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Lectures on Physics Chapter V: Moving charges, electromagnetic waves, radiation, and near and far fields

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The special problem we try to get at with these lectures is to maintain the interest of the very enthusiastic and rather smart people trying to understand physics. They have heard a lot about how interesting and exciting physics is—the theory of relativity, quantum mechanics, and other modern ideas—and spend many years studying textbooks or following online courses. Many are discouraged because there are really very few grand, new, modern ideas presented to them. Also, when they ask too many questions in the course, they are usually told to just shut up and calculate. Hence, we were wondering whether or not we can make a course which would save them by maintaining their enthusiasm. This paper is a draft of the fifth chapter of such course. It offers a comprehensive overview of the complementarity of wave- and particle-like perspectives on electromagnetic (EM) waves and radiation. We finish with a few remarks on relativity.
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Lectures on Physics - Chapter V
Moving charges, electromagnetic waves,
radiation, and near and far fields
Jean Louis Van Belle, Drs, MAEc, BAEc, BPhil
26 December 2020
Abstract
The special problem we try to get at with these lectures is to maintain the interest of the very
enthusiastic and rather smart people trying to understand physics. They have heard a lot about how
interesting and exciting physics isthe theory of relativity, quantum mechanics, and other modern
ideasand spend many years studying textbooks or following online courses. Many are discouraged
because there are really very few grand, new, modern ideas presented to them. Also, when they ask too
many questions in the course, they are usually told to just shut up and calculate. Hence, we were
wondering whether or not we can make a course which would save them by maintaining their
enthusiasm. This paper is a draft of the fifth chapter of such course. It offers a comprehensive overview
of the complementarity of wave- and particle-like perspectives on electromagnetic (EM) waves and
radiation. We finish with a few remarks on relativity.
1
Table of Contents
Table of Contents .......................................................................................................................................... 1
1. Potentials, forces, fields, and field energy ................................................................................................ 2
Newton’s First Law (action = reaction) ..................................................................................................... 2
Potential energy ........................................................................................................................................ 4
Yukawa’s nuclear potential and force ....................................................................................................... 8
Kinetic energy ......................................................................................................................................... 13
Forces, charges, fields, and potentials once more ................................................................................. 15
The scalar and vector potential .............................................................................................................. 17
The least energy and least action principle ............................................................................................ 19
Magnetic and electric dipoles ................................................................................................................. 19
3. The measurement of the position of a charge ....................................................................................... 20
The concept of an elastic collision .......................................................................................................... 21
Compton scattering ................................................................................................................................ 21
Photon and electron spin ........................................................................................................................ 23
Is non-perturbative measurement of the position of a charge possible at all? ..................................... 24
4. Charges in motion ................................................................................................................................... 26
5. Charges, energy states, potentials, fields, and radiation ........................................................................ 28
6. Photons and fields ................................................................................................................................... 30
7. The near- and far-fields ........................................................................................................................... 33
8. Complicated trajectories ......................................................................................................................... 34
9. Relativity ................................................................................................................................................. 36
2
Lectures on Physics Chapter V :
Moving charges, electromagnetic waves,
radiation, and near and far fields
Jean Louis Van Belle, Drs, MAEc, BAEc, BPhil
1. Potentials, forces, fields, and field energy
Let us quicky review the basics of field theory with an introduction of the concepts, vector operators
and, finally, Maxwell’s equations and the Planck-Einstein relations.
Newton’s First Law (action = reaction)
A charge carries potential and kinetic energy. What about fields? There is field momentum and energy
too, but fields and particles are different. Think about this: if the field has no charge to work on, should
we talk about potential or potential energy? Potential only, right? The physical dimension of energy is
force times distance: 1 joule (J) = 1 newton (N) · 1 meter (m). If the force has no charge to work on, is the
force really there? The Lorentz force, for example, is written as F = qE + q(vB). We can, therefore, write
the electromagnetic field vector as:
Is there a force if there is no charge to work on? What about photons? They are particle-like, aren’t
they? And they carry energy according to the Planck-Einstein relation:
1. Photon energy: E = h·f
2. Photon momentum: p = h/λ = h/c·T = h·f/c = E/c
Besides linear momentum, they also carry angular momentum or spin: plus or minus one unit of ħ, to be
precise (photons do not have a zero-spin state
1
). We may, therefore, want to think of photon absorption
and emission by an electron as occurring through a spin flip of the electron: electron spin is ħ/2, so a
1
Most if not all courses in quantum mechanics spend one or more chapters on why bosons and fermions are
different (spin-one versus spin-1/2) and will present nice graphs showing how spin-one particles have three energy
levels in a magnetic field (one for spin-up, zero-spin and spin-down) but, when it comes to the specifics, then the
only boson we actually know (the photon) turns out to not be a typical boson because it cannot have zero spin.
Richard Feynman, for example, first talks about this fact in a footnote only (Feynman’s Lectures, III-11, footnote 1),
before he offers a rather comprehensive analysis of how this angular momentum might be transferred to an
electron in section 4 of Lecture III-17. We will let you look it up. It is really one of those things which makes
students nod their head in agreement with Prof. Dr. Ralston’s criticism of his own profession: Quantum mechanics
is the only subject in physics where teachers traditionally present haywire axioms they don’t really believe, and
regularly violate in research.” (John P. Ralston, How To Understand Quantum Mechanics, 2017, p. 1-10)
3
flip from up to down or vice versa would absorb one unit of ħ. That is rather convenient because we can
then stick to a conservation law for angular momentum which works with units of ħ (not half-units
2
).
In 1995, after a long and distinguished career (including a Nobel Prize for Physics), W.E. Lamb Jr. wrote
the following on the nature of the photon:
“There is no such thing as a photon. Only a comedy of errors and historical accidents led to its
popularity among physicists and optical scientists. I admit that the word is short and convenient.
Its use is also habit forming. Similarly, one might find it convenient to speak of the “aether” or
“vacuum” to stand for empty space, even if no such thing existed. There are very good
substitute words for “photon”, (e.g., “radiation” or “light”), and for “photonics” (e.g., “optics” or
“quantum optics”). Similar objections are possible to use of the word “phonon”, which dates
from 1932. Objects like electrons, neutrinos of finite rest mass, or helium atoms can, under
suitable conditions, be considered to be particles, since their theories then have viable non-
relativistic and non-quantum limits.”
While we understand Lamb’s skepticism (quantum physics has effectively suffered from an overdose of
unproductive new concepts), we do not fully share it: a photon effectively carries a discrete amount of
energy and (linear as well as angular) momentum and are emitted and/or absorbed by charged particles
so as to transfer that energy, momentum, and spin from/to the field. In other words, the idea of a
photon embodies the Planck-Einstein relation and it is, therefore, a useful concept.
3
At this point, we want to reintroduce Newton’s oft-forgotten first law of physics: action equals reaction.
An electron which absorbs a photon absorbs the (physical) action of the photon. Conversely, an electron
which emits a photon will experience a recoil effect. The Planck-Einstein relation basically states such
2
The Uncertainty Principle is often written as σpσx ħ/2 or, the complementary viewpoint, σEσt ħ/2. See, for
example, the proof of the Kennard inequality using wave mechanics in the Wikipedia article on the Uncertainty
Principle. The σ is the standard deviation (of the uncertainty on the (linear) momentum, position, energy, and
time, respectively. The proof assumes a random but normal distribution of the observations, so the [μ + σ] interval
would catch about 68% of the observations. If the distribution is normal, the observation can have any value, but
the [μ + 2σ] interval captures a more acceptable 95% of the observations and, therefore, gives a better intuitive
idea of the spread (aka dispersion, variability, or scatter) of the distribution. In any case, our interpretation of
quantum physics sticks to statistical determinism. We agree with H.A. Lorentz who, at the occasion of the
(in)famous 1927 Solvay Conference (just a few months before his demise), asked the presenters of the new ideas
(Louis de Broglie, Max Born, Werner Heisenberg, Erwin Schrödinger, with Niels Bohr offering final remarks) why
felt they needed to “elevate indeterminism to a philosophical principle.” To be precise, our worldview is
deterministic but our theories must also account for the observer effect: electrons have a non-zero radius of
interference and any probing of their position will inevitably lead to a change in that position and, therefore,
introduce an uncertainty after the measurement: we knew where the electron was, but we cannot be so sure now.
3
Lamb apparently finds the idea of “neutrinos of finite rest mass” a more useful concept. We think neutrinos have
no rest mass and act as the carriers of nuclear force or energy. In other words, we think they are the photons for
the nuclear force. We wrote an, admittedly, rather speculative paper on this: Neutrinos as the photons of the
strong force?
4
action (whose unit is N·m·s: force·distance·time
4
) comes in units of h or if angular momentum is
involved units of ħ/2π.
5
Potential energy
The potential energy of a charge depends on (1) its charge (which does not change with motion) and (2)
its position. The idea of position is simple enough but we must remind the reader, once again, of the
subtle difference between the potential and the potential energy of the charge that we would be
bringing it to measure the potential. If we write the charge at the center as q and the charge that we are
bringing in as Q, then the electrostatic or Coulomb force between them will be given by Coulomb’s
inverse square law:

We will soon have a rather nasty game of tracking signs of charges and direction of forces, so let us do
the dimensional analysis upfront here. The electric constant k is equal to 1/40 and its physical
dimension is [1/40] = N·m2/C2. Hence, [ kQqr2] = (N·m2/C2)·(C2·m2) = N, and so we do get the right
unit: newton, the force unit.
Now, we know we get the electric field the electrostatic field E from separating q and Q:

The dimensions here come out alright too: [kQ/r2] = (N·m2/C2)·(C·m2) = N/C (newton per coulomb). But
the field is not the potential. The potential must be expressed in energy units (newton times meter). But
wait! what if Q is not the unit charge? If Q is a multiple of the unit charge, then the potential energy
of Q will be a multiple of the potential because F is proportional to both q and Q. This solves one of the
little questions we started out with: if the field has no charge to work on, should we talk about potential
or potential energy?
We should talk of potential only, of course. Potential is potential energy divided by charge: V(r) = U(r)/Q.
For the gravitational field, we should write: Vg(r) = U(r)/m. The difference is subtle but we note this
subtlety is not always noticed, and we do not always respect these subtleties ourselves! We ourselves
will write the potential often as V(r) = kqe2/r2 rather than as V(r) = kqe/r2. We request the reader to
forgive us when that happens.
Note that Q and m are the ‘charges’ that we are bringing into the field. The force law, in both cases, is
what is referred to as an inverse square law because of the 1/r2 factor. The physical proportionality
constant (k and G respectively) will always ensure the physical dimensions on both sides of the equation
come out alright. Their physical dimension should, therefore, always be written as N·m2/[charge]2. For
the electromagnetic force, the charge will be the unit (electric) charge (e).
4
We think the German Wirkung is far more evocative than the English action.
5
The h and ħ units are, obviously, not very different in terms of physical concept: think of the difference between a
radius and the circumference of a circle, measuring time in radians, and other equivalents when going from linear
to circular motion and vice versa.
5
For the gravitational force, we have the unit mass (kg). Most textbooks (and Wikipedia) write the
gravitational constant G as (approximately) 6.67410−11 m3kg1s2. That does not look like N·m2/kg2,
does it? When defining mass as inertia to acceleration (a change in its state of motion as a result of a
force acting on it), we should also define the unit mass (kg) in terms of force. This is Newton’s second
law: a force of 1 N will give a mass of 1 kg an acceleration of 1 m/s2. Hence, 1 kg = 1 N·s2/m. Inserting
this into the structure of the proportionality constants for all inverse-square force laws, and then
converting N back to kg so as to get that we get that m3kg1s2 expression, we get the right dimension:



