# Ontologies in physics—5: Energy-based analysis of EM force-fields

1. A recap of the physics of EM force-fields:

Let me recap the most salient parts of the discussion we’ve so far had, in this series, concerning the physics of the electrostatic forces and fields. As before, we will continue to ignore the specifically dynamical effects in EM. Thus, the positions of charges are fixed in space at any instant.

1.1 Coulomb’s Law:

Suppose there are two charges. Then there arise two forces: $\vec{F}_{12}$ which is exerted by $q_1$ on $q_2$ at $\vec{r}_2$, and $\vec{F}_{21}$ which is exerted by $q_2$ on $q_1$ at $\vec{r}_1$. They happen to be equal in magnitude but opposite in directions, exactly as if they were a pair of the direct contact-forces described in the NM ontology. They obey the inverse-square relation for separation, exactly as in gravity.

1.2 A mathematical function of hypothetical forces distributed over space:

We keep any one of the two charges (say $q_1$) fixed, and imagine what would happen when the other charge (say $q_T$) were to be placed at all different points in the infinite space, one position at a time. We thus generate an infinite set of $\vec{r}_2 \leftrightarrow \vec{F}_{1T}$ pairs. Another name for this mapping is: the mathematical force-field, a function of space. This function is only a mathematical convenience. It represents a mathematical generalization of Coulomb’s law to cover the entire space.

Let the force-field function for (i.e. associated with) $q_1$ be denoted as $\vec{F}_{1}(\vec{r} - \vec{r}_1)$. Here, the variable $\vec{r}$ successively assumes all points in the entire space; it is inspired from but only abstractly represents, the position vector for the second charge. It does not represent the actual position of an actual charge unless we fix the position of the second charge via the initial/boundary conditions of a problem, and substitute such a position vector in place of $\vec{r}$. So, in the absence of BC/IC’s of a problem, $\vec{r}$ remains just a dummy variable whose sole purpose to allow us to define the field function. The field is, of course, determined by Coulomb’s law:
$\vec{F}_{1}(\vec{r} - \vec{r}_1) = \dfrac{1}{4\,\pi\,\epsilon_0}\;\dfrac{q_1\,q_T}{r^2}\,\hat{r}$, where $r = |\vec{r} - \vec{r}_1|$ and $\hat{r} = \dfrac{(\vec{r} - \vec{r}_1)}{r}$.

1.3 Maxwell’s field idea:

Maxwell took the idea of the mathematical field functions from the Continental physicists, and synthesized a set of coupled differential equations that together captured all the known laws of electromagnetism (in a form that didn’t have Lorentz’ force law as a separate equation). At the same time, Maxwell also followed the conceptual lead provided by Faraday, and asserted that the field function wasn’t just a mathematical device; it represented something that existed physically. Accordingly, the field was to be seen as a continuously existing condition existing in a mechanical aether. Thus, the ontological change is from “mathematical field function” to “physical force fields”.

1.4 Lorentz’ idea of the EM aether:

The aether is electrical in nature, not mechanical. The actually existing field quantity is the electric vector, $\vec{E}$, which is defined as $\vec{E}_{1}(\vec{r} - \vec{r}_1) = \dfrac{\vec{F}_{1}(\vec{r} - \vec{r}_1)}{q_T}$. He also recast Maxwell’s original equations into a simplified form of a set of four equations, and further added an equation (known by his name) so as to make classical EM description complete.

1.5 Our modification to Lorentz’ idea:

The charges $q_1$ and $q_2$ are not arbitrary; their magnitudes are always equal to the electronic charge (within the algebraic sign). The quantity $q_T$ appearing in field definition itself is always $e$, the fundamental constant of electronic charge. Hence, there is no need to scale $\vec{F}$ to $q_T$. Accordingly, the field which actually exists in “empty space” is that of $\vec{F}$.

I have left out most of the ontological points from this recap.

2. Superposition of force-fields:

Fields due to multiple charges superpose: their net effect is given by the algebraic sum of the independent fields respectively produced by the charges in question. Thus, if there are two charges $q_1$ and $q_2$ present in an isolated system, then we have these two fields in it:
$\vec{F}_{1}(\vec{r} - \vec{r}_1) = \dfrac{1}{4\,\pi\,\epsilon_0}\;\dfrac{q_1\,q_T}{r^2}\,\hat{r}$, where $r = |\vec{r} - \vec{r}_1|$ and $\hat{r} = \dfrac{(\vec{r} - \vec{r}_1)}{r}$,
and
$\vec{F}_{2}(\vec{r} - \vec{r}_2) = \dfrac{1}{4\,\pi\,\epsilon_0}\;\dfrac{q_2\,q_T}{r^2}\,\hat{r}$, where $r = |\vec{r} - \vec{r}_2|$ and $\hat{r} = \dfrac{(\vec{r} - \vec{r}_2)}{r}$.
The total field in the system is then given by:
$\vec{F}_{\text{sys}} = \vec{F}_{1}(\vec{r} - \vec{r}_1) + \vec{F}_{2}(\vec{r} - \vec{r}_2)$.
Notice that both the terms on the right hand-side denote a field that actually exists, but whose magnitude is to be calculated in reference to $q_T$. Thus, now, the test charge $q_T$ is the third charge.

The first two charges form the actual system; the third charge is a device of calculations that provide the measure of the forces which anyway exist even in its absence.

3. Force-fields as physical, not mathematical entities—same maths but different ontology:

In going from the mathematical field function to Maxwell’s fields, the maths remains exactly the same, but there is a remarkable change in the ontology. The difference in the two ideas can perhaps be better illustrated via an error of thought.

3.1 A wrong imagination:

Suppose we define a field for a single charge. Let’s reproduce the equation for convenience, but let’s note the test charge explicitly:
$\vec{F}_{1}(\vec{r}_T - \vec{r}_1) = \dfrac{1}{4\,\pi\,\epsilon_0}\;\dfrac{q_1\,q_T}{r^2}\,\hat{r}$, where $r = |\vec{r}_T - \vec{r}_1|$ and $\hat{r} = \dfrac{(\vec{r}_T - \vec{r}_1)}{r}$.

Since Coulomb’s law is symmetrical, it might be tempting to introduce a similar force field function, now for the test charge $q_T$; it could be given as:
$\vec{F}_{T}(\vec{r}_1 - \vec{r}_T) = \dfrac{1}{4\,\pi\,\epsilon_0}\;\dfrac{q_1\,q_T}{r^2}\,\hat{r}$,
where $r = |\vec{r}_1 - \vec{r}_T|$, $\hat{r} = \dfrac{(\vec{r}_1 - \vec{r}_T)}{r}$, and $\vec{r}_1$ is the variable position of the first charge, which now acts as a test charge for the test charge [sic] $q_T$.

Further, it may then also be tempting to think of a net or total force field for the system. We might try to give it as the vector sum of the two:
$\vec{F}_{\text{sys}} = \vec{F}_{1}(\vec{r} - \vec{r}_1) + \vec{F}_{T}(\vec{r} - \vec{r}_T)$

Inasmuch as the above two equations basically refer to an actual presence of an actual test charge, the whole idea becomes plain wrong! Let alone the physical fields, even the purely mathematical idea of fields does not allow for an actual existence of a test charge. The mathematicians begin to derive their (field-) “functions” using the device of a test charge, but they also unhesitatingly drop it from all considerations once it has served the purpose of giving them their prized equations. They feel no guilt about it, because they almost never notice the practice of context-dropping which is so wide-spread in their community.

But of course, even purely mathematically, the whole scheme is anyway weird because if the test charge were to not exist, you couldn’t define the mathematical field function. But if it were to regarded as actually existing, then it would have to have its own field too, but such a field would not remain single-valued. It would go on acquiring different values as you shifted $q_T$ to different locations. So, the total (system-wide) field wouldn’t any longer be a $1:1$ mapping. It would become a $1:\infty$ mapping.

Thus, despite its neat appearance, the “equations” involving the test charge are invalid. They do not define a definite function; something definite doesn’t come out of this procedure.

We need not take the errors or the context-drops further, because Faraday and Maxwell anyway had something entirely different—far better—in mind anyway!

3.2 Faraday and Maxwell’s idea regarding fields:

Following Faraday and Maxwell’s field idea, what physically exists are only (i) the first charge $q_1$, and (ii) the field condition $\vec{F}_{1}(\vec{r} - \vec{r}_1)$ it produces in the aether, that’s all!

The test charge is not there even as a mathematical device; it’s not present even just “virtually” (whatever it might mean). Following their logic (even if not their original writings), the test charge is simply non-existent in their description.

