In the next post, we will look into an alternative description, one that is couched in terms of energies and quantities derived from them. While both the approaches are physics-wise equivalent, the energy-based approach helps simplify calculations. In fact, most physicists get so thoroughly used to the energy-based approach in their PG years, that they even come to forget that ever was a force-based approach (which is quite unlike engineers—we engineers never come to forget forces, including reaction forces at supports).
We have to look at some of the basics of the energy-based approach to physics, simply because the Schrodinger equation is inductively derived using only the energies. However, since we are covering this material purely from the viewpoint of the Schrodinger picture of QM, we will try to keep the variational calculus ideas as much to the background as possible.
Thus, in the next post, we will go over the following sequence of ideas: the potential energy number of an electrostatic system; the potential energy field in the spatial region of a system; the potential energy field of a single elementary point-charge; the potential of an elementary point charge. We will also try to look into the issue of how the background object comes to support forces within itself. Finally, there also will be an issue of justifying the inclusion of the background object in an ontology—its necessity. We will try to cover it right in the next post. If not, we will do that in a subsequent post, and then we will be done with our EM ontology.
See you the next time, bye for now, and take care….
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(Hindi) “geet tere saaz kaa, teri hi aawaaz hoon”
Singer: Lata Mangeshkar
Lyrics: Rajinder Krishan