The fact that mass serves both as the ‘charge’ for the gravitational force as well as a definition for force
led Einstein to develop his geometric approach to gravity (general relativity theory), in which masses
move through space along (non-linear) geodesics in (curved) space, without force acting on them.
We think such geometric approach does not fundamentally solve the question: what is gravity? Many
physicists hoped to be able to analyze the gravitational force as a residual force of the two more
fundamental forcesthe electromagnetic force and the nuclear force, which is supposed to hold
protons (and neutrons and protons) together.
6
This would then ensure we rewrite the gravitational
force in terms of the charges upon which these forces are acting, i.e. the electric and nuclear charge,
respectively. The structure of the electromagnetic force is given by Maxwell’s equations and is,
therefore, known. Unfortunately, it does not look like we will find the equivalent of Maxwell’s equations
for the nuclear force any time soon. The problem is complicated because the electromagnetic force will
act on an electric charge (q), while the nuclear charge should act on a nuclear charge (which we will
soon write as gN), but these two forces are not supposed to act on photons, and we know light gets bent
when it passes large masses, so how does that work then?
7
We have no clue.
Let us get back to our (non-uniform) static electric field here whose strength is measured as the force
per unit charge: E = F/Q. So we have a force measured in N, a field in N/C, and a potential (energy) in
6
This is a rather simplistic analysis because two protons do not bind in any stable way. Diproton the nucleus of
2He isotope is, effectively extremely unstable. A neutron is needed to glue them together in a more stable
configuration: 3He. We think of a neutron as consisting of a proton and a deep (nuclear) electron. It is an idea
which Rutherford thought of when he first hypothesized the existence of neutrons (see our paper on Rutherford’s
contributions to the Solvay Conferences) and which explains proton-neutron reactions as well as the instability of
the neutron outside of the nucleus but, while some physicists are making decent progress in modeling deep
electron orbitals (e.g. Jean-Luc Paillet and Andrew Meulenberg, Highly Relativistic Deep Electrons and the Dirac
Equation, 2019-2020), we have come not across a decent neutron model yet (we are not talking an analysis in
terms of quarks and gluons here, of course). If the n = p + e hypothesis holds, then protons would actually be held
together by electrons (see our (speculative) paper on electrons as gluons) and then we should probably prioritize
the development of a decent model of the nucleus of heavy water: deuteron. This looks like the classical three-
body problem, which we cannot solve analytically but numerical approaches would be good enough. In addition,
chemists already analyzed how valence electrons hold atoms together in molecules long before the structure of
the atom was revealed, so we should not despair!
7
Arthur Eddington, Frank Watson Dyson, and their collaborators effectively observed that electromagnetic
radiation (light) follows a curved path near a massive object during the total solar eclipse on May 29, 1919 (Dyson,
F. W., Eddington, A. S., Davidson C. (1920), A determination of the deflection of light by the Sun's gravitational field,
from observations made at the total eclipse of 29 May 1919, Philosophical Transactions of the Royal Society 220A
(571581): 291333. This confirmed Einstein’s general theory of relativity.
6
N·m = J. All express the same reality of the field. To simplify the analysis, we will assume both charges
are like unit charges (both positive or negative). We can then simplify and write:


To calculate the potential energy, we have to calculate an integral. This integral defines energy as a
force over a distance
8
, and so we calculate the energy that we will expend on a charge qe by integrating
from the point of reference (PE = 0 at an infinite distance) to the point for which we want to calculate
the potential, which we call point 1 (see Figure 1)
9
:
 
Figure 1: Doing work in a radial field
This is line integral with a vector dot product (F·ds) because we can follow any trajectory to move the
charge from infinity to (1). To go from 1 to 2, for example, we may first follow an equipotential line,
which is the trajectory given by the arrows in Figure 2. We should get the same results even if the
integral is indefinite. We will probably want to use the usual definite integral, however, which assumes
the angle between the line of force and the trajectory is 180(), which ensures the cos in the F·ds =
F·ds·cos = F·ds·cos vector dot product is always equal to minus one. F·ds·cos is, therefore, equal to
F·ds·cos, so we can write:
8
The measurement unit for energy is newton·meter: force times distance. Because this is a very large unit in
quantum physics, it is more common to use the electronvolt (eV). One eV is the energy gained or lost by an
electron when it moves an electric potential difference of one volt in vacuum. The conversion factor is 1 eV =
1.602176634×1019 J. This numerical value is equal to the numerical value of the electron charge expressed in
Coulomb (1.602176634×1019 C) because the volt is an SI unit (1 V = 1 J/C): it is the potential difference between
two points that will impart one joule of energy per coulomb of charge that passes through it. The
1.602176634×1019 value is an exact value since the 2019 redefinition of the system of SI units.
9
We use an integral because the field is non-homogeneous or non-uniform: the field strength and, therefore, the
force, effectively varies from point to point. It may seem to be rather tedious to bring charges to infinity and then
back again but, as we will see in a moment, we will be more interested in measuring potential differences between
two points 1 and 2. That will be easier in practice. However, the theory requires we do think about infinity.
Velocities are always finite, fortunately! Indeed, a particle with infinite velocity would be everywhere and,
therefore, nowhere, as we would not be able to say where it was, where it is right now, and where it will be. We
therefore think the finite speed of light (c) and relativity theory makes a lot of sense!
7






We get the primitive or antiderivative function U(r) = kqe2r1 from the force formula:

The minus sign causes quite some headache here because the antiderivative does not have the minus
sign of the force formula: d(kqe2r1)/dr = kqe2r2 but reappears because the direction of the force and
the direction of motion are opposite (cos = 1).
10
Any case, the bottom line of this story is that the potential energy of a charge depends on (1) its charge
(which does not change with motion) and (2) its position (which we wrote as a radial distance r
11
):

No velocity or mass factors here. These factors are discussed below. Before we do that, let us quickly
introduce the idea of electric dipoles so as to introduce the idea of superposed fields and explain why
the electromagnetic fields do not rip us apart. Indeed, electrons and anti-electrons annihilate each
other, but electrons and protons do not: they stick together in electrically neutral atoms or (multi-
particle) ions.
Ions carry charge and, therefore, are associated with intense field strengths. Homogenous but overall
neutral distributions of positive and negative charge do not: the fields compensate and, therefore,
absent. However, this compensation may not be totally effective. Neutral composite particles or
electrically neutral materials may still have some polarity left because of an asymmetry in their
structure. Think of water molecules here (H2O), which will freeze forming a regular crystal structure.
Such asymmetries can be analyzed in terms of non-uniform distribution of charges and give rise to
(electric) dipole fields. The function for the magnitude of the dipole potential denoted as φ(r) below –
has the same (physical) proportionality as the function for the electrostatic potential of a single charge
denoted as V(r) below
12
but, importantly, the magnitude goes down with the square of the distance
(as opposed to the 1/r function for electrostatic potential) and also depends on the angle vis-à-vis the .
13
Also note the dipole moment p·cosθ = q·d·cosθ factor versus the charge q in the numerator
14
:
10
Keeping track of signs and dimensions is not always easy! You will often read the minus sign is just a matter of
convention and it is but several conventions must be followed here, and there is only one way to do it right,
and that is by consistently following all conventions! Note that the electron charge is negative but this minus sign
cancels with the minus sign of the second charge that we are bringing in.
11
When solving practical problems, such as finding solutions to Schrödinger’s equation for atomic electron
orbitals, one will sometimes want to switch from Cartesian to spherical coordinates.
12
We will write V(r) or U(r) instead of φ(r) only if we need to distinguish different potentials.
13
For a succinct analysis, see: Feynman’s Lectures, Vol. II, Chapter 6 (The Electric Field in Various Circumstances).
14
The p is not momentum but the dipole moment of the two (opposite) charges (+q and q). To be precise, the
dipole moment is the product of the charge and the distance between them: p = qd. The formula for the dipole
moment is an approximation only: it neglects the higher-order terms of a binomial expansion and is, therefore,
8


Why are we mentioning this? It is because of what we said above about gravity: (almost) every physicist
would like to think of the gravitational field as some kind of residual field. Such theory would show that
various 1/r, 1/r2, 1/r3 and lower-order potentials of electric and nuclear charges
15
combine to, somehow,
leave a gravitational potential. The gravitational potential is currently written as a function of the
gravitational constant (G) and the mass which causes the field (M). It is a simple 1/r function as well:
If protons (and neutrons) would also carry a nuclear charge gN, then the added fields would probably
look very much like dipole fields as well, except that we are not talking opposite charges here (an
electric dipole consists of a +q and a q charge). Hence, for all practical purposes, the (far) field around
two protons would very much look like the added fields of two gN charges at exactly the same spot.
What law would we have: 1/r or 1/r2 or something else? Let us think about that using Yukawa’s potential
for the nuclear force.
Yukawa’s nuclear potential and force
The idea of the Yukawa potential is not badin principle, at least. We write it as follows
16
:

The a is a range parameter. According to Aitchison and Hey, we should use a value around 2 fm, which is
about the size of deuteron, i.e. the nucleus of deuterium, which consists of a proton and a neutron
bound together.
17
To make sure you understand what Yukawa tried to model, we will remind you of the
formula for the electrostatic (Coulomb) potential:

The structure of these two formulas is exactly the same, except for the e
r/a function and the υ0 factor,
which is usually forgotten. We need it, however, to ensure the physical dimension of both sides of the
only good enough as an approximation at larger distances. We will talk about the difference between near and far
fields later in this paper.
15
See footnote 35.
16
The Wikipedia article uses a mass factor but we prefer the original formula given in Aitchison and Hey’s Gauge
Theories in Particle Physics (2013). It is a widely used textbook in advanced courses and, hence, we will use it as a
reference point. We may also refer the reader to Feynman’s remarks on it (II-26-6, the nuclear force field) because
these are online and free. Note that we write the potential as V(r) = kqe2/r2 rather than as V(r) = kqe/r2. We
mentioned the subtle difference between potential and potential energy already, and that we would not always
respect these subtleties ourselves! We are in good company here, however, because Aitchison and Hey do the
same!
17
To be precise, the charge radius of deuteron is about 2.1 fm.
9
equation is the same. It is, therefore, similar to the physical dimension of the electric constant ε0:
instead of C2/N·m2, we write: [υ0] = Y2/N·m2. It does the same trick as ε0 for the electric potential (it gives
us a U(r) expressed in joule or N·m), and it would be a mistake to leave it out. Indeed, we started off by
saying that the idea of a nucleon charge is something new: we associate some potential with it.
However, we should not think of it as electrostatic charge. We have no positive or negative charge, for
example: all nucleons positive, negative, or neutral
18
share the same charge and should attract each
other by the same (strong) force. Hence, we need to define some new unit for it. We thought of the
Einstein, but that name is used for some other unit already.
19
In my previous papers on the topic of the
Yukawa potential
20
, we suggested the Yukawa but I now think there is too much association between
that name and the presumed unit of the Yukawa potential.
21
We, therefore, propose the dirac.
22
However, for reasons of consistency we will continue to use the charge symbol we used in previous
papers: Y.
23
Now that we have fixed the dimensions, what numerical value should we take for υ0? We have no idea,
but now that we are discussing these things in very much detail, should we wonder about the 4π factor?
Do we need it? It is common to both potentials (and to the forces, which we calculate in a minute)
because it is the 4π factor in the formulas for the surface area (4πr2) and the volume (4πr3) of a sphere.
24
Feynman often substitutes qe2/4πε0 by e2, which is a unit with a weird but simple physical dimension:
the C2 in the numerator and denominator cancel out and we are left with N·m2 only, which is great
because we need the force to be expressed in newton, of course!
25
So we will do the same here and we
hope the reader will be able to distinguish e2 and e (Euler’s number). Writing gN2/4πυ0 as N2 (again, we
hope this causes not too much confusion in the mind of the reader!), we get the following formulas
now:
18
Negative? We only have neutrons and protons, don’t we? Yes, but we can imagine anti-atoms and, hence, anti-
protons. Protons and anti-protons will annihilate each other, but two anti-protons should stick together by the
same nuclear force.
19
Believe it or not, but the Einstein is defined as a one mole (6.022×1023) of photons. It is used, for example, when
discussing photosynthesis: we can then define the flux of light or the flux of photons, to be precise in terms of x
micro-einsteins per second per square meter. For more information, see the Wikipedia article on the Einstein as a
unit: https://en.wikipedia.org/wiki/Einstein_(unit). If we would truly want to honor Einstein, I would suggest we re-
define the Einstein as the unit of charge of the nucleon.
20
See: The nature of Yukawa’s force and charge and Who needs Yukawa’s wave equation? (June 2019). Our
treatment here is a shortened and revised version of Neutrinos as the photons of the strong force (October 2019).
The main revision consists of the use of gN and qe instead of gN2 and qe2 in the potential and force formulas.
21
The Wikipedia article on the Yukawa potential associates the 1/m unit with it, but that makes no sense
whatsoever to us.
22
We note that Dirac’s colleagues at Cambridge seem to have defined the dirac as ‘one word per hour’ but we
think there is no scope for confusion here.
23
This matches the upsilon (υ) it is not a μ (mu)! we use for the proportionality factor.
24
Gauss’ Law can be expressed in integral or differential form and these spherical surface area and volume
formulas pop up when you go from one to the other. Hence, you should not think of this 4π factor as something
weird: it is typical of spherically symmetric fields.
25
See, for example, Feynman’s calculation of the Bohr radius (a) using the p·a = h relationa rather precise
expression of the Uncertainty Principle, that is! Note that we will effectively get force formulas both for the
Coulomb as well as for the nuclear force with 1/r2 in the denominator, so we get something expressed in newton
alright!
10

What about the distance a? We will want to think of it as a natural distance unit (think of the typical
inter-proton distance in an equally typical nucleus
26
) and we may, therefore, equate a to 1 and measure
distances in units of a. However, we will wait with that for the time being.
If we have a potential, we can calculate the force. In fact, we should calculate the force, because we
should not be thinking in terms of terms of equating potentials here but in terms of equating forces.
27
To
do this, we should use this force formula:



Let us think about the minus signs here. The forces should be opposite, right? Right, but the formula
takes care of that: the total force is, effectively, equal to dU/dr + dV/dr and we will want it to be zero
so the dU/dr = dV/dr is quite alright. As you can see, we should really keep our wits with us here, so let
us remind ourselves of what we are trying to do here. We are thinking of two protons here, and these
two protons carry an electric charge (qe) as well as what we vaguely referred to as a nuclear charge (gN).
The electric charge pushes them away from each other, but the nucleon charge pulls them together. At
some in-between point, the two forces should be equal but opposite. So we should find some value for a
force expressed in newton. A force is a force, even if we know it acts on two different unit charges: qe
versus gN. We express one in Coulomb units, and the other in this new unit: the dirac. Sounds good? Let
us go through the calculations, then. The Coulomb force is easy to calculate:




This is just Coulomb’s Law, of course! The calculation of the nuclear force is somewhat more
complicated because of the er/a factor
28
:


 

26
We will let the reader google actual measurements of such distances! The beauty of our argument is that
unlike Aitchison and Hey and others writing on the topic we do not want to predetermine the range parameter a.
27
The 1/r and er/a/r functions do not cross anyway, so we should not try to equate them. In fact, the 1/r2 and (r/a
+1)·er/a ]/r2 do not cross either! Note that we can compare forces only because the nucleon carries both electric
charge as well as nuclear charge. The associated fields are, therefore, different: newton/coulomb versus
newton/dirac, to be precise.
28
We need to take the derivative of a quotient of two functions here. Needless to say, we invite the reader to
carefully check all logic and double-check the calculations and if needed to email us their remarks and/or
corrections.
11

 





This gives us the condition for the nuclear and electrostatic forces to be equal:



What can we do with that? We know what e2 is (we know what the electron charge is and we can,
therefore, calculate it), but what about N2? We have one equation and two unknowns here, so we
cannot calculate anything, right? Should we convert back to qe and gN? Not sure, but let us see if we get
something more meaningful:




Nope. But Eureka!
29
r must be equal to a if the two forces are equal, right? Right. So, perhaps, we
cannot determine a, yet, but we can determine N2 or gN2. When r = a, the condition above becomes:

When substituting ε0 for ε0 = qe2/2hc
30
, we get the following equation for the squared nuclear charge
or, more appropriately, perhaps, the nucleon charge:



This is Van Belle’s formula
31
: the product of two pure numbers (Euler’s number and the fine-structure
constant) and two physical constants (Planck’s constant and the speed of light). Let us first check the
physical dimensions to see if we have done our work correctly:
[gN2] = [ehcυ0] = (N·m·s)·(m/s)·(Y2/N·m2) = Y2
[gN2/4πυ0] = [ehc] = (N·m·s)·(m/s) = N·m2
We can now compare the value of N2 to the value of e2, which is equal to:
29
Archimedes is said to have exclaimed this in the bathtub when he found a way to distinguish fake from real gold
for the tyrant who paid him, but Scientific American thinks the story is fake news. We think Archimedes must have
had several aha moments. We do too.
30
We refer to the 2019 revision of SI units here, which we think of as being very significantmore significant than
CERN’s experiments on testing the quark hypothesis or the Higgs field.
31
I told my kids I want it on my tombstone.
12

 
Hence, we get the following:






This does not make sense! Not at all, really! What is the conclusion?
We are not sure. Either we made a terrible logical or calculational mistake or, else, Yukawa’s potential
formula does not make sense. If we made no logical or calculational mistake and we invite the reader
to carefully check then the latter should be the case. :-/
We plotted the functions, and find that the 1/r and er/a/r potential functions never cross, so we should
not try to equate them. We also tried various range parameters, but the 1/r2 and [(r/a +1)·er/a ]/r2 do
not cross either!
32
What should we do? Try another potential function, perhaps? One that falls off with
1/r2 or 1/r3, perhaps?
We are not sure. We think Yukawa’s approach might be too simple because two protons do not bind in
any stable way. A diproton the nucleus of 2He isotope is, effectively extremely unstable. A neutron is
needed to glue them together in a more stable configuration: 3He. So perhaps we should think of a
neutron as consisting of a proton and a deep (nuclear) electron which binds both. It is an idea which
Rutherford thought of when he first hypothesized the existence of neutrons and which would explain
proton-neutron reactions as well as the instability of the neutron outside of the nucleus.
Indeed, the n = p + e hypothesis holds, then protons would actually be held together by electrons, and
then we should probably prioritize the development of a decent model of the nucleus of heavy water:
deuteron. This looks like the classical three-body problem, which we cannot solve analytically but
numerical approaches would be good enough. In addition, chemists already analyzed how valence
electrons hold atoms together in molecules long before the structure of the atom was revealed, so we
should not despair!
33
What about Van Belles formula, then !? Utter nonsense.  If anything, it becomes the definition of the
electric constant if you replace Eulers number by one (as you should) and forget about the whole idea
of a nuclear charge in Yukawas now (in)famous nuclear potential function:

If anything, Van Belles formula proves nuclear charges make no sense because we cannot possibly mix
charges and fields. Why? Because they lack a comparable physical dimension. The 2019 revision of SI
units firmly cements and buttresses classical physics. Forget about all of the weirdness ! 
32
We recommend the Desmos graphing tool because it is easy to use.
33
We offered some references in footnote 6, but here is another idea: magnetic forces holding nucleons together:
Paolo Di Sia, A solution to the 80-year old problem of the nuclear force, October 2018.
13
Kinetic energy
The kinetic energy of a charge does not depend on charge or position but on (1) its velocity (a function
which may non-zero first-order, second-order etc. derivatives
34
) and (2) the inertia to a change to its
state motion, which is measured as the (relativistic) mass of the charge. A proton and an anti-electron,
for example, have the same charge but a proton is about 1,836 times more massive.
We should immediately note that their magnetic moments are also very different, which matters when
discussing angular momentum and rotational inertia.
35
Magnetic moments are important: they explain
why electrons want to pair up in atomic or molecular orbitals
36
despite the strong repulsive force
between them. However, we will talk about that later as we were talking kinetic energy here.
The relativistically correct formula for kinetic energy defines kinetic energy as the difference between
the total energy and the potential energy: KE = E PE. The total energy is given by the E = mc2 = m0c2
which, as Feynman shows, can be expanded into a power series using the binomial theorem
37
. He does
so by first expanding m0:

We can then multiply with c2 again, to obtain the following series:

The potential energy here is equal to m0c2: the rest mass, including any energy it may acquire from any
potential energy it may acquire because it is one or the other field. As mentioned, the kinetic energy
must then be the difference:
34
The first-order derivative is the acceleration (a = dv/dt). The second-order derivative is usually referred to as jerk
(we do not like that word) or (our preference) jolt (j = da/dt = d2v/dt2). Third- and higher-order derivatives carry
names like snap, crackle, and pot (which are the cartoon mascots of Kellogg's cereal Rice Krispies). Peter
Thompson whose design of devices to control the moves of tools, printers, amusement park rides, aircraft
autopilots and telescope mounts thinks engineers are entitled easily remembered mnemonics so we do not want
to challenge the seriousness of these names.
35
The electron’s mass, radius, magnetic moment, and spin (angular momentum) can easily be explained in terms
of a ‘mass without mass’ model if we think of an electron as a tiny ring current (see our paper on the mass, radius
and magnetic moment of electrons and protons following the PRad measurement of the proton radius). In other
words, the electron’s mass can be explained in terms of electromagnetic mass only. For the proton, however, we
get annoying 1/2 or 1/4 factors when trying to do that. These can only be explained away by assuming that a
proton, besides electromagnetic mass, has nuclear mass too. This is why we believe a much stronger nuclear force
must be present. Force acts on charge and, therefore, a proton must carry both electric as well as nuclear charge.
Nuclear charge would be denoted as gN and we invented the dirac unit to provisionally talk about that (which we
will do later) but to honor Yukawa’s invention of a nuclear potential – we reserve the symbol Y for it.
36
Atoms with incomplete shells will, therefore, react chemically so as to share their valence electrons. If the
nuclear charge is real, there may also be a nuclear equivalent of the magnetic moment, but we have not even
started to think about that.
37
Feynman’s Lectures, I-15-8 and I-15-9 (relativistic dynamics).
14