If we are to follow Maxwell’s scheme, the variable $\vec{r}_T$ becomes $\vec{r}$; it actually ranges over the entire space in defining the field function. The variable $q_T$ is factored out by division of $\vec{F}$ to $\vec{E}$. As to the force $\vec{F}$ itself, it is not a force on a charge, but a force that a small CV within the aether exerts on its neighbouring portions.

Thus, when we make a transition from the mathematical field function to the physical field, the test charge ceases to have any existence in reality—and any relevance in mathematics. Instead, each small CV of the domain itself is described to carry some such a “forceful” or “force-producing” condition at all times. If a second charge (say a test charge) is then introduced in that CV, it merely feels the condition which is already present there.

The force-condition represented by the Maxwellian field is always present. Its existence is not conditioned upon the existence of the second charge. In fact it’s the other way around. The force on the second charge is conditioned upon the existence of the field in its neighbourhood, and the field everywhere in space is produced by the distant first charge.

So, a test charge $q_T$ begins to appear in the Maxwellian description only from the time it is physically introduced at one (and only one) point in the field. The charge then passively senses the already existing field condition. The field condition was always existing there even in the absence of the test charge. There is no special status to the test charge—it is as good as any other charge.

4. Ontological implications of the fact that in the fields-based view, charges don’t interact with each other—at all!

In the field-theoretical view, the charges never interact with each other—directly or indirectly. They interact only with a field—in fact, with only that local portion of a field which is directly adjacent to them. The interaction of a charge and a field, or vice versa, occurs only via the direct contact.

I don’t know if Maxwell or Lorentz thought this way, but taking their idea to its logical end, if two charges are “somehow” brought in a direct contact, I think, they wouldn’t even force each other at the point of touch. They simply wouldn’t “know” what to do with the other charge—through the touch. Nature has “taught” them to deal with (i.e., either force or be forced by) only a field—in fact only that surface portion of the field which is in direct contact with them. The charges don’t “know” how to force (or be forced by) anything other than a field—be it a chargeless EM object, a charged EC object, or even a CV of the field that is not in direct touch to them. A direct charge-to-charge interaction is not at all defined in the fields idea of Maxwellian EM. Neither is an action that skips the intervening portions of the aether.

The logical consequence is this:

If the charges are finite-sized, e.g. spherical in shape, then they would touch each other only at one point. They will continue exchanging forces with the field at all other points on the the spherical boundary because  these other points remain in touch with the field. But the point of touch would contribute nothing.

If charges are point-particles, and if they are brought progressively closer, then they would exert ever increasing forces (attractive or repulsive) on each other. The forces would even approach infinity as the separation goes on decreasing. However, at the point of an actual “touch”, the forces should simply disappear, because the intervening field no longer can fit in between them.

If the charges in question are two point-particles, then a direct contact can only occur when both are literally at the same point-position. The implication is that one electron and one proton, when placed at the same point, could possibly exert no forces on each other. Further, since both their fields are singularly anchored in their respective positions, and since the electric charge they carry also is identical in magnitude though opposite in sign, their respective fields would cancel each other at every point in space. Thus, the net field would be zero—the entire infinitely spread field would simply disappear. The MIT notes [^] illustrate this situation via a simulation.

Thus, we have two points of view here:

1. According to the mainstream physics (as in the MIT notes), when two opposite point-charges occupy the same location, there is no net force left anywhere in the entirety of space. The isolated point where both the point-charges are present is excluded from analysis anyway.
2. Additionally, we can say something more following our ontological insights: There would be no force between the two point-charges either. That’s because charges interact only with fields, and no space is left for intervening field to occupy if both charges are “on top of each other.”

Funny.

Just one more point. When the interacting point-charges are elementary (as electrons and protons are), in both of the aforementioned viewpoints, the charges do not get discharged even when if they are of opposite polarities and even if they are present at the same point. However, inasmuch as the only way for a charge to make its presence felt is via its interactions with a field, two opposite charges existing behave as if they were temporarily discharged.

On the question of whether charges, when temporarily discharged as described above, continue to retain their attribute of inertia or not. To say that the inertia does not get affected is to ascribe a non-electrical attribute to the charges. I gather that Lorentz had put forth some idea of the entire mass being only electrical in nature. I have not thought about it so far, and so, do not take any definitive position about this issue.

This paragraph inserted via an update on 2019.10.07 11:39 IST: I do tend to think that inertia does exist as an attribute separate from the electrical charge, and so, even when two opposite charges occupying the same location get temporarily “discharged,” they still retain their respective inertias. That’s because I think that explaining mass in electrical terms alone has a certain weakness. Think: Once the charges are effectively “discharged,” the whole space ceases to have a net force-field. This implies a discontinuity in the existence of the internal energy too (at least in a naive argument about it all), which would violate the first law of thermodynamics for that state. Further, suppose that the two charges do re-emerge from the discharged state. The question is: Why should their “re-charging” occur only in the neighbourhood of the point of the “discharge”? In an infinite space, they could have emerged anywhere else too. All in all, the whole thing gets more and more complicated and unsatisfactory. So, it seem better to regard the electrical charge and the mechanical inertia (the so-called “rest mass” of special relativity) as two independent attributes. Anyway, please disregard the more speculative discussions like these. (Update on 2019.10.07 11:39 IST over.)

To summarize what we can definitely say:

In the fields idea, every single charge causes local force conditions to come to exist in all parts of aether. On the other side of the equation, any given charge also gets forced due to the field generated by all the other charges; due to Gauss’ theorem, it cannot however feel the force of the field generated by itself. The field in its neighbourhood is all that a given charge “knows” about. It has no other mechanism to come to “know” if there are any other charges in existence anywhere or not; it even can’t “tell” whether the same aether has some other parts at some other locations or not. A lone charge doesn’t require a second charge (as a test charge) in order to cause its field to come into existence (which is of $\vec{E}$ for arbitrary charges, and of $\vec{F}$ for elementary charges). A lone charge too causally and inevitably creates a physical field purely out of its own independent existence, and vice versa. A charge and its field always go together, even if the field is the attribute of the aether, and not of the EC Object which causes it. A charge-less field is an ontological impossibility, and so it a field-less charge. For the latter reason, fields cannot be specified arbitrarily.

With that extra clarification, let’s resume our coverage of the energy-analysis aspects, building on what we saw the last time [^].

5. Potential energy of a spring-mass system as a variable but single number:

5.1 Analysis with one ball fixed:

Consider two NM objects, say two steel balls, connected by a spring. We idealize the balls as point-particles having mass, and the spring as having stiffness but no mass.

For convenience, keep the left ball fixed in space; the ball on the right can be moved. This mass-spring arrangement forms our thermodynamic system. Let the variable $\vec{r} = \vec{x} - \vec{x}_0$ denote the separation of the movable ball from its neutral position $\vec{x}_0$.

The resistive force that the spring (within the system) exerts on the second ball is given by $\vec{F}_{12} = -k \vec{r}$. Work is done by some other force in the environment on the system when the second ball is moved against this resistive force. The action results in an increase in the internal energy of the system, given by:
$\text{d}U = -\text{d}W = \vec{F}_{\text{env}} \cdot \text{d} \vec{r} = (-\ \vec{F}_{12}) \cdot \text{d} \vec{r}$
Notice that we’ve put a negative sign in front of $\vec{F}_{12}$ because $\vec{F}_{\text{env}}$ acts in opposite direction to the spring force $\vec{F}_{12}$. Using Hooke’s law for the spring force, we get:
$\text{d}U = -\text{d}W = k \vec{r} \cdot \text{d} \vec{r} = k r \text{d}r$
After integrating between two arbitrary $r$ values $r_1$ and $r_2$, we find that
$U(r_2) - U(r_1) = \dfrac{1}{2}k \left( r_2^2 - r_1^2 \right)$ where $U(r_i)$ is the energy at the position indexed by $i$.

Now, we note that $\vec{F}_{12} = 0$ for $r = 0$. This is because the neutral position $x_0$ remains fixed in space because the left hand-side ball is fixed.

Further, for convenience, we also choose to set $r_1 = 0$.

Finally, and separately, we also choose to have $U(r_1) = 0$.

Notice that $\vec{F}_{12}$ is physically zero at $r = 0$, whereas $r_1$ being set to zero is just a mathematically convenient choice for the origin of the coordinate frame. But these preconditions still do not mandate that $U(r_1)$ also has to be regarded as zero. If $\vec{F}_{12}$ is zero, only $\text{d}U$ is zero in the infinitesimal neighbourhood of $\vec{r}_1$. But $\text{d}U$ being zero at $r_1$ does not mean that $U$ also has to be zero there. (There is an infinity of parallel lines all having the same non-zero slope such that none of them passes through the origin.) In short, the constant of integration could have been any number. But we deliberately choose it to be zero, purely out of convenience of arithmetical manipulations.

