# 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.