We recognize, of course, the non-relativistic or classical m0v2/2 term but, as you can see, when
relativistic speeds come into play, the higher-order terms may matter too!
Let us come back to the concept of rest mass. We mentioned it should include any energy it may acquire
from any potential energy it may acquire because it is one or the other field. This, of course, assumes the
mass carries (net) charge, because forces act on charges, remember? So how does that work, then?
The matter is trickyliterally! Let us compare the gravitational and electric force once more. They both
fall off non-linearly with distance. To be precise, they follow the inverse square law, so the force is
proportional with the inverse of the squared distance (1/r2). For the rest, the gravitational and Coulomb
force have pretty much the same structure:

We just have a minus sign for the Coulomb force because like charges (both positive, or both negative)
will repel each other while opposite charges will attract each other: the minus sign takes care of that.
We might now throw many more definitions and equations but, before we do so, let us think about
what is relative, or not. We can appreciate that mass is a relative concept depending on velocity, in
other words but charge is surely not:
Regardless of the state of motion of our q1 and q2 charge, we will always measure the two charges as
being equal to q1 and q2, and such charge will be a positive or negative multiple of the elementary
charge e which depending on your convention will be the charge of either the proton or the
electron.
38
In contrast, relativity theory tells us the masses M and m will increase as their velocity increases. The
formula for the relativistic mass increase multiplies the rest mass m0 by the Lorentz factor to obtain the
relativistic mass mv:

Isn’t there a bit of an inconsistency here? We hope not!
Also, the formula makes us wonder: what velocity? Velocity is not a scalar quantity: it is a vector
quantity and we may, therefore, distinguish between a velocity in the x-, y-, and z-directions,
respectively. Should we, therefore, think of the mass concept as a vector itself, with a magnitude and a
38
In general, we take e to be the charge of the proton so as to avoid another minus sign. At the same time, protons
are very massive and, therefore, it is easier to talk about moving electrons than moving protons.
15
direction? H.A. Lorentz did so because he defined mass as the ratio of force and acceleration (as
opposed to the definition which we use now, which is the time rate of change of the momentum).
39
The background here is the relativistic force law, which defines a force as that what changes the
(relativistic) momentum of an object: F = dp/dt. If we take F to be the force in the direction of motion,
then we can write this in terms of the magnitudes of F and p (F = dp/dt). The derivation is somewhat less
straightforward than you might expect at first but by using the product and chain rule, and with some
re-arranging, you should be able to prove the formula
40
:


 
 

This relativistically correct definition of a force does away with the definition of a force as the product of
mass and acceleration (and of mass as the ratio of force and acceleration), which we get by substituting
the rest mass m0 by the relativistic mass m0 in Newton’s F = m·a formula.
41
It, therefore, confirms the
definition of mass as a simple scalar quantity measuring the inertia of a charge to a change in its state of
motion.
42
This should be sufficient for the time being. Let us turn back to the fields.
Forces, charges, fields, and potentials once more
We will review the concepts introduced already once more but give you a bit of toolbox to work with
them. You will remember we measured the potential at some point in space x by doing work against the
force field from a reference point. We explained this for static fields, using the example of the
gravitational field, or the electric field surrounding a charge. Things get more complicated when allowing
these fields to vary in time, which we will do in a moment, but let us first stick to the example of a
simple radial field surrounding a pointlike charge. We then have an electric potential which depends on
the (radial) distance from the point charge only, which we denote by x in Figure 2.
39
This explains Einstein’s mentioning of the concepts of ‘longitudinal’ versus ‘transverse’ mass for a slowly
accelerating electron in his 1905 article on special relativity: On the electrodynamics of moving bodies, p. 22. The
Wikipedia article on relativistic mass offers a good overview of the history of ideas here.
40
In case you want to check your calculations, we recommend the online (free) LibreTexts Physics textbook.
41
From the above, it should be clear this is not the correct force law: it yields F = mv·a= m0·a. This equation does
not have the required cube of the Lorentz factor (F = 3m0·a).
42
It is obvious that, if we would write m as a vector quantity (m), we would have a vector dot product m·v = mxvx +
myvz + mzvz = px + py + pz. Hence, it is quite convenient we may consider mx, my, and mz to be equal to one and the
same scalar quantity mv, because then we can simply write px, py, and pz as px = mvx, py = mvy, and pz = mvz,
respectively.
16
Figure 2: Doing work in a radial field
The calculation is quite subtle and we need to keep track of signs and respect conventions. The
assumptions in Figure 2 is that the radial field is caused by an electron at the x = 0 point, and that we are
measuring the potential energy by moving another electron a like charge qe to or away from the
charge at the center. The reference point is x = , where the charge will have zero potential energy:
think of an electron that is free of the force. When moving it towards the center to point (1) or (2), for
example we are doing work against the force. The amount of work that needs to be done which is
nothing but the energy (energy is force over a distance
43
) we will expend on it is calculated using an
integral
44
:
 
This is line integral with a vector dot product (F·ds) because we can follow any trajectory to move the
charge from infinity to (1). To go from 1 to 2, for example, we may first follow an equipotential line,
which is the trajectory given by the arrows in Figure 2. We should get the same results even if the
integral is indefinite. We will probably want to use the usual definite integral, however, which assumes
the angle between the line of force and the trajectory is 180(), which ensures the cos in the F·ds =
F·ds·cos = F·ds·cos vector dot product is always equal to minus one. F·ds·cos is, therefore, equal to
F·ds·cos, so we can write:






We get the primitive or antiderivative function U(r) = kqe2r1 from the force formula:
43
The measurement unit for energy is newton·meter: force times distance. Because this is a very large unit in
quantum physics, it is more common to use the electronvolt (eV). One eV is the energy gained or lost by an
electron when it moves an electric potential difference of one volt in vacuum. The conversion factor is 1 eV =
1.602176634×1019 J. This numerical value is equal to the numerical value of the electron charge expressed in
Coulomb (1.602176634×1019 C) because the volt is an SI unit (1 V = 1 J/C): it is the potential difference between
two points that will impart one joule of energy per coulomb of charge that passes through it. The
1.602176634×1019 value is an exact value since the 2019 redefinition of the system of SI units.
44
We use an integral because the field is non-homogeneous or non-uniform: the field strength and, therefore, the
force, effectively varies from point to point.
17

The minus sign causes quite some headache here because the antiderivative does not have the minus
sign of the force formula: d(kqe2r1)/dr = kqe2r2 but reappears because the direction of the force and
the direction of motion are opposite (cos = 1).
45
Apart from the signs, we also need to keep track of
physical dimensions but the physical proportionality constant ensures the force and potential are
expressed in force and energy units (N and N·m = J) alright.
46
We did actually not define the field yet but, as mentioned, we are talking a (non-uniform) static electric
field here whose strength is measured as the force per unit charge: E = F/qe. We, therefore, have a force
measured in N, a field in N/C, and a potential (energy) in N·m = J. All express the same reality of the
field.
Let us wrap up this introduction on definitions and concepts by noting a few more things. First, it is
obvious that we can measure the potential difference between point 1 and 2 in the graph either by
directly moving the charge from 1 to 2 or, alternatively, by (i) moving a charge first from infinity to 1, (ii)
moving a charge from infinity to 2, and (iii) calculating the difference:
 
 
This energy difference will be negative because, this time around, we are moving the charge from lower
to higher potential. Hence, we get work out of the system this time around (W is negative).
Second, because we will usually think of a field vector E whose components Ex, Ey, Ez we will want to
measure along the three directions of our three-dimensional reference frame, we will want to introduce
differential vector operators using the gradient operator = (/x, /y , /z). This is just a
generalization from a one- or two-dimensional analysis to a three-dimensional Cartesian space and,
from the above, the reader should now understand the ubiquitous formula for the derivation of a field
from a potential:




This is quite wonderful: we have a scalar function U(x) or U(r) from which we can derive the electric field
and, therefore, the force F = q·E on a charge. We will refer to U as the scalar potential in the future, and
we will usually denote it by φ.
The scalar and vector potential
What about the magnetic force or field? Can we derive it from the potential too? The total
electromagnetic force is the Lorentz force and consists of two components: F = qE + q(vB). The vB
45
Keeping track of signs and dimensions is not always easy! You will often read the minus sign is just a matter of
convention and it is but several conventions must be followed here, and there is only one way to do it right,
and that is by consistently following all conventions! Note that the electron charge is negative but this minus sign
cancels with the minus sign of the second charge that we are bringing in.
46
The electric constant k is equal to 1/40 and its physical dimension is [1/40] = N2m2/C2. Hence, [ kqe2r2] =
(N·m2/C2)·(C2m2) = N (the force unit) and [kqe2/r] = N·m = J (the energy unit).
18
vector cross product shows the magnetic force is more complicated. We will soon show we can
associate a very different potential a vector field instead of a scalar field with the magnetic field,
which we will refer to as the vector potential A. The magnetic field will then be given by the vector cross
product of the gradient operator and the vector potential A:
B = A
This is all very consistent, and we will soon be using more vector operators, so let us introduce them
here:
The divergence, which is the dot product of the gradient (·) and a field vector and, therefore,
yields a scalar quantity. We have already applied it to the electric field:



 
 

The curl, which involves a vector product with the gradient (). We gave the example of the
magnetic field, which is the curl of the vector potential: B = A. Note that we actually have
three equations here: one for each of the components (Bx, By, and Bz) of B.
The Laplacian, which is the dot product of the gradient and the gradient and, therefore,
involves second-order derivatives:










We are well on our way here, and so we will introduce a final remark. We can only define the scalar
potential uniquely by choosing a reference point where φ = 0. We chose that point to be infinity in the
example of the radial (electric) field: φ() = U() = 0 and, therefore, E() = φ() = U() = (0, 0, 0) =
0. This is what is referred to as choosing a gauge for the fields. If we change the gauge, we get a
different scalar potential φ’. When one combine various theorems of vector calculus
47
, one can show
the physics do not change when we change A and φ together by the following rules:
1. A’ = A + 
2. φ' = φ – /t /t = φ – φ’
While this seems to introduce yet another field () for which we have derivatives both with regard to
space () as well as with regard to time (/t) it is actually a condition which removes all ambiguity
in the description. The Lorentz gauge
48
connects both the scalar and vector potentials:


For a time-independent scalar potential we get the usual ·A = 0 choice because the time derivative is
zero: φ/t = 0 ·A = 0.
47
We are referring to the Gauss and Stokes theorems here.
48
The Lorentz gauge does not refer to the Dutch physicist H.A. Lorentz but to the Danish physicist Ludvig Valentin
Lorenz.
19
The least energy and least action principle
If we let the charge that we brought in move freely, it will follow a trajectory which respects both the
least energy as well as the least action principle. The minimum energy principle is, effectively, not
sufficient to determine the trajectory which our charge should follow. The minimum or least action
principle is effectively needed to determine the path or trajectory of our charge: it tells us the charge
will lower its total energy (kinetic and potential) by moving along a path which minimizes the (physical)
action 
.
49
The Planck-Einstein relation tells us (physical) action comes in units of h.
50
To move from one point to
another, some energy is needed over some distance or what amounts to the same some momentum
during some time. We think this energy and/or momentum is extracted from the field and effectively
comes in units of h:
h = 6.62607015×1034 Nms = ΔE·Δt = Δp·Δx
Does this imply motion itself comes in discrete bits, and that the energy and momentum in the fields
must come in discrete amounts too? We think it does, but that should not necessarily lead one to
conclude that motion itself is not continuous, or that continuous fields do not exist. We think quantum
field theory is a bit of an aberration.
Magnetic and electric dipoles
We have forces and fields because we have charges out there.
51
A mass will attract some other mass; a
charge will attract or repel another charge. Think of mass as a charge that comes in one color only. In
contrast, electric charges comes in two colors: positive and negative. The force formula looks pretty
much the same, however. We just have a different proportionality factor to sort out the physical
dimensions. We should also note that we have electric charges only, even if we will soon throw the
magnetic field into the equations. A magnetic charge which is also referred to as magnetic monopole
would also come in two colors south and north poles but no one has ever managed to create a
magnetic monopole.
We should note, however, that electrically neutral particles such as neutrons or atoms have a magnetic
moment , which is attributed to an internal current. To be precise, the magnetic moment is equal to
the product of the current (I) and the area of the loop (A).
52
A magnetic field will, therefore, also act on
49
For a full development of the least action principle both from a classical as well as a quantum-mechanical
perspective we refer to Feynman’s Lectures, Volume II, Chapter 19 (The Least Action Principle). We think its
central place right in the middle of the middle Volume is no coincidence.
50
One might prefer the German Wirkung to refer to physical action.
51
We need at least two to build the theory because a force is always a force between two charges. Fortunately,
forces and fields can be added. Indeed, of all fundamental principles in physics, I find the principle of superposition
principle is surely one of the most convenient, as it offers the hope that thinking of many charges interacting
altogether should not be too difficult.
52
To explain the magnetic moment of an electron, we think of it as a single pointlike charge circulating in a loop at
the speed of light itself. Furthermore, we think the frequency of rotation is given by the Planck-Einstein relation (f =
E/h), while the circumference of the loop is given by the Compton wavelength λC = h/mc. The radius r is, therefore,
equal to ħ/mc. We can now calculate the electron magnetic moment as:








20
neutrons or atoms. It will align the magnetic moment in the up or down direction of its spin, which is
ħ/2 for the electron but which, for other particles, may also take values ħ/2 n·ħ (e.g. +3/2, +1/2,
1/2, 3/2). It may or may not come as a surprise to the reader that the Stern-Gerlach experiment must
actually be carried out using neutral particles because the force on a charge would be far stronger than
the force on the magnetic moment.
53
However, we will not digress on this, except for noting positive
and electric charges may also combine to form an electrically neutral electric dipole. The study of dipole
fields is (almost) a study field on its own, but in this paper we will only study the electric and magnetic
forces acting on a (net) charge, which are given by the first and second term of the Lorentz force: F = q·E
+ q·(vB).
The vector dot product vB tells us that the magnetic force will be perpendicular both to the velocity
vector v as well as to the magnetic field vector B, whose properties (magnitude and direction) we will
examine later as part of a more general discussion on the fields. Back to the forces.
3. The measurement of the position of a charge
A charge to make matters more specific, we will be talking an electron (e), but it might also be a
proton or any other charged particle moves along a trajectory which we describe in a three-
dimensional Cartesian space (an empty mathematical space) by measuring its position in terms of its
distance from a chosen point of origin to the moving charge at successive points in time t. The observer
will use an atomic clock or stopwatch to measure and mark the exact time at each position. We assume
this measurement does not affect the observer, the charge that is being observed, or the space
inbetween them. In other words, we assume there is no exchange or conversion of energy while
measuring time.
We have our clock and, therefore, a time unit. Now, we should think about how to measure distance. If
you are a DIYer, you will think of a household distance measurement laser measuring the distance to
your wall by (1) sending regular bursts of photons (the laser beam) to the wall and (2) receiving some of
them back in the receiver. The clock of the device then calculates the distance back and forth by
multiplying the time between the sent and receive moment to give you a distance expressed in meter
instead of light-seconds: c: λ = c·t. Consider it done, right? Yes, but such reasoning assumes the
photon(s) will travel back and forth in a straight line to the charge and then back to the detector(s).
54
The reasoning also involves the assumption of an elastic or instantaneous collision between the charge
and the photon(s)
55
, and the charge will, therefore, acquire some extra kinetic energy or momentum
which, measured from the zero point (KEe = 0 and pe = 0) is just the kinetic energy and momentum of the
electron after the collision (KEe’ > 0 and pe’ 0). So how does that work, then?
53
To measure the magnetic moment of an electron, an ion trap (Penning trap) may be used.
54
A straight line of sight assumes the absence of gravitational lensing. Arthur Eddington, Frank Watson Dyson, and
their collaborators effectively observed that electromagnetic radiation (light) follows a curved path near a massive
object during the total solar eclipse on May 29, 1919 (Dyson, F. W., Eddington, A. S., Davidson C. (1920), A
determination of the deflection of light by the Sun's gravitational field, from observations made at the total eclipse
of 29 May 1919, Philosophical Transactions of the Royal Society 220A (571581): 291333.
55
If there is photon-electron interaction, such interaction will take some time which should, therefore, be
subtracted from the total travel time to calculate the exact travel time (excluding interaction time).
21
The concept of an elastic collision
We can calculate the extra momentum from the momentum conservation principle while noting we
should write the momenta as vectors so as to take the direction of the momentum into account. We
can, unfortunately, not predict the angle between the incoming and outgoing velocity vector for the
photon (v and v, respectively). What we do know, however, is that all of the momentum of the
incoming photon (p) is being transferred to the outgoing photon (p) and the electron (pe’). Because
(linear) momentum is conserved simultaneously in the x-, y- as well as the z-direction, we should write
this as a vector equation:
pe’ = p p
The situation is depicted below (Figure 3). As for the precision of the measurement, we agree with the
interpretation of the Compton wavelength as the relevant “distance scale within which we can localize
the electron in a particle-like sense.”
56
All of this, of course, assumes the electron is travelling freely: it is
not bound to an atomic or molecular orbital, in other words.
Figure 3: The scattering of a photon from an electron (Compton scattering)
Now, the linear momentum of a photon is equal to p = mc = E/c = h·f/c = h/c·T = h/λ. Hence, if the
electron momentum changes, then the wavelength, cycle time, and frequency of the incoming and
outgoing photon cannot be the same. Hence, if our household laser distance sensor would also be able
to measure this wavelength or frequency change, we would be able to tell how the collision has changed
the state of motion of the charge not approximately, but exactly. We would, in effect, be able to
calculate the velocity change (direction as well as magnitude) of the electron by using the ve’ = pe’/me’
formula.
57
We have come to the conclusion that a fully elastic interaction between an electron and a photon (1)
transfers momentum and kinetic energy and (2) causes a wavelength shift between the outgoing and
incoming photon. Let us quickly model this before we turn back to our distance measurement problem.
Compton scattering
Electron-photon interaction is modelled by the equation for Compton scattering of photons by
electrons:


56
See our analysis of Compton scattering in our paper on the difference between a theory, a calculation, and an
explanation, April 2020.
57
We have no need for the Uncertainty Principle here!
22
The λC = h/mec in this equation is referred to as the Compton wavelength of the electron and is, a
distance: about 2.426 picometer (1012 m). The 1 cosθ factor goes from 0 to 2 as θ goes from 0 to π.
Hence, the maximum difference between the two wavelengths is about 4.85 pm. This corresponds,
unsurprisingly, to half the (relativistic) energy of an electron.
58
Hence, a highly energetic photon could
lose up to 255 keV while the electron could, potentially, gain as much.
59
That sounds enormous, but
Compton scattering is usually done with highly energetic X- or gamma-rays.
When the photon does not interact with the electron, there is no scattering angle (we may enter it as 0
in the equation) and the wavelength shift between the incoming and outgoing photon which is
actually just traveling through will, therefore, vanish (1 cos0 = 1 1 = 0). In contrast, when the
photon bounces straight back, the scattering angle will be equal to π (see Figure 3) and the
wavelength shift between the incoming and outgoing photon will, therefore, attain its maximum value,
which is equal to λC(1 cosπ) = λC(1 + 1) = 2λC.
60
This is, as mentioned, a rather formidable result. Two
more things should be noted here:
1. The wavelength shift ∆λ = 2λC is independent of the energy of the incoming photon.
2. The outgoing photon will have longer wavelength and, therefore, lower energy, and the kinetic
energy of the electron must change so as to explain the energy difference between the
incoming and outgoing photon:
Ee’ + E = Ee + E E = Ee’ Ee = E E
Bref (French for ‘in short’), if the electron is moving in free space, then the probing of an electron by
photons i.e. the measurement of a position will result in a change of its trajectory.
Of course, one may argue the electron may not be in a free but in a bound state: such bound state may
be an atomic or molecular orbital in a crystal lattice.
61
In such case, the photon will be temporarily
absorbed as the electron first absorbs and then emits a photon with exactly the same wavelength.
However, because such absorption and emission of a photon with linear momentum should also respect
the conservation of (linear) momentum, the electron should first absorb the incoming momentum of
the incoming photon and then return it to the outgoing photon. In such case, we will observe what is
referred to as reflection. Such reflection may be specular or diffuse. In the case of specular reflection,
the outgoing photon which we may now refer to as a reflected photon will emerge from the
58
The energy is inversely proportional to the wavelength: E = h·f = hc/λ.
59
The electron’s rest energy is about 511 keV.
60
See the exposé of Prof. Dr. Patrick R. LeClair on Compton scattering. Prof. LeClair’s treatment is precise and
offers plenty of other interesting formulas, including the formula for the scattering angle of the electron which,
as mentioned above, is fully determined from the wavelength shift and the scattering angle . We offer a concise
discussion of his derivation and arguments in our paper on the difference between a theory, an explanation, and a
calculation.
61
The Rutherford-Bohr model of an atom gives us the following formula for the energy level:

We, therefore, get the following formula for the energy difference between two states with principal quantum
number n2 and n1 respectively:

23
reflecting surface at the same angle to the surface normal as the incident ray, but on the opposing side
of the surface normal in the plane formed by the incident and reflected rays, as illustrated below.
Figure 4: Specular reflection of light photons
Most materials will reflect the light diffusely: just like specular reflection, diffuse reflection depends both
on the properties of the surface as well as of the properties of the crystal lattice.
62
Before returning to
the problem of distance measurement, we should add one final remark on specular reflection: photon
interference experiments reveal a phase shift between the incoming and outgoing photons upon
reflection which is equal to π, and we must assume such phase shift is relevant in the context of
Compton scattering too.
63
A phase shift effectively suggest the electron will effectively take some time
to absorb and re-emit the photon. The time which corresponds to a phase shift equal to 180° (π) can be
calculated from the representation of a photon as the vector sum of a sine and a cosine oscillation:
 