With these choices, we now get to the simpler expression:
$U(r) = \dfrac{1}{2}k r^2$

Obviously, by differenting the above expression, we can see that:
$\vec{F}_{12}(r) = -\ \nabla U(r)$

The spring-mass system is a $1D$ example, but similar examples can be constructed where a conservative force varies in $3D$ space. Hence the more general expression of $\nabla$.

At any specific position $x$ of the second ball, there is a single unique number for the potential energy of the system $U$. This fact holds for all conservative forces.

(A conservative force is a position-dependent force such that if the particle undergoes arbitrary displacements, and then is brought back to the original position, not only the force it experiences at that position but also its entire dynamical state (including its velocity, acceleration and all the infinity of the higher derivatives) is exactly as it was initially. In this consideration, we neglect all the other agents, and consider only one force at a time. Friction is not a conservative force. A conservative force is frictionless.)

5.2 Analysis for the case when both balls undergo arbitrary displacements:

The above description can be generalized in the situation in which both the balls are movable. Here is an outline.

Let $x_{L_0}$ and $x_{R_0}$ respectively denote the absolute positions of the balls on the left and the right in the undeformed state, and let their positions in the deformed state respectively be $x_{L_1}$ and $x_{R_1}$. The two forces exerted by the spring on the two balls are given as: $\vec{F}_L = -k(x_{L_1} - x_{L_0})$ and $\vec{F}_R = -k(x_{R_1} - x_{R_0})$. An environmental force does work on the system by acting on the left ball (i.e., against $\vec{F}_L$), and another environmental force does work on the system by acting on the right side-ball (i.e., against $\vec{F}_L$). The quantum of the two work done on the system are: $-\ \text{d} W_L = -(\vec{F}_{L}) \cdot \text{d} \vec{x}_L$, where $\text{d}x_L$ is the infinitesimal change in the position of the left hand-side ball. Similarly, $-\ \text{d} W_R = -(\vec{F}_{R}) \cdot \text{d} \vec{x}_R$. The total work done on the system is then given by their algebraic addition: $-\ \text{d} W_{\text{sys}} = -\ \text{d} W_L + (-\ \text{d} W_R)$. Integrating, we get the increase in the internal energy $U_{\text{sys}}$ associated with the two finite displacements of the balls.

Proving that for such a system (both balls movable), the internal energy $U$ so obtained is identical to the case in which only one ball is movable, is left as an exercise for the interested reader.  (You need to just conduct the integrations and note that the separation between two balls is the algebraic sum of the individual displacements at each end of the spring. Thus, the basic idea is that the potential energy of the system is a number which is a function of only the separation vector of the two balls taken together.)

The above sketchy outline for a two-ball system was very general in the sense that the environmental forces $-\vec{F}_{L}$ and $-\vec{F}_{R}$ had to be equal, but they didn’t have to displace the respective balls through an equal distance. (Think of an additional spring on the left and right of our system, and assume that their stiffnesses are neither equal to each other nor to the stiffness of the middle spring. In this case, the forces on our (middle) spring will remain equal, but the two balls would have been displaced to different extents.)

If we further assume that both the balls also displace through the same distance, then we can split the total increase in the internal energy of the system into two equal components, one each for a moving boundary at a ball. Thus,

$x_{L_1} - x_{L_0} = x_{R_1} - x_{R_0} = \Delta x$, $U_{\text{sys}} = U_L + U_R$ and $U_L = U_R$, so that

$U_{\text{sys}} = \dfrac{1}{2}\sum\limits_{i=1}^{2} k (\Delta x)^2$.

The assumption that each ball in a pair displaces through the same distance is justifiable if the spring itself is without inertia and so its motion doesn’t involve transmission of momentum, so that the interactions of the spring with the environmental forces (through the massive point-particles of balls) remains independent of the position of the spring or the state of deformation it is in.

6. Electrostatic potential energy of a pair of EC objects as a variable but single number (a global attribute):

6.1 Preliminaries:

EM forces superpose. Therefore, for most ontological purposes, discussions in terms of a single pair of charges is enough. But note that at least two charges must be considered. Having just one charge (and its force-field) won’t do, because a physically isolated single charge experiences no forces—not from its own field. (Here, we don’t have to make an appeal to a spherical symmetry; it’s enough to invoke Gauss’ divergence theorem; looking up the proof is left as an exercise for the reader.)

6.2 Potential energy of a pair of charges:

Consider a two charge system once again. As before, the first charge remains fixed in space; this is purely for convenience. The second charge can be placed at different points. The thermodynamic boundary exists at the two points and nowhere else.

As the position of the second charge changes, we obtain a different configuration. Work must be done on the system to move the second charge. (Force must be exerted by the environment and on the system in order to keep the first charge where it is. However, since the first charge does not undergo displacement, no work is done on the system by this force.) In EM systems, the sign of the work can get further confusing, because forces can be both attractive or repulsive. This is in addition to the standard thermodynamic sign convention. Read the description below accordingly. (Best is to work with the algebraic term $q$ all throughout, and then to put specific charge values only in the end.)

By the first law (and neglecting heat), $\text{d}U = -\delta W$, where the $\delta W$ is the work done by the system. Work crosses system boundary because when a charge shifts, the boundary fixed at it gets shifted too. (This is electro-statics. So, the shift is in the variational sense. There is no motion, no displacement in time; just the fact of a difference between two separation vectors.)

Work done by the system is defined through $\text{d}W = \vec{F}_{12} \cdot \text{d}\vec{r}$. So, if the second charge is brought nearer, the work done on the system i.e. the increase in the internal energy is:
$\text{d}U = -\ (\vec{F}_{12}) \cdot (-\text{d} \vec{r})$
After integrating we find that
$U_f - U_i = \dfrac{1}{4\,\pi\,\epsilon_0} \dfrac{q_1\,q_2}{r}$

As the absolute position of the second charge $\vec{r}_2 \rightarrow \infty$, the difference in the internal energies in two infinitesimally close configurations $\text{d}U \rightarrow 0$. In other words, $U_f$ and $U_i$ approach the same value. Note, it’s the difference which approaches zero, not the respective values of the individual $U_f$ and $U_i$ terms. If one of them is specified as a boundary condition “at” infinity, the other value would be infinitesimally close to the same value.

Here, we introduce a convention: As the second charge goes infinitely away, the force is anyway dropping to zero. So it makes sense (a good convention) to choose $U_{\infty} = 0$ rather than any other number. This logic is very similar to how, in the spring-mass system too, we chose a zero potential energy at the spatial position where the force became zero. In case of spring, zero force was achieved at a definite point of space. Here, the zero force occurs in a limiting process. But the idea behind choosing a $0$ value for $U$ is similar: $U$ is zero “where” $\vec{F}$ is zero. Note, this is only a mathematical convention, not a physical fact. We simply don’t have any evidence to know what specific energy value there is as the distances become very large.

With the above convention, the increase in the internal energy of the system in bringing both the charges from infinity to their present positions ($\vec{r}_1$ and $\vec{r}_2$) becomes the same as the absolute internal energy of the system (and not a change in it). This internal energy is given by: $U_r = \dfrac{1}{4\,\pi\,\epsilon_0} \dfrac{q_1\,q_2}{r}$, where $r = |\vec{r}_2 - \vec{r}_1|$ and hence $U_r$ are variable, but at any instant, both their values are fixed. Thus the potential energy of a system is just a single number at any point of time, no matter where the charges are.

6.3 Other points:

• What if the two charges are of different polarities? Working out whether the internal energy increases or decreases (from its zero at the infinitely large separation) is left as an exercise for philosophers. (No, this is not an insult. This exercise is about physics, not philosophy.)
• By the way, note a fact established by the known physics (but not by mathematics):
The net charge of the universe is not only conserved, it also is zero.
• If you imagine mechanical supports so as to ensure fixed positions for the two charges (the supports shift with the charges), then what about the changes in the internal energy of the system due to the work done against or by these support forces? This is left as an exercise to XII standard students/JEE aspirants.

7. Field of electrostatic potential energy ($U$):

7.1 Mathematical function for potential energy:

Following the same logic as for superposition of forces, the mathematical function for potential energy of a system containing two charges can be given as:
$U_1(\vec{r}_T) = \dfrac{1}{4\,\pi\,\epsilon_0} \dfrac{q_1\,q_T}{r}$, $r = |\vec{r}_T - \vec{r}_1|$, and
$U_2(\vec{r}_T) = \dfrac{1}{4\,\pi\,\epsilon_0} \dfrac{q_2\,q_T}{r}$, $r = |\vec{r}_T - \vec{r}_2|$

The mathematical function for the total potential energy of the system is then given by the simple algebraic sum of the two.
$U_{\text{sys}}(\vec{r}_T) = U_1(\vec{r}_T) + U_2(\vec{r}_T)$
The same logic can be extended to $n$ number of charges.

Similarly, force can be obtained from the mathematical energy function as: $\vec{F}_{12} = -\ \nabla U(r)$

7.2 Potential energy field as physically existing:

Then, once again, we can follow Faraday and Maxwell’s lead, and assert that what the mathematical expression for potential energy field gives is a quantity of something that actually exists in the physical world out there.

The infinitely extended field is sensitive (and unique) to each specific physical configuration of charges. If a single charge under consideration is fixed in space, then a unique potential energy can be assigned to each point of space. If it moves in space, so does the force-field and the potential energy-field associated with it.

7.3 Potential energy field of a single charge:

Again, following the logic seen in the case of spring and two masses, we can say that electrostatic potential energy for a pair can be split into two equal components, one each specific to a charge in the pair. Thus:

$U_{\text{sys}} = \dfrac{1}{2}\sum\limits_{i=1}^{2} \dfrac{1}{4\,\pi\,\epsilon_0} \dfrac{q_1\,q_2}{r}$.

7.3 Potential energy field of a $n$ charges:

Generalizing, it is easy to prove that for a system of $n$ charges:

$U_{\text{sys}} = \dfrac{1}{2}\sum\limits_{i=1}^{n} \sum\limits_{j \neq i; j=1}^{n} \dfrac{1}{4\,\pi\,\epsilon_0} \dfrac{q_i\,q_j}{r}$, where $r = |\vec{r}_j - \vec{r}_i|$

This form is especially useful in building simulations like the molecular dynamics.

8. An ontologically very interesting point—differences in internal energy storage, work actually done or extractable, and potential energy:

We must note a very important point here. It concerns the issue of the ontological understanding of what precisely the terms potential energy and internal energy mean.

8.1 In direct-contact systems, a moving boundary can sweep over the entire volume where the internal energy is stored:

Strictly speaking, as touched upon in the last post, the term potential energy means that part of the internal energy which can be converted into work.

Work done on a system increases its internal energy; some part of the stored internal energy can also be later on converted into work; when the forces are conservative, the increase in the internal energy of a system due to work done on it is, in its entirety, available for conversion into work.

When we talk of a work done on a system (or by it), we are basically referring only to the system boundaries—not at all to its internals. Work is not at all defined at any points other than at the instantaneous position(s) of system boundary(ies). This fact introduces a tricky issue.

In idealized systems like the ideal cylinder-piston arrangement or ideal spring-mass system, two features come into play: (i) being idealized, the forces can be treated as being conservative, and (ii) the moving objects, and hence the system boundaries, during their motions, can come to sweep the entirety of the spatial region where internal energy is stored. It is the second factor which needs special commenting.

These systems are such that during a single work-extraction process, the moving boundary can trace over and exhaust the entirety of the spatial region where the internal energy is defined. During such a sweep of the boundaries, the entirety of the earlier increases in the internal energy of the system can be converted back to work (assuming conservative forces i.e. neglecting friction).

Therefore, when work has been maximally extracted from such systems (e.g. when the spring returns to its neutral length), there is no further storage of internal energy at all left in such a system.

8.2 In fields-based systems with point-particles, a moving boundary cannot sweep over the entire volume where the internal energy is stored:

The fields-based systems show a remarkable difference in this second respect. They store far more energy than can ever be converted into work. If the fields are singular, they must be seen as storing infinitely more energy than what can ever be extracted from them.

The basic reason for this characteristic is that no matter where the objects (point-boundaries) are kept or how they are shifted, there always is an infinitely greater portion of the domain all parts of which are still left carrying energy, and this entire portion is left untouched because the boundaries are properly defined only at points (at most at surfaces), but not over the entire volume.

To repeat, a system of two (or more) distinct EC Objects, shows the following three feature: (i) work can be exchanged only at the system boundaries, (ii) the system boundary exists only at the two (or more) EC objects (point-particles), (iii) but the force- or energy-field logically “internal” to the system definition exists at all points of the infinite space at all times. Since the work-extraction process can happen only at the points where the charges are, such a process can never come to have covered, through a finite and completed process of work extraction, the entirety of space.

Therefore, there will always be far more internal energy than can ever be converted to work. In other words:

The total internal energy stored in a system via the EM force-field is far greater than the quantity of potential energy (i.e. the change in the internal energy) which is acquired by the system when two movable EC objects themselves are displaced.

8.3 Putting it mathematically:

Mathematically, the whole matter can be put very simply:

In case of the ideal spring-mass system (point-masses, massless springs, no friction), the total increase in the internal energy of the system $\Delta U$ equals the volume integral of the internal energy density over the entire swept volume (i.e. the regions where the masses can move) $\iiint\limits_{\Omega} \text{d}\Omega \rho$, where $\rho$ is the internal energy density.

In contrast, in case of fields-based systems (whether gravitational or static EM), $\Delta U \neq \iiint\limits_{\Omega} \text{d}\Omega\ \rho$, where $\rho$ is the density of the local increases in the internal energy due to the presence of an EM field; in fact, the left hand-side is far smaller (infinitely smaller if fields are singular inside the system) as compared to the right hand-side.

So, in the EM systems (as also in gravitational systems), internal energy exists everywhere in space. However, the potential energy of the two EC Objects—and hence of the system—refers to only those local energies which exist in their immediate neighbourhood, the energies which are acquired by these discrete objects. Hence, only this much energy is available for any conversion to work at the moving system boundaries.

8.4 Consequences in calculations—and in further physics:

Now, what do we do with this additional energy which is exists “internally” to the system but is not available even potentially for extraction into work?

Well, in most calculations, we can just ignore it. As they say, the datum for any potential energy is arbitrary. That’s because the datum for the changes in the internal energy are arbitrary. What we are interested in are the changes in the internal energy of the system, not in its absolute value or datum. So, the unextractable portion of internal energy is perfectly OK to have. In EM, the unextractable portion of $U$ makes for a minor fact which is necessary just to bring logical completeness to the physical description. In fact, it is a trivial fact because in EM, ultimately, the fields have a relatively indirect role to play; the actions of the fields are of no consequence unless they result in some action on the charges. (EC Objects have inertia; the field doesn’t.)

However, the situation changes very significantly when we it comes to quantum mechanics. We will pursue the QM ontology in the next post.

9. Electrostatic potential ($V$) of an EC object:

9.1 The field of the “voltage”:

The 19th century physicists wanted to get you confused even further.

• Even for the action-at-a-distance systems (i.e. the fields-based systems), they happily equated the internal energy with the potential energy.
• Further, they defined not just a potential energy field, but also an electrostatic potential field—i.e. the one which has the word “energy” dropped from its name.
• Finally, they (and everyone else) used (and continue to use) the same symbol $V$ to denote both.

The potential field $V$ is the equivalent, in energy-based analysis, of the electric vector field $\vec{E}$. The potential function is defined by dividing the potential energy function due to a single charge by the magnitude of that charge so that what we have is a “pure” function that is independent of any other charge (i.e., a field defined for a shiftable unit charge), as shown below:
$V_1 = \dfrac{U_1}{q_2} = \dfrac{1}{4\,\pi\,\epsilon_0} \dfrac{q_1}{r}$

The motivation of defining such a thing? Well, if the second charge is not the unit charge, then the potential for the first charge ($V_1$) still remains the same.

Of course the potential energy function for the second charge does depend on its charge:
$U_2 = V_1 q_2$

9.2 A personal comment:

As we saw, if the charges are elementary, then $\vec{E}$ is not necessary. Further, if we assume that fields are not arbitrary but are always anchored into point-particle positions, then it is always possible to split up the total internal energy $U$ into contributions arising from internal charges; see the equation in section 7.3 of this post. The whole business of having yet another field—of the electric potential—is thereby avoided.

9.3 Multiplying confusions:

QM text-books further help compound the confusions. [In an update on 2019.10.07 12:15 IST, I deleted a misleading statement here.]

Notation-wise, $V$ should be reserved for electric potential (or its field), because $V$ is also the SI symbol for the unit of potential i.e. voltage (after Volta).

In turn, something like $U$ might be used to indicate the total internal energy field, defined as the total energy content of the aether due to all the charges present in the system. The potential energy field affecting the dynamics of a given charge $q_{i}$ (and arising due to the other charges) may be denoted using something like $\Pi$; this quantity has the relation $\Pi_{q_{i}} = U_{\text{sys}} - U_{q_{i}}$, where $U_{q_{i}}$ is the field-component contributed to the internal energy field of the system by $q_{i}$. The potential energy of the system, seen as a global attribute, can continue to remain a single number that depends on the set of point-positions of charges (their configuration).

Using a Greek letter for the potential energy is not a very good choice. But we can’t use $E$ for the potential energy because the symbol is already taken by the electric force field. We can’t use $P$ because it stands for pressure, and also for protons—again confusing.