One may think of these two oscillations as representing the electric and magnetic field respectively
when measured using natural time and distance units (c = 1), in which case the (maximum) amplitude of
the magnetic field B = E/c will be measured as being identical. The multiplication by i then represents
the orthogonality of the E and B vectors and the phase difference between the cosine and sine accounts
for the phase shift between them, which we know to be equal to 90° (π/2): sin( π/2) = cos(π/2). Any
case, if the phase shift is equal to  = (·t) = ·t = (E/ħt = π, then we can calculate t as being
equal to t = πħ/E= (1/2)/f = T/2. Unsurprisingly, this is half the cycle time of the photon.
Photon and electron spin
So far, we have been discussing spin-zero photons and electrons. Both photons and electrons have spin.
To be precise, besides linear momentum, a photon will also have an angular momentum, which is either
+1 or 1 and which is expressed in units of ħ: hence, this amount of (angular) momentum should be
conserved as well throughout the process.
64
In contrast, an electron has spin ħ/2 only. The absorption
62
We refer the interested reader to the rather instructive Wikipedia article on diffuse reflection (from which we
also borrowed the illustration) for more details.
63
See, for example, K.P. Zetie, S.F. Adams, and R.M. Tocknell, How does a Mach-Zehnder interferometer work?,
Phys. Educ. 35(1), January 2000.
64
We find the fact that photons are spin-one particles without a zero-spin state rather striking, especially because
it is usually not mentioned very explicitly in (most) physics textbooks. Richard Feynman, for example, hides this
fact in a footnote (see: Feynman’s Lectures, III-11, footnote 1), and the context of this footnote is rather particular.
To explain the interference of a photon with itself in the one-photon Mach-Zehnder interference experiment, we
actually do assume a beam splitter will actually split in two linearly polarized photons which each have spin-1/2
24
of the full angular momentum of a photon must, therefore, involve a spin flip of the electron, going from
+ ħ/2 to ħ/2 and then back again so as to return the full amount of spin to the outgoing photon. In
fact, one may speculate the temporary spin flip of the electron explains why the electron in a
configuration where all sub-shells (which are identified using not only the principal quantum number n
but also the orbital angular momentum number l) have been filled by a pair of electrons with opposite
spin has to go from one orbital to the next: the line-up of its spin violates the Pauli exclusion principle,
according to which two electrons in the same subshell must have opposite spin.
65
By now, it should be sufficiently clear that the probing of a free electron using radiation (photons) is
perturbative. We may, therefore, try to find another way to observe the charge’s state of motion.
Is non-perturbative measurement of the position of a charge possible at all?
One potentially non-destructive observation might be the use of potential meters: as we will see in a
moment, a moving charge will change the electric and magnetic field all over space and, hence, by
measuring how the electric and magnetic field changes at one or various positions.
However, the measurement of a change in the electric and/or magnetic field will inevitably involve a
charge as well because we must observe be able to observe the force on a charge in order to measure
the change in potential. A change in electric potential, for example, will result in a simple Coulomb
force:







We note that the magnitude of the force falls off following the inverse square law (F 1/r2) while the
(electric) potential (V) diminishes linearly (V 1/r). This makes sense because the energy flux is inversely
proportional to the square of the distance as well. What about the magnetic force? The calculation of
the magnetic field and force is more complicated because it depends on the (relative) motion of both
charges and because it will not only act on the (moving) charge but also on its magnetic moment. We
will let the reader review the relevant equations here, but it should be clear that any force acting on a
charge will change its state of motion.
To be precise, a force acting on a charge will cause it to accelerate. The acceleration vector is given by
Newton’s force law (a = F/me) and the total energy expended will be equal to the (line) integral E =
LF(r)·dr.
66
At this point, we should invoke the principle of least action as used in both classical as well as
in quantum mechanics.
67
This principle states that, in free space, the charge will lower its total energy
(kinetic and potential) by moving along a path which minimizes the (physical) action
only. Each of these two linearly polarized photons then has the same frequency but carries only half of the total
energy of the incoming photon:
. We work out this hypothesis in our realistic or classical explanation of
one-photon Mach-Zehnder interference.
65
The electron subshells define the fine structure of the atomic or molecular spectrum. The fine spectral line can
be further split into two hyperfine levels because of the coupling between electron spin and nuclear spin. See our
paper on a classical explanation of the Lamb shift.
66
The reader should note that we have a line integral here and, therefore, we are integrating a vector dot product
(F·dr) over a curve or a (non-linear) line.
67
See: Feynman’s Lectures, The Least Action Principle (Vol. II, Chapter 19).
25

. To be precise, potential energy must be converted to kinetic energy and vice versa: the
total energy of the charge (the sum of KE and its PE) does, therefore, not change: only its components KE
and PE, which depend on its velocity and its position respectively, will change. However, in free space,
they will add up to a constant. In other words, the situation which we have been considering up to this
point, is that of a charge whose energy state does not change.
This now changes because we will want to use a change in the energy state of the electron to measure a
change in the electric and magnetic fields or, if one prefers a more elegant representation perhaps, the
scalar as well as vector potential.
68
Quantum-mechanically, this implies the electron we use to measure the change must following another
trajectory. This trajectory will differ from its geodesic its trajectory in free space, that is by an
amount of physical action equaling one or more units of physical action h. Now, this amount of physical
action the product of (1) a force, (2) a distance and (3) some time must be extracted from the fields,
somehow. We refer the reader to Feynman’s treatment of the topic for a complete analysis of how a
field loses energy to matter or, in the reverse case, how it gains energy, which is just a negative loss.
69
As
Feynman puts it, we now need to change the energy conservation law and restate it as follows:
“Only the total energy in the world – which includes the energy of both matter and fields is
conserved. The field energy will change if there is some work done by matter on the field or,
conversely, by the field on matter.” (Feynman, Vol. II, page 27-8).
The reader may think this conversion from potential into kinetic energy field energy goes down while
the energy of the electron which we use to measure the change in potential goes up should not affect
the state of motion of the electron which we are trying to observe. However, this would imply that,
sooner or later, all potential energy in the world gets converted to kinetic energy or vice versa: in other
words, all of the charges would deplete all potentials and we would be left with kinetic energy and a
matter-world only: no fields.
We, therefore, add an additional conservation lawand it will be the final one: in addition to the
conservation of total energy as well as (linear and circular) momentum in the world, the total amount of
physical action in the world must be conserved. If we, therefore, extract a unit of h from the fields, the
fields will, somewhere, extract a unit of h from matter. If we only have two charges in the world the
one we want to observe and the other one which we want to use to measure any change in potential
which the first charge is causing then the fields will have to extract one unit of h from the charge for
every unit of h they are transferring to the charge we use to measure the changing potential(s).
68
We have reasoned in terms of electric and magnetic fields so far, but we may rewrite Maxwell’s equations in
terms of the scalar and vector potential. This may or may not simplify the math: the electric and magnetic field
vectors E(x, t) and B(x, t) effectively have three spatial coordinates each (Ex, Ey, Ez, Bx, By and Bz respectively), while
a description in terms of the scalar and vector potential (x, t) and A(x, t) involves four numbers only (φ, Ax, Ay, and
Az).
69
See: Feynman’s Lectures, Field Energy and Momentum (Vol. II, Chapter 27)
26
Can we prove this? No, but we think Feynman’s derivation of the equation of continuity for probabilities
comes (very) close to proving this and we, therefore, like to interpret this equation as the conservation
law for physical action.
70
We, therefore, think the extraction of an equivalent energy E = h·f from the fields must not only involve
the absorption of a photon this photon will, of course, have the same energy E = h·f by the charge we
use to measure the changing fields: it must also involve the emission of a photon by the charge we are
trying to observe. Hence, this will, once again, result in a change of the state of motion of the charge we
are trying to observe. We may, therefore, say that the two charges will be interconnected or to use
more formidable language that the two charges will be coupled or entangled both classically as well as
quantum-mechanically.
Should Einstein worry about ‘spooky action at a distance’ (spukhafte Fernwirkung) here? Not
necessarily, because this spooky action should still respect the principle that nothing can travel faster
than the speed of light: the total effect will push both electrons away from each other and may,
therefore, be said to involve the exchange of a photon. In order to distinguish such photon from the
photons we first wanted to use to observe the charge directly, we will refer to such photons as virtual
photons. We will, however, assume such virtual photons cannot travel any faster than any other
electromagnetic wave. In fact, we will assume such virtual photons are just like real photons nothing
but a pointlike electromagnetic oscillation which propagates at the speed of lightnot approximately,
but exactly.
4. Charges in motion
From the extremely discussion above, the reader should just note the crux of the argument: any
measurement of a position will inevitably involve a delay except if the charge and the observer happen
to be at the same position, of course). One must, therefore, distinguish between the actual position of a
charge and its retarded position. This is illustrated in Figure 5. Assuming the observer is positioned at
point (1), he can ascertain the position of the charge x by measuring its distance (denoted as distance r’
which, just like the position is a retarded rather than an actual (right now) distance. This concept is
illustrated below (Figure 5).
Figure 5: The concepts of retarded time, position, and distance (Feynman, II-21, Fig. 21-1)
70
See: Feynman’s Lectures, The equation of continuity for probabilities (Vol. III, Chapter 21, section 2). This
equation is closely related to the distinction between kinematic momentum and dynamical momentum
(Feynman’s Lectures, II-21-3).
27
The retarded position is written as x and should be written as a function of the retarded time t r’/c:
x = x(t r’/c)
The actual position x(t) can only be measured in the future but can be extrapolated by making
continuous measurements of the position right now. Such measurements then allow to associate a
velocity vector v = dx/dt with the charge. Just like position, the velocity function will (or should) have the
retarded time as its argument: v = v(t r’/c). A first-order approximation of the actual position x(t) is
then given by the expression:



This first-order approximation may, of course, be complemented by adding second- and higher-order
terms by measuring acceleration and using higher-order time derivatives of the position variable.
However, let us first stick to the use of the velocity vector. The continuous measurement of the position
assumes the measurement of the infinitesimal distance:


We may, therefore, write the velocity vector as:





Figure 5 shows a unit vector er’ from the retarded position (2’) directed towards the observer (1). One
might also draw a unit vector from (1) to (2’), which makes it easier to appreciate that the vector r’ can
be written as r’ = r’· er’ and, more importantly, that x r’· er’. The retarded velocity vector can,
therefore, also be approximated by:




Moving to differential notation, one can, therefore, write the retarded velocity vector function as:




Position, time and, therefore, motion is relative. However, charge is not relative and different observers
should, therefore, also agree on a measurement unit for charge, which we may equate to the
elementary charge e. This is the charge of a proton or the (negative) charge of the electron. A charge fills
empty spacetime (all of it) with a potential which depends on position and evolves in time. This
potential is, therefore, also a function of x, y, z, and t.
71
Two equivalent descriptions are possible:
A description in terms of the electric and magnetic field vectors E(x, t) and B(x, t); and
A description in terms of the scalar and vector potential (x, t) and A(x, t) respectively.
71
We will no longer be worried about the relativity of the reference frame and assume the reader will understand
what is relative and absolute in our description.
28
The field vectors E and B have three components
72
and we, therefore, have six dependent variables Ex(x,
t), Ey(x, t), Ez(x, t), Bx(x, t), By(x, t), and Bz(x, t). In contrast, the combined scalar and vector potential give
us four dependent variables (x, t), Ax(x, t), Ay(x, t), and Az(x, t) only, which may appear to be simpler.
73
However, in this paper, we will stick to a description of the fields in terms of the E and B fields.
5. Charges, energy states, potentials, fields, and radiation
We are now ready to analyze Feynman’s rather particular formulas for the E and B field vectors at point
(1):