So, all in all, Schrodinger’s equation should be re-written to have $\Pi$ in place of $V$ in the expression of its Hamiltonian. It’s a field—a $3D$ field. It is different for each charge in the QM system.

Regardless of the notation, remember, $V$ in the Schrodinger equation has the unit of joule, not of volt.

10. Physics of the electric field: “stresses” and “strains” in the aether:

No, we are not done covering the planned topics for this post yet. The next point concerns some plausible mechanism whereby the Lorentz Aether (LE) might be imagined as storing the internal energy associated with the physically existence of the EM fields.

I will not go into the specifics of it except for noting that we imagine the electrostatic electric force field ($\vec{E}$ or $\vec{F}$) at a point as a result of a kind of a stress field in the LE. This imagination involving a stress field needs some commentary.

First of all, notice that, strictly speaking, you can’t have point-forces in a continuum. That’s because if each point of a continuum were to have a finite force, since there are an infinity of points within any finite volume, every finite portion of the aether would end up having an infinitely large force. (The mathematical argument here is identical in spirit to why Born’s rule applies to small CVs, but not to points.)

That is the reason why the only force-like quantities permissible within a continuum are: (i) a volume density of force, or (ii) a surface intensity of force (as in pressure, a type stress).

In between the volume- and surface-defined quantities, I believe that the $\vec{E}$ field (rather, the $\vec{F}$ field) arises out of internal surface intensities (i.e. stresses) rather than volume force densities. My reasoning is the following:

The EC Object is ontologically a different kind of an object, and forces are exchanged between the EC Object and the Lorentz Aether. Stronger: The existence of an EC Object is essential (even tantamount) to having a force-field within the aether; an EM force-field cannot arise without there being a charge which may be seen as causally producing it—or vice versa, if you will. The two always go together; none can exist without the other.

Now, two different objects that are in a direct contact can be separated from each other only by a surface, not a volume. So, if a $\vec{F}$ arises at a point, it must be seen as the limit of a surface-integral of some surface-intensity of a force on the boundaries of a small CV (control volume). That’s nothing but a stress; it’s not a body force.

So, though from a mathematical viewpoint, internal surface forces are perfectly interchangeable with internal volume forces, from an ontological perspective, the surface force-intensities are primary.

Accordingly, the field energy $\dfrac{1}{2} \epsilon_0 E^2$ noted in the MIT notes [^] may be seen as the strain-energy density, with $\epsilon_0$ playing a role analogous to that of Young’s modulus in elasticity.

But carefully note that all said and done, this analogy still must be regarded as only a convenience in visualization. As of today, we don’t know the physics of exactly how the EM force fields come to be, even if we know that they must be there. There must be some physics to them, but it’s summarily unknown to us as of today.

Of course, personally, I do visualize a stress field producing the electrostatic $\vec{F}$ field—it’s easy to keep in mind the caveat noted just above.

OK. Enough is enough. (With ~7,500 words, this has become the longest post ever written on this blog.)

11. A preview of the things to come:

Since there was a gap in writing of this series (due to flash floods in Pune which affected us too), I happened to lose the “flow” of my thoughts. So, there must be some odd points or notings here and there that got slipped out of the mind during writing this post. Not an issue. I anyway plan to pull this entire series together and convert it into a stand-alone essay (after due revisions, rearrangement of material, etc). I am in fact toying with the idea of sending the resulting PDF to some philosophy journal too, eventually. But a journal article isn’t an immediate goal. May be in 2020, after the first paper on my new approach on QM gets written.

Coming back to this series: Much of our work in this series is already over. We will require, I guess, just one more post, for covering the next topic, viz. the ontology of QM. The reason that the QM ontology will get over so quickly is because there are hardly any tricky issues still left to be touched on. Most of them actually got covered right during the progression from the NM ontology to the EM ontology. So, just one more post should be able to cover the QM ontology.

The only portion still remaining then would be: a few general philosophical remarks regarding the necessity of having a physical background object—the Lorentz Aether, and, as my small but new contribution, a few additional mathematics- and physics-based points that go in the same direction. We will see if it would be possible to cover this portion too right in the next post (on the QM ontology). If yes, we will be free to discuss visualization for my new approach to QM.

Alright then, bye for now, take care, and see you the next time.

A song I like:

(Hindi) “toraa mana darpana kahalaaye”
Lyrics: Sahir Ludhianvi
Music: Ravi
Singer: Asha Bhosale
[Credits listed in a random order.]

PS (2019.10.05 13:30 IST): Minor modifications/additions or streamlining may get affected, but the basic points will remain as they are.

History: First published on 2019.10.05 13:30 IST. Some minor portions added/deleted (noted inline) on 2019.10.07, by 12:29 IST. Further clarified a bit on 2019.10.07 21:53 (the ‘net connection was very flaky throughout the day). Now will (really) leave this post as is.

# Ontologies in physics—3: EM fields in terms of forces; space; and related ontological issues

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 on the view of these matters I have come to develop. There’s a risk here that in writing without fresh referencing, I might be mixing my own views with the original ideations. This is not at all to my liking. But I have to make do with precisely that 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 [^]. There are many great books on history (some being listed above). 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:

1.1 The background object of 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, Maxwell and Newton’s view.

[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.]

1.2 The physical space, and mathematical spaces:

When you point your finger to some place in between two EC objects, i.e., if you point out the “empty space” in between them, what you are actually pointing to is 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”. (And there is something as the physical space.)

In our view, physically, the concept of space denotes the fact that physical objects (both the EC objects and the background object) have spatial attributes or characteristics like extension, location, and also other spatial attributes like the topological ones. The concept also includes the physically existing spatial relations between all objects. The physical space is the sum total of all the spatial attributes or characteristics of all the physically existing objects, with all the interrelations between them. Mathematically, the concept of space denotes a quantitative system of measuring the sizes (or magnitudes) of those 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 fact assumed by all of physics. A primary fact is one which cannot be analyzed as implied by or arising from other facts. So, objects exist, full-stop. Every object exists with all the attributes that it has; each object has a certain identity.

Attributes exist (as part of identity of an object) only in some specific quantities or sizes. There can be no size-less attribute. As a simple example, a pen has the spatial attributes of length, diameter, shape, etc. Each pen exists with a specific quantity or measure of length and other attributes. 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 the science that develops the methods using which physical sizes of comparable objects (i.e. objects having the same attributes but to different measures) can be quantitatively related to each other. Mathematical concepts refer to mathematical objects, not physical, even though these concepts are reached only after observing the size-wise relations among physical objects. Mathematical objects are a result of objectifying the methods invented by us for measuring the existing sizes of the physical objects. The same set of physically existing objects (or their attributes, characteristics, properties, etc.) can give rise to an indefinite number of mathematical concepts.

Coming back to the concept of space: There can be many mathematical spaces, but they all refer to the same physical space. A mathematical space is an abstract quantitative system of measuring extensions and locations of physical objects. It is a mathematical concept which is developed in reference to the facts subsumed by the concept of the physical space.

The referents of the concept of the physical space are perceived directly, but not the physical space itself. The physical space is a concept, not a directly perceivable concrete.

What you directly see, the sense you get in your direct perception that there is some “solidity” or “volume-ness” to the physical world, is not the physical space itself, but the evidential basis for that concept.

1.3 The background object

The background object is a physical substance, but having a “non-material” nature, speaking vaguely—the exact statement here is that it is not an NM object.

The physical space is just one way to characterize the background object. It is possible to spatially characterize the background object because all its parts 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 possesses 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. Force-conditions are being sustained in it at all times, which it makes it active in the same sense that a foundation stone of a building is: no overall motion, but still a transmission of forces through it, at all times.

Neglecting gravity, the background object does not interact in any way with the NM-objects. It is for this reason that no inertia or mass can be ascribed to it. This is one of the easy reasons why it can’t be regarded as an NM-object. However, as we shall see later, the background object does possess something like a state of a “stress” within itself. (The nature of these stress-strains is different from what you have in case of the purely NM-objects.) It is worth noting here that even in the NM-ontology (i.e., in solid- and fluid-mechanics), the equations defining the stress/strain fields do not have mass appearing in them. So, the point here is that even if the background object were to have mass, it still wouldn’t matter because it doesn’t enter into dynamical equations involving it.

2. A few words on the difference between the 19th century view of space and ours:

2.1 What space meant to the 19th century physicists:

The 19th century physicists (like Maxwell) took space in the same sense as Newton did—in the NM-ontological sense of the term.

Space, to them, is a directly given (i.e. directly perceptible) absolute, having an independent physical existent apart from any NM-objects there may be. They would regard objects as filling or occupying some parts (some regions) of this already given absolute space.

In our opinion, this view is somewhat Platonic in nature. Faraday and Maxwell received this view of a physical, “absolute” space from Newton.

2.2 Our view: Space as a concept derived from the spatiality shown by physical objects:

We take the view that only physical objects are what is directly given in perceptions. (Perception here also includes observations of EC objects made in controlled EM experiments.) It is true that we directly perceive something space-like—the solidity or volume-ness of objects. It is also true that we also directly perceive the particulars of configurations of objects, including their directly evident locations. But that’s about all. We don’t perceive space as such; only the volume-ness or extended-ness of objects, their spatiality.

This point is somewhat similar to saying that what we directly perceive are blue objects (of different hues). We do directly see that blue objects show blueness. But the concept “blue” is only an abstraction from these objects; it is a product of a concept formation. The blue is not out there; only blueness is, and that too, only as a characteristic or attribute of those actual objects which do possess it.

People call the “empty” region “space” simply because they are unable to think of an invisible (or untouchable etc.) object that can exist in between two visible (or perceptible) NM objects. But realize that this “emptiness” also is a part of what we directly perceive. It is not as if our perceptual fields comes to have an in-principle hole (or complete absence) in the places where there is this absolute “nothing”ness of the “empty space.” Perceptions don’t have “holes” because there is something of what we call “space.” Our perceptual field cannot contain any evidence for the existence of a literal nothing in the world out there—a nothing that supposedly is at par with the things that do exist in the world out there.

So, what we directly perceive are only objects having spatiality, but never “space” itself. Space is something which we conceive of, based on these percepts.

2.3 Filling space with objects—what it means in our view:

It is only at this point in development—after we already have the concept of space—that we are able to trace the concrete-to-abstract relationship in reverse, and say that objects can be seen as coming to occupy some region of space that was initially empty (of similar, NM-type of, objects).

In our view, this statement (that objects fill space) involves an application of a concept, a backward motion in conceptual linkages. Objects can fill the physical space only because there are other NM-objects that help you isolate the “empty” region, which can then be filled with some other NM-objects.

The reverse-tracing of conceptual linkages becomes possible only if you have it in the first place, if you start from an abstract level. Perceptual level does not permit such reverse-tracing.

2.4 Space couldn’t possibly survive literal annihilation of all objects:

To Newton, Faraday and Maxwell, if all physical objects were to be annihilated, then the absolute space would still be left behind. To us, this is an impossibility. There cannot be spatial attributes without there being objects that have spatial attributes, and there cannot be the concept of space without our grasping a certain aspects like extended-ness or volume-ness that they have, and then abstracting a concept out of such physical features.

3. Causality and interactions:

3.1 Our view of causality

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.

3.2 Characterizing all the causes and effects operative in 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.

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).

3.3 Causality as a concept far more basic than an orderly progression in time:

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 ($\vec{E}$) as a mathematical generalization from Coulomb’s law of force:

4.1 Coulomb’s law as operative at only two distinct points, and nowhere else:

It is the electric charge which gives rise to the Coulomb forces between two EC objects.

Consider an EC object of a charge $q_1$ fixed at a position $\vec{r}_1$. Consider a “test charge” of magnitude $q_T$ at some arbitrarily chosen but fixed position $\vec{r}_T$.

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 $q_1$ always stays fixed.

The force exerted by $q_1$ on the test charge $q_T$ is given by Coulomb’s law:
$\vec{F}_{1T} = \dfrac{1}{4\,\pi\,\epsilon_0} \dfrac{q_1\,q_T}{r^2}\;\hat{r}_{1T} \qquad\qquad r = |\vec{r}_T - \vec{r}_1|$.

The equation of the law tells us that the force $\vec{F}_{1T}$ is a function of both $q_T$ and $\vec{r}_T$. 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 $q_1$ alone. In short, we want to isolate the action of the first (given) charge from that of the test charge.

4.2 How to isolate the actions of the first charge from that of the test charge?: The electric vector ($\vec{E}$)

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 $\vec{r}_{T} \leftrightarrow \vec{F}_{1T}$ 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 $(x,y,z)$ 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 $q_{T}$ of the test charge, we now propose a new vector quantity called the electric vector, denoted as $\vec{E}$. It is defined as:
$\vec{E}_{1} = \dfrac{\vec{F}_{1T}}{q_T}$
What we have effectively done here, by dividing the force at $\vec{r}_T$ by $q_T$, is to suppose that $q_T$ always remains a unit charge. Since neither position nor charge-magnitude of the test-charge have to be specified, we have dropped the $T$ subscript.

Whereas the electrostatic force field depends on the magnitude of the second 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 $_{1}$ to $E$ still remains.

Actually, the second charge ($T$) 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 the unit charge, respectively.

4.3 The electric vector’s field as a mathematical device of calculations:

A field is just a simple function of the spatial coordinates. You plug in any specific position into the field-function, and you get the electric field vector ($\vec{E}$) 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 which would be present at a point, then 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: $\vec{E} = \lim\limits_{q_0 \rightarrow 0} \dfrac{\vec{F}}{q_0}$. Taking this limit is not at all necessary. In fact I fail to see even its relevance. If at all a limit has to be conducted, then it could be for a vanishing size (diameter etc.) of the test charge, so that the point of definition of $\vec{E}$ becomes unambiguous. But taking the charge to vanishingly small charge simply does not seem to bring in anything of 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 (at least in the vicinity of the test charge). 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:

6.1 A possible reason why the continental physicists didn’t  go for a physically existing field:

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 development program primarily relied on the underlying idea of energies, and so they would certainly require the potential energy, not forces. Actually, the concept of energy was yet to be isolated as such—they simply used the mathematical functions that we today regard as defining the quantities of energy.

Strictly speaking, having a field of potential energy was not necessary for formulation of laws of physics in their program. Their laws could have been formulated with just a single number for the potential energy of the entire system. (The number would vary with positions of discrete bodies, but it would nevertheless always be a single number.) That is, as far as their laws were concerned.

However, the field idea would be convenient when it came to applying their formulation. Problems having continuous boundary conditions naturally got simplified with the working idea of a function of all possible spatial coordinates. Thus, a field came be supposed for the potential energy. They would see forces as gradients of potential function; forces were secondary or derived quantity for them. Indeed, the problems they worked on, during the development of the potential field concept, came exclusively from gravity.

Thus, all in all, the field was primarily a mathematical idea for them—just a device of calculations, and that too, only for gravity, even if electromagnetic laws also were being discovered during the same period.

6.2 Faraday makes the ontological breakthrough:

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 therefore exist; only point-wise forces (action at a distance) do. 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 fact is that the field idea can only be introduced as based on 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 vector field as an attribute of the background object:

Let’s now note a subtle ontological point.

7.1 An EC object is the cause of the $\vec{E}$ field:

As the position of a given charge (the “first charge” or $q_1$) 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 thereby 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 shifts with its position further establishes the idea that the electric force field is a causal effect of that object.

7.2 The $\vec{E}$ field is not an attribute of an EC 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.

7.3 The existence of $\vec{E}$ implies an interaction between two physically existing objects:

If an object is causally responsible for producing an effect, but if that effect is an attribute of some other object, clearly we have two different objects interacting here.

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 do not determine the distribution pattern of 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 Faraday’s lines of force is implied by (and implies) the existence of the second charge. However, their very existence does not depend on 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….

An important update on 2019.09.17 12:26 IST:

I think that the sections 7. and 8. above have come out as being somewhat misleading.

The present write-up suggests that what physically exists is the $\vec{E}$ field. However, if the magnitude of the second charge $q_2$ or $q_T$ is not $1$, then it is not clear precisely what physical process occurs at $\vec{r}_2$ such that it results in a multiplication of $\vec{E}$ by $q_2$ to produce the force $\vec{F}_{12}$ at that point. This force, after all, must physically exist. But it is not clear what physics is there for the multiplication by $q_2$.

So the proper conclusion to draw seems to be that the actually existing field is always that of the force $\vec{F}_{1T}$ and not of $\vec{E}_{1}$; that the latter is only a mathematical device, a convenience in formulating Maxwell’s system of equations. But what will be the implications for an isolated system having only a single charge? for bringing a second charge into it from infinity? … Need to think through.

… I will think about it for a while and then, if necessary, I will come back and update the above description appropriately.  … In the meanwhile, any thoughts / suggestions?

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Music and Singer: Vasant Aajgaonkar

# Ontologies in physics—2: Electromagnetic fields as understood by Faraday and Maxwell

In this post, we shall begin going through the ontology assumed in—or rather, demanded by—the physical phenomena which are covered by the classical (i.e. Maxwellian) electrodynamics. We call this ontology EM for short.

We will mostly be talking in reference to electro-statics. However, note, this ontology just as well applies also to electro-dynamics.

1. A list of the ontological objects used in the physics theory of electromagnetism (EM):

The EM objects basically are of only two types: (1) massive and electrically charged point-particles (of one of two polarities: positive or negative), and (ii) a background object (sometimes loosely identified with the fields induced in it, by calling it “field”; other times identified with an aether of a certain kind by Maxwell; more, below).

Notably, there are no separate magnetically active objects in this ontology even though magnetism has been known as a force for more than at least one millenium. Magnetism is an effect produced by the electrical charges.

2. Electrically Charged (EC) objects:

The ontology of NM (seen in the last part in this series) is basically that of the uncharged bodies. What the EM ontology now does is to further endow these same objects with an additional attribute of the electric charge. This extra attribute considerably modifies the entire dynamical behaviour of these objects, hereafter called the EC objects for short.

For the elementary charged objects (basically, here, only the electrons and protons), the phenomenon of the induced charge/polarity does not come into picture—the charge of each elementary EC object always remains with it and its quantity too remains completely unaffected by anything or any action in the universe.

The electrical charge is just as inseparable an attribute of an EC object as its mass is. As an EC object moves in space, so does its charge too. An electrical charge cannot exist at any spatial location other than that of the massive EC object which possesses it.

Just as in NM, the EC objects too can be abstractly seen as if all their mass, and now charge too, were to be concentrated at a single point. We call such EC objects the point-charges.

3. Electrostatic forces between point-charges—Coulomb’s law:

Coulomb’s law is an empirically derived quantitative relationship. There is no theoretical basis beyond the fact that such behaviour was actually observed to occur in carefully conducted experiments. The statement of the law, however, is mathematically sufficiently refined that it would be easy to suspect whether it was not derived from some other a priori basis. As a matter of fact, it was not. What does Coulomb’s law describe?

Consider an isolated system of two point-charges fixed in space at some finite distance apart. [Help yourself by drawing a free-body diagram, complete with the structural support symbols for each of the charged bodies too. No, these supports, though they look like the electrical ground, are actually mechanics symbols; they don’t discharge the charges by grounding.] It is experimentally found that both the charged bodies experience forces of certain magnitudes and directions as given by Coulomb’s law.

Let the electrical charges of the two point-charges (EC massive point-particles) be $q_1$ and $q_2$, and let their positions be $\vec{r}_1$ and $\vec{r}_2$.

Let the separation vector going from the first point-charge to the second be given by $\vec{r}_{12} = \vec{r}_2 - \vec{r}_1$. (There is no typo in the last equation.) Similarly, let the separation vector going from the second point-charge to the first be given as $\vec{r}_{21} = \vec{r}_1 - \vec{r}_2$. (Again, no typo.)

[Note, a separation vector measures the difference in the two vector positions of two different bodies at the same time, whereas a displacement vector measures the difference in the two vector positions of the same body at two different times. … There is another related idea: The variation in position is a vector that measures the difference in the two vector positions of the same body, without reference to motion (and hence time), but as imagined in two possible and different configurations within two description of the same system. We will not need it here.]

Let $\vec{F}_{12}$ be the force that $q_1$ exerts on $q_2$ at $\vec{r}_2$ (and nowhere else). Similarly, let $\vec{F}_{21}$ be the force that $q_2$ exerts on $q_1$ at $\vec{r}_1$ (and nowhere else).

Coulomb’s law now states that:

$\vec{F}_{12} = \dfrac{1}{4\,\pi\,\epsilon_0} \dfrac{q_1\,q_2}{r^2}\;\hat{r} = - \vec{F}_{21}$,
where
$r = |\vec{r}_{12}| = |\vec{r}_2 - \vec{r}_1| = |\sqrt{\vec{r}_2 \cdot \vec{r}_1}| = |\vec{r}_1 - \vec{r}_2| = |\vec{r}_{21}|$, and $\hat{r} = \vec{r}_{12}/r = - \vec{r}_{21}/r$.

A few points are especially noteworthy:

1. The two electrical forces so produced are always equal in magnitude, regardless of how big or small $q_1$ and $q_2$ may be with respect to each other. Even if $q_1 \ll q_2$, the first object still ends up exerting just as much of an electrical force on the second object as the electrically much stronger $q_2$ exerts on the first.
2. The forces appear on the two EC objects even though they are not in a direct physical contact—exactly as in Newtonian gravity. They also obey an inverse-square law, once again exactly as in Newtonian gravity.
3. The forces don’t at all depend on the respective masses of the bodies; they are only for the electrostatic interaction.In reality, the two point-charges in the fixed configuration would also experience the force of gravity, not to mention the imaginary forces exerted on them by the mechanical supports. However, we ignore gravity in this entire description, and we exclude the support forces from our system—they are regarded as at all times belonging to the environment. (If a charge moves, so does its support, and along with it, the system–environment boundary. In fact a motion of a boundary is necessary for any system to exchange energy with its environment in the form of the mechanical work. We will need to take into account the mechanical work when it comes to defining the electrical potential energy of the system.)

4. Why the big ontological issue of Action-at-a-Distance remained unresolved before EM came on the scene:

The issue slipped under the carpet the last time (while discussing NM) now once again raises its (ugly or beautiful) head: How come two charges are able to exert forces on each other even if they are separated by nothing but the empty space, i.e., when there is no direct contact of the NM-kind between them?

People had thought a lot on this question right since the time that the issue came up in the context of gravity. In fact even Newton himself had once speculated whether an invisible string might not extend between two gravitating bodies like the earth and the Moon. However, Newton was only too well aware of the limitations of the available experimental evidence. The observational data such as that by Kepler indicated very high speeds for transmission of gravitational forces; the data were not refined enough to capture any effects of a finite but high speed for the forces. So, Newton refrained from adopting any definitive position concerning either a mechanism for the transmission. (Unlike Faraday, the poor fellow could not sprinke asteroids between the earth and the moon, and thereby grow confident enough about ideas like tubes of force or space-filling fields.)

The people from the other side—from the energetics approach to formulating mechanics—were not as deeply interested in the ontological matters anyway. In the market-place of ideas, their main market-differentiator was not a superior or more refined physics but a superior method of calculating solutions when the boundary conditions became complex, e.g., too numerous, as with continuous and curved constraints. (The physics implied by the mathematics of both the approaches—Newton’s momentum-based approach and the energy-based approach—was exactly identical. What differed between the two was mathematics—the methods of calculations.)

So, there arose a feedback circle of sorts: people who didn’t care about ontology and foundations of physics, but wanted to do maths, got attracted to the energetics program; people who already were in the program directed their energy in pursuing their strong point further. So, what they kept on developing was maths. Given this feedback circle, any ontological problem concerning the action-at-a-distance couldn’t have benefitted from them anyway. In actuality, it didn’t.

That’s why even if a term for the gravitational potential energy $V$ had appeared as early as ~1773 in Lagrange’s writing (which was devoid of not just ontology but also of even a single geometrical diagram), and even if luminaries like Laplace (~1799), Gauss (about the same time but published in 1840) and Green (1828) happily developed the potential theory for gravitation, they all were perfectly happy working with just “a mathematical function” of “coordinates” for $V$—not with an actually existing physical field. It was Green who described $V$ as the “potential function”.

Of course, these physicists couldn’t have used the term “potential energy” for $V$. The discovery of the first law of thermodynamics and the conceptual clarity on what precisely the concept of energy itself meant, was still only in latency; the explicit identification was at least 20–30 years away.

In any case, the mathematically oriented physicists on the continent didn’t pursue the issue of the physical meaning of potential energy a lot. The quizzical end-result was this: There wasn’t just an equation for the gravitational (and later electrostatic) potential energy of a system, there also was this distinctly further development of an equation for the gravitational potential of a single gravitating body. Thus, the Continental physicists had succeeded in mathematically isolating the interacting system of two charges into components specific to single charges: the potential field due to each, taken in isolation. The potential energy of the system could be found by taking the potential of either charge and multiplying it with the other charge. They were really advanced in maths. They successfully manipulated equations and predicted results. But they didn’t know (or much care) about what ontology their concepts or procedures suggested.

That’s why the ontological issue concerning the action-at-a-distance remained unresolved.

5. Faraday’s lines of force and Maxwell’s fields:

Historically, the crucial step in developing the idea of the field was taken by Faraday. I will not go into the details simply because they are so well known: sprinkling of iron filings near magnets, tubes of force (called “lines” of force by Faraday himself), the broad laws governing their behaviour (attractive force goes with tension, repulsive with sideways pressure), etc.

What is important is to note that Faraday did explicitly advocate the idea that the lines of force physically existed; they weren’t just a device of calculations the way the gravitational potential was to the Continentals. At the same time, he also believed that even the apparently empty space in between two lines of force was also filled with more such lines, that they filled the entire 3D space.

Maxwell got impressed by Faraday’s idea. He even met Faraday, and then, set out to translate the idea into more precise mathematical terms. He arrived at his mathematical description of the electromagnetic phenomena by imagining not just the tubes of force but even many further mechanical mechanisms, invented by him, in order to give a mechanical explanation of the known EM laws (Coulomb, Biot-Savart, Ampere, Faraday, and others.) Eventually, the system of explanations of EM phenomena using mechanical means (essentially, the NM-objects) became too complex. So, in his final synthesis, he simply dropped these complex details, but retained only the differential equations.

However, importantly, he continued to keep the abstraction of the fields in his final synthesis too.

The idea of the field solved the problem of action at a distance. Maxwell put forth the idea that all of space (except for the regions where charged objects are present) is filled with a physically existing field. Charged bodies are in continuous contact with the field, and therefore, are able to induce a condition of force in space—which is the field of force. Maxwell imagined that the force field consists of non-uniform mechanical stresses and strains. (He used the term “displacements” for these mechanical strains; hence the term “displacement current,” which is still in use.) When the same field comes in contact with some other charge, it experiences a net force due to the presence of these mechanical stresses at its boundary. That’s how the forces get transmitted.

Maxwell was smart. What he started out (or invented as he went along) were rather complicated physical mechanisms (all of which were made from NM-objects). But he had a definite sense of which point-quantities to abstract away, using what kind of limiting arguments, and how. Thus showing a refined and mathematically informed judgment, he arrived at an essentialized description of all the electromagnetic phenomena in terms of point-properties of a continuum.

6. The ontological view implied by Maxwell’s ideas:

In Maxwell’s view, the entire universe could be analyzed in purely mechanical terms. This means that all the objects he employed in his synthesis were essentially only NM-objects. These came in two types: (i) point-charges, and (ii) a mechanical continuum for the field.

The two interacted (exchanged forces) using the only mode that NM-objects were allowed: using direct physical contact. The interaction proceeded both ways: from charges to the continuum and from continuum to charges.

The point-charges pressed forces on the continuum of the field at their common boundary, which resulted in there being a mechanical field of stresses and strains inside it. These stress-strain states extended everywhere in the continuum, “up to” infinity.

The continuum, in term, generated forces on the surfaces of the boundary between itself and any charged object embedded in it.

The specific stress-strain field generated by a charge was spherically symmetric around that charge. Hence, it didn’t result into any net force acting on the same charge. However, due to the inverse-square nature of Coulomb’s law, and the fact that charges were point-particles, the field they generated was necessarily non-uniform at all other points. Hence, the continuum did generate a net force on the other charges.

Ontologically, there was little difference between Faraday meant (or strived to indicate), and what Maxwell directly put forth, using mathematical concepts.

7. The basic weakness of what precisely Faraday, and also Maxwell, meant by a field—its ontology:

Maxwell’s proposal of fields had a very great virtue, and a very great conceptual (ontological) weakness.

The virtue was an advocacy of a physically existing condition in what earlier was regarded as completely empty space. This condition was identified with the mathematically defined fields. The idea of fields was not just satisfactory from the viewpoint of broad philosophic principles (we will touch on them in a short while), it also gave a reasonable-sounding solution to the physics problem of action-at-a-distance.

The weakness was that the field, even if defined very carefully (in direct reference to empirically observed electromagnetic laws, and with rigorous mathematical abstraction), still was characterized, explained, and defended as a specifically mechanical kind of a physical existent. To describe the ontology of such fields using our scheme and notation, these EM fields were “pure” NM-objects.[Professional physicists often don’t have very good ontological clarity, but they are referring to the same underlying physical fact when they say that a mechanical aether provides a reference frame that obeys the Galilean-invariance but not the Lorentz-invariance.]

This particular weakness immediately led to conceptual challenges for Maxwell. Eventually, it also led to a lasting confusion for all, a confusion that persists till date (at least in the discussion of EM and aether).

The difficulties posed by the weakness were actually insurmountable. Here is one example. If the field had to be mechanical in nature as Maxwell said, then it would have to possess seemingly impossible combinations of physical properties. It would have to be an infinitely rigid object, and yet allow other massive objects (of the NM-object kind) to pass through them without hindrance. Et cetera. For an interesting history of how creative solutions were sought, and even were supplied see [TBD]. (A candidate explanation: The field acts like a metal ball placed on a block of snow: The ball passes through the block’s thickness even while keeping the block solid everywhere else, but the ball gets reflected when thrown with a sufficiently high speed. Another example I can think of: a jet-plane (say in a tail-pin) that hits the ocean surface. When the speed is great, it first rebounds as if it had hit a solid surface, rather than sinking in the water as it eventually does when it loses speed.) However, such explanations did not bear out—no mechanism would if it produces a mechanical aether in the end.

If Maxwell’s theory were to be less successful, this weakness would be less consequential. However, this was not the case. Maxwell’s was one the most fundamental and most successful theories of his time. (It was what QM and Relativity are to us.) Since the weakness rode on a very strong theory, it irretrivably put people in the mindset that all fields, including the luminiferous aether (which transmits Hertzian waves through it), had to be necessarily mechanical in nature.

8. The consequences of the conceptual weakness of Maxwell’s idea of the field—its ontology:

The weakness had to come out sooner or later, and in the 19th century, it came out rather sooner.

In particular, if the aether was to be space-filling, then the earth would have to move through it during its elliptical motion around the Sun. If this aether was mechanical in nature, then the interaction of the earth with the aether would have to be mechanical in nature. It would be analogous to the motion of a finger through the tub-water. The aether would exert a drag force on the earth’s motion, which could be detected via light interference experiments.

However, experiments such as those by Michaelson-Morely showed that such a drag was not actually present. This momentous experimental finding led to the following situation.

The physicists and mathematicians of the energetics program (mostly on the continental Europe) had labored for almost 70 years to develop the mathematics of fields, but without regarding it as a physical entity. But they had produced excellent mathematics which greatly clarified presentation of physics and simplified calculations. Even Maxwell’s theory had its mathematics developed on the basis of these developments.

The aether-induced drag was only a deductive inference made from Maxwell’s theory. Maxwell’s theory itself was founded on very well established experimental findings. Another deductive inference, namely that light was an EM wave with the speed of $c$, had been borne out by experiment too. So, all these aspects had to be kept intact.

But the drag implied by Maxwell’s aether was not to be found in the experiments. So, this part of Maxwell’s theory had to be corrected for. The simplest way to do that was to drop the whole idea of the aether from the theory!

After all, in his own development, Maxwell himself had started out with a laundry list of different kinds of physical mechanisms for different aspects of electromagnetics. However, eventually, he himself had come to drop all these mechanical features, because it was hard to get all these mechanisms to work together in a simple manner. So, he had instead decided to abstract out just his mechanical field from them. This field was then identified with the luminiferous aether.

But the aether-wind gave problem. So, why not take just another step of abstraction, and entirely do away with the very idea of the aether itself? Why not regard it as just a mathematical entity? Why can’t space once again be completely empty of any physical being, just the way it had been right since Newton’s times—and even during the entire development of the potential theory and all?

If the situation is to be framed as above, then there can be only one logical way out of it. The physicists came to choose precisely that. Without challenging the specifically mechanical nature of the aether (because no one could think of any other kind of a nature for a physical aether, since none could figure out any good philosophical arguments for having a non-mechanical aether), physicists in the late 19th- and early 20-th century simply decided to remove this whole idea from physics.

In the meanwhile, Einstein was advocating a denial of the absolute space and absolute time anyway. If the space itself was not absolute but depended on the relative motion of the observer, i.e., if space itself depended on motion, then was there any point in filling it with anything?

For the rest, pick up virtually any of the hundreds of the pop-sci books on the relativity theory and/or on Einstein’s (IMO at least in part undue) glorification. … If Maxwell to be less successful as a physicist, his wrong ontological views would be much less consequential. Ditto, for Einstein.

9. A preview of the things to be covered (concerning the EM ontology):

The essential error, to repeat, was to conclude that since Maxwell was successful, and since his field was mechanical, therefore every field has to be taken as being mechanical—i.e., as if it had to follow the NM ontology. Wrong.

Next time, we will cover the correct ontological view to be taken of the Maxwellian fields. We will also look into a few issues about ideas like the electrostatic potential, the electrostatic potential energy, and their fields. We look into the details of these concepts only because they are relevant from an ontological point of view. [And yes, there is a difference between just potential and potential energy—just in case you had forgotten it.] This discussion will also help us prepare for the correct ontological view which is to be adopted when it comes to the quantum mechanical phenomena. Hopefully, the whole portion would be over in two more posts, at most three.

See you next time. Bye for now, and take care…

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(Hindi) “o sanam, tere ho gaye hum…”
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