We refer to Feynman’s Lectures
74
for a clear and complete derivation of these functions out from
Maxwell’s equations for a single charge q moving along any arbitrary trajectory, as illustrated in Figure 5.
The point to note is that the electric and magnetic field at point (1) now will be written as a function of
the position and motion of the charge at the retarded time t r’/c. The relevant distance is, therefore,
also the retarded distance r’, which is the distance between (1) and (2’) – which is not the charge’s
position at time. The latter position is point (2): it is separated from position (2’) by a time interval equal
to t (t t’/c) and a distance interval which depends on the velocity v of the charge q which will be
generally much less than c.
The second and third term in the expression for E(1, t) are, obviously, equal to zero if the charge is not
moving, in which case the charge comes with a static (i.e. non-varying in time) Coulomb field only: in this
case, the retarded field is just the Coulomb field tout court. The scalar product which defines the
magnetic field is equal to the product:
(1) the magnitude of the unit vector er’, whose origin is (2’) and which points to (1) and whose
magnitude is equal to 1;
(2) the magnitude of the electric field vector at point (1) at time t;
(3) the cosine of the angle between er’ and E(1, t).
The latter factor the cosine of the angle between er’ and E(1, t) is, obviously, zero if the second and
third term are zero, which is just a confirmation of the fact that static electric fields do not come with a
magnetic field. However, our charge is moving, and the first- and second-order derivative of the er’ will,
72
Boldface symbols denote vector quantities, which have both a magnitude and a direction. Scalar quantities only
have a magnitude. However, depending on the reference point for zero potential energy, the potential energy of a
charge in a potential field may be negative. Potential energy is just like a distance measured as a difference. The
plus or minus sign of the potential energy, therefore, depends on the direction in which we would be moving the
charge.
73
Feynman makes extensive use of the scalar and vector potential for formulas, and they also appear in most
quantum-mechanical equations. One, therefore, needs to become intimately familiar with them: the scalar and
vector potential are, in many ways, more real than the electric and magnetic field vectors.
74
Richard Feynman, II-26, Solutions of Maxwell’s equations with currents and charges.
29
therefore, not be equal to zero, which modifies the electric field E at point (1) at time t, and which also
gives us a non-zero magnetic field B:
(i) The second term corresponds to what Feynman refers to as a compensation for the retardation
delay, as it is the product of (a) the rate of change of the retarded Coulomb field multiplied by
(b) the retardation delay (the time needed to travel the distance r’ at the speed of light c). In
other words, the first two terms correspond to computing the retarded Coulomb field and then
extrapolating it (linearly) toward the future by the amount r’/c which is right up to time t.
75
It
should be noted that one might think this second term is (also) inversely proportional to the
squared distance r’2, but the r’ and r’2 in the numerator and denominator respectively leaves us
with an 1/r’ factor only.
(ii) The third term the second-order derivative d2(er’)/dt2 is an acceleration vector which
because of the origin of the unit vector er’ is fixed at point (2’) – can and should be analyzed as
the sum of a transverse component and a radial component.
76
Needless to say, this second-
order derivative of d2(er’)/dt2 will be zero if the charge moves in a straight line with constant
velocity v. In other words, the third term will vanish (be zero) if there is no acceleration.
In the chapters where Feynman first introduces and uses these equations (Vol. I, Chapters 28 and 29 as
well as Vol. II, Chapter 21), the assumption is that the transverse piece of the acceleration vector is far
more important than the radial piece, but such statement crucially depends on the assumption that the
charge is moving at a more or less right angle to the line of sight, which is not necessarily the case.
Feynman corrects for this assumption in Chapter 34 of Vol. I, in which he gives the reader a full
treatment of all ‘relativistic effects’ of the motion of a charge.
Feynman also associates the third term with radiation which, as we now know, consists of a stream of
photons carrying energy. We must, effectively, assume the charge does not only generate a potential
but moves in a potential field itself. Its energy, therefore, must also continually change. To be specific, in
free space, we must assume the charge will lower its total energy (kinetic and potential) by moving along
a path which minimizes the (physical) action 
. This is just an application of the
(classical) least action principle.
77
Of course, in classical physics, potential energy must be converted to kinetic energy and vice versa: the
total energy of the charge (the sum of KE and PE) does, therefore, not change: only its components KE
and PE, which depend on its velocity and its position respectively, but they add up to a constant. In
other words, the situation which we have been considering up to this point, is that of a charge whose
energy state does not change.
Such energy state may be the energy state of a free electron or of an electron in a bound state, i.e. an
electron in an atomic or molecular orbital. If and when an electron moves from one energy state to
75
We apologize for quoting quite literally from Feynman’s exposé here, but we could not find better language.
76
In the chapters where Feynman uses these equations (Vol. I, Chapters 28 and 29 as well as Vol. II, Chapter 21),
he assumes the transverse piece is far more important than the radial piece, but such statement crucially depends
on the assumption that the charge is moving at a more or less right angle to the line of sight, which is not
necessarily the case. Feynman corrects for this assumption in Chapter 34 of Vol. I, in which he gives the reader a
fuller treatment of the ‘relativistic effects in radiation’.
77
See: Feynman’s Lectures, The Least Action Principle (Vol. II, Chapter 19).
30
another, as it does when hopping from one atomic or molecular orbital to another. Indeed, as the
electron moves as a proper current in a conductor
78
whose direction is from high to low potential it
should emit photons which will be packing a discrete amount of energy which is given by the Planck-
Einstein relation:
E = h·f = h/T
The frequency f of the photon is, obviously, the inverse of the cycle time T, and the Planck-Einstein
relation may, therefore, also be written as h = T. Because the drop in potential from one atomic or
molecular orbital in a crystal structure i.e. along the conductor is extremely small, power lines
whether they be high-voltage DC or low-voltage AC lines emit only extremely low frequency (ELF)
radiation. Such low-frequency radiation is associated with heat radiation at very low temperature: a
photon frequency of 300 Hz, for example, is associated with a wavelength that is equal to λ = c/f
(3108 m/s)/(300 s1) = 1106 m = 1000 km.
79
Hence, yes, we finally dropped the word: radiation. Electrons who stay in the same energy state in a
bound atomic or molecular state, for example do not emit radiation and, hence, do not lose energy.
Likewise, the orbital motion (spin) of the charge inside a stationary charge does not cause any radiation
and, therefore, the energy does not leak out.
This, then, combines Maxwell’s equations with the Planck-Einstein relation which tells us energy comes
in quantized packets whose integrity is given by Planck’s quantum of action (h). We now have the trio of
physical constants in electromagnetic theory (classical as well as quantum physics): c, e, and h.
6. Photons and fields
1. In 1995, W.E. Lamb Jr. wrote the following on the nature of the photon:
“There is no such thing as a photon. Only a comedy of errors and historical accidents led to its popularity
among physicists and optical scientists. I admit that the word is short and convenient. Its use is also habit
forming. Similarly, one might find it convenient to speak of the “aether” or “vacuum” to stand for empty
space, even if no such thing existed. There are very good substitute words for “photon”, (e.g., “radiation”
or “light”), and for “photonics” (e.g., “optics” or “quantum optics”). Similar objections are possible to use
of the word “phonon”, which dates from 1932. Objects like electrons, neutrinos of finite rest mass, or
helium atoms can, under suitable conditions, be considered to be particles, since their theories then have
viable non-relativistic and non-quantum limits.”
80
The opinion of a Nobel Prize laureate carries some weight, of course, but we think the concept of a
photon makes sense. As the electron moves from one (potential) energy state to another from one
78
For a distinction between the concepts of current, electrical signal, and (probability) amplitudes, see our paper
on electron propagation in a (crystal) lattice (November 2020).
79
ELF radiation is usually defined as radiation with a (photon) frequency below 300 Hz. Typical field strength near a
high-voltage power is typically 2-5 kV/m (1 V/m = 1 J/C·m = 1 N/C) for the electric field strength and up to 40 T (1
T = 1 (N/C)·(s/m), with the latter factor reflecting the 1/c scaling factor and the orthogonality of the E and B
vectors) for the magnetic field but as the equations show diminish rapidly with distance. The typical range for
low-voltage lines is 100-400 V/m and 0.5-3 µT, respectively. See, for example:
https://ec.europa.eu/health/scientific_committees/opinions_layman/en/electromagnetic-fields07/l-2/7-power-lines-
elf.htm
80
W.E. Lamb Jr., Anti-photon, in: Applied Physics B volume 60, pages 7784 (1995).
31
atomic or molecular orbital to another it builds an oscillating electromagnetic field which has an
integrity of its own and, therefore, is not only wave-like but also particle-like.
The photon carries no charge but carries energy. We should probably assume its kinetic energy is the
same at start and stop of the transition. In other words, at point t1 and t2, (KE)1 and (KE)2 are assumed to
identical in the (physical) action equation which we introduced above:

 

This, of course, does not mean that the 
integral vanishes: it only does so when assuming the
velocity in the KE = mev2/2 formula
81
is zero everywhere, which cannot be the case because when
everything is said and done the electron does move from one cell in the crystal lattice to another.
However, we will leave it to the reader to draw possible KE, PE and total energy graphs over the electron
transition from one crystal cell to another. Such graphs should probably be informed by a profound
analysis of the nature of the photon.
We mentioned a photon carries energy, but no charge. While carrying electromagnetic energy, a photon
will only exert a force when it meets a charge, in which case its energy will be absorbed as kinetic energy
by the charge. In-between the emission and absorption of the photon, we should effectively think of the
photon as an oscillating electromagnetic field and, hence, such field can usefully be represented by the
electric and a magnetic field vectors E and B. The magnitudes should not confuse us: field vectors do not
take up any space, although we may want to think of them as a force without a charge to act on. Indeed,
a non-zero field at some point in space and time which we describe using the (x, y, x, t) coordinates
tell us what the force would be if we would happen to have a unit charge at the same point in space and
in time. This is reflected in the electromagnetic force formula: the Lorentz force equals F = q·(E + vB).
Hence, the electromagnetic force is the sum of two (orthogonal) component vectors: q·E and q·vB.
The velocity vector v in the equation shows both of these two component force vectors depend on our
frame of reference. Hence, we should think of the separation of the electromagnetic force into an
‘electric’ (or electrostatic) and a ‘magnetic’ force component as being somewhat artificial: the
electromagnetic force is (very) real because it determines the motion of the charge but our cutting-
up of it in two separate components depends on our frame of reference and is, therefore, (very)
relative. We should refer to our remarks on the relative strength of the electric and magnetic field,
however: the reader should not think in terms of the electric or magnetic force being more or less
important in the analysis and always analyze both as aspects of one and the same reality.
Let us get back to our photon: we think the photon is pointlike because the E and B vectors that describe
it will be zero at each and every point in time and in space except if our photon happens to be at the (x,
y, z) location at time t.
[…] Please read the above again: our photon is pointlike because the electric and magnetic field
81
We use the non-relativistic kinetic energy formula here because the drift velocity of the electron is very low.
Also, the rather low energy levels involves ensure a particle with rest mass of about 0.51 MeV/c2 should not reach
relativistic velocity levels. The non-relativistic formula simply defines the kinetic energy as the difference between
the total energy and the potential energy.
32
vectors that describe it are zero everywhere except where our photon happens to be.
2. At the same time, we know a photon is defined by its wavelength. So how does that work? What is
the physical meaning of the wavelength? It is, quite simply, the distance over which the electric and
magnetic field vectors will go through a full cycle of their oscillation. That is all there is to it: nothing
more, nothing less.
That distance is, of course, a linear distance: to be precise, it is the distance s between two points (x1,
y1, z1) and (x2, y2, z2) where the E and B vectors have the same value. The photon will need some time t
to travel between these two points, and these intervals in time and space are related through the
(constant) velocity of the wave, which is also the velocity of the pointlike photon. That velocity is, of
course, the speed of light, and the time interval is the cycle time T = 1/f. We, therefore, get the equation
that will be familiar to you:


We can now relate this to the Planck-Einstein relation. Any (regular) oscillation has a frequency and a
cycle time T = 1/f = 2π/ω. The Planck-Einstein relation relates f and T to the energy (E) through Planck’s
constant (h):
The Planck-Einstein relation does not only apply to matter-particles but also to a photon. In fact, it was
first applied to a photon.
82
Think of the photon as packing not only the energy E but also an amount of
physical action that is equal to h.
3. We have not talked much about the meaning of h so far, so let us do that now. Physical action is a
concept that is not used all that often in physics: physicists will talk about energy or momentum rather
than about physical action.
83
However, we find the concept as least as useful. Physical action can express
itself in two ways: as some energy over some time (E·T) or alternatively as some momentum over
some distance (p·). For example, we know the (pushing) momentum of a photon
84
will be equal to p =
E/c. We can, therefore, write the Planck-Einstein relation for the photon in two equivalent ways:

We could jot down many more relations, but we should not be too long here.
85
82
The application of the Planck-Einstein relation to matter-particles is implicit in the de Broglie relation.
Unfortunately, Louis de Broglie imagined the matter-wave as a linear instead of a circular or orbital oscillation. He
also made the mistake of thinking of a particle as a wave packet, rather than as a single wave! The latter mistake
then led Bohr and Heisenberg to promote uncertainty to a metaphysical principle. See our paper on the meaning
of the de Broglie wavelength and/or the interpretation of the Uncertainty Principle.
83
We think the German term for physical action Wirkung describes the concept much better than English.
84
For an easily accessible treatment and calculation of the formula, see: Feynman’s Lectures, Vol. I, Chapter 34,
section 9.
85
We may refer the reader to our manuscript, our paper on the meaning of the fine-structure constant, or various
others papers in which we explore the nature of light. We just like to point out one thing that is quite particular for
33
7. The near- and far-fields
The picture above is quite clear and consistent: a conductor or a crystal lattice emits electromagnetic
waves as photons, who should be thought of as self-perpetuating through the interplay of the electric
and magnetic field vector.
86
The direction of propagation equals the line of sight (more or less
87
) and a
crystal lattice (conductor) acts as a series of point sources or oscillators. By modulating the voltage (AC
or DC), frequency and and taking into account the spacing and properties of the crystal lattice one
gets photon beams in all directions, whose intensity and energy depends on the above-mentioned
factors and important can carry a signal through frequency or amplitude modulation (AM or FM). We
take, once again, an illustration from Feynman to show how this works (Figure 6). It should be noted
that the interference pattern that emerges does not result from random indeterminism but from an
interplay of regular and statistically determined photon emissions from each of the crystals in the
conducting lattice. As such, the addition or superposition of photons, electromagnetic waves and
probabilities amounts to the same with the usual caveat for the photon picture, of course, which as
particles do not engage in constructive or destructive interference. The complementarity of the
different viewpoints, perspectives or representations of the same reality is, therefore, clear.
Figure 6: The intensity pattern of a continuous line of oscillators (Feynman, I-30, Fig. 30-5)
However, by way of conclusion, we must probably say something about the oft-used distinction
between near- and far-fields. In order to do so, we ask the reader to, once again, carefully look at the
relevant equation(s) for the E and B field vectors:
the photon: the reader should note that the E = mc2 mass-energy equivalence relation and the p = mc = E/c can be
very easily related when discussing photons. There is an easy mathematical equivalence here. That is not the case
for matter-particles: the de Broglie wavelength can be interpreted geometrically but the analysis is somewhat
more complicatednot impossible (not at all, actually) but just a bit more convoluted because of its circular (as
opposed to linear) nature.
86
We skipped a discussion on photon spin: we think of photon spin as angular momentum, and it is always plus or
minus one unit of h. Photons do not have a zero-spin state.
87
Because the lattice consists of several layers, one may think an electron may not always move to the crystal cell
right next to it. This is true, it may deviate to left, right, up, or down while moving through the lattice. On the other
hand, the conducting electrons will repel each other and will, therefore, tend to travel on the surface of the crystal,
which is in agreement with standard theory.
34







The reader will note the magnitude of the (retarded) Coulomb effect (the first term) diminishes with
distance following the inverse square law ( 1/r’2) while the second term involves only inverse
proportionality ( 1/r2).
88
Finally, the third term does not fall off with distance at all! It is this what gives
rise to the very different shape of the near-field versus the far-field waves, with a transition zone in-
between.
In terms of the shape of the electromagnetic waves, one should probably think of the first effect
(retarded Coulomb effect) as a spherical wavefront, whose energy density effectively diminishes as per
the inverse square law, while the second effect is a plane wavefront
89
8. Complicated trajectories
It is important to note that the trajectory of a charge will usually not appear as a straight line in the
reference frame of the observer: we think of an electron in an atomic or molecular orbital as following a
non-linear trajectory. Such non-linear trajectory may be repetitive or cyclical. An example of such
cyclical trajectory is the motion of an electron in a Penning or ion trap, in which a quadrupole electric
field confines the electron while an axial magnetic field causes orbital motion. In addition, when thinking
of an electron as a ring current itself, two frequencies of cyclical motion corresponding to two modes
must be defined, as illustrated below.
90
In addition, the magnetic moment of his ring current will cause precession in the magnetic field. Such
precessional motion will cause the axis of rotation to rotate itself.
91
One should, therefore, appreciate
88
The coefficient r’/c and the 1/r’2 in the argument of the first-order derivative combine to give us a rather
straightforward 1/r’ factor.
89
The direction of the field vector(s) may be parallel or orthogonal to the direction of propagation, which gives rise
to the distinction between longitudinal and transverse waves. The reader may also remember lenses can change
plane waves into spherical waves and vice versa but such fact is not very relevant for the discussion here.
90
The illustration was taken from the Wikipedia article on the Penning or ion trap, but we do not expect the reader
to review this in depth.
91
Such precessional motion will be described by a precession frequency and an angle of precession. For more
detail, we may refer the reader to course F47 Cylotron frequency in a Penning trap, Heidelberg University, Blaum
Group, 28 September 2015.
35
that a description of motion x = x(t r’/c) will usually be quite complicated not only involving the
velocity, acceleration and jolt or jerk vectors v = dx/dt, a = dx2/dt2, j = dx3/dt3, but, possibly, even higher-
order derivatives. The use of spherical coordinates to describe the position using radial distance from
the origin (r) and a polar and azimuthal angle (usually denoted by and respectively) may or may not
make calculations generally easier.
92
The distance from the (0, 0, 0) origin to the x = (x, y, z) position is the norm of x and is given by the
Pythagorean Theorem:

The same expression is, obviously, valid in the moving reference frame:

Distances may be measured in light-seconds (299,792,458 m) instead of meter by dividing all distances
by c. This amounts to measuring the distance in the time that is needed for light to travel from the origin
to the position x = (x, y, z) and, therefore, gives a distance measurement in seconds.

It is tempting to think of the c2t2 = x2 + y2 + z2 expression as an expression of the Pythagorean Theorem
but this can only be done if t is effectively defined as the time that is needed to travel from the origin to
x, in which case the expression above can effectively be written as:

However, we are modeling motion and, hence, the time variable is the time which we associate with an
object in motion (usually a charge) and we are, therefore, concerned with the equation of motion only:
x = x(t) = (x(t), y(t), z(t))
By way of conclusion and to warn the reader against using relativity theory without much appreciation
of what might actually be going on, we make a few remarks on relativity theory.
92
For an overview of how the Lorentz transformation of the position and time variables used to describe motion
works for spherical coordinate frames, see: Mukul Chandra Das and Rampada Misra, Some studies on Lorentz
transformation matrix in non-cartesian co-ordinate system, Journal of Physics and Its Applications, 1(2) 2019, pages
58-61.
36
9. Final remarks on relativity
Because there is no preferred origin, the coordinate values (x, y, z, t) and (x, y, z, t) have no essential
meaning: we are always concerned with differences of spatial or temporal coordinate values belonging
to two events, which we will label by the subscript 1 and 2. This difference is referred to as the
spacetime interval s, whose squared value is given by:
(s)2 = (x)2 + (y)2 + (z)2 (ct)2
= (x2 x1)2 + (y2 y1)2 + (z2 z1)2 c2(t2 t1)2
In the context of this expression, c should be thought of as an invariable mathematical constant which
allows us to express the time interval t = (t2 t1) in equivalent distance units (meter). The same
spacetime interval in the moving reference frame is measured as:
(s’)2 = (x’)2 + (y’)2 + (z’)2 (ct’)2
= (x’2 x’1)2 + (y2 y1)2 + (z2 z1)2 c2(t’2 t’1)2
The spacetime interval is invariant and (s)2 is, therefore, equal to (s)2. We can, therefore, combine
both expressions above and write:
(x)2 + (y)2 + (z)2 (ct)2 = (x’)2 + (y’)2 + (z’)2 (ct’)2
[(x)2 (x’)2]+ [(y)2 (y’)2] + [(z)2 (z’)2] = c2[(ct)2 (ct’)2]
This equation shows two observers in relative motion one to another can only meaningfully talk
about the spacetime interval between two events if they agree on (1) the reality of the events
93
, (2) a
common understanding of the measurement units for time and distance as well as a conversion factor
between the two units so as to establish equivalence.
Because the speed of light is an invariant constant the only measured velocity which does not depend
on the reference frame
94
it will be convenient to measure distance in light-seconds (the distance
travelled by light in one second, i.e. 299,792,458 meter exactly
95
). This, of course, assumes a common
definition of the second which, since last year’s revision of the international system of units (SI) only, can
be defined with reference to a standard frequency only. This standard frequency was defined to be
equal to 9,192,631,770 Hz (s1), exactly
96
, which is the frequency of the light emitted by a caesium-133
93
Both observers need to agree on measuring time along either the positive or negative direction of the time scale
because the order of the events (in time) cannot be established in an absolute sense. Even if one reference frame
assigns precisely the same time to two events that happen at different points in space, a reference frame that is
moving relative to the first will generally assign different times to the two events (the only exception being when
motion is exactly perpendicular to the line connecting the locations of both events).
94
The velocity of light does not depend on the motion of the source.
95
It was only in 1983 about 120 years after the publication of Maxwell’s wave equations, which showed that the
velocity of propagation of electromagnetic waves is always measured as c that the meter was redefined in the
International System of Units (SI) as the distance travelled by light in vacuum in 1/299,792,458 of a second.
96
This value was chosen because the caesium second equaled the limit of human measuring ability around 1960,
when the caesium atomic clock as built by Louis Essen in 1955 was adopted by various national and international
agencies and bodies (e.g. USNO) for measuring time.
37
atom when oscillating between the two energy states that are associated with its ground state at a
temperature of 0 K.
97
The rather long introduction on relativity illustrates that two observers need to agree on (1) the use of a
(physical) clock to count time and (2) the invariance of the speed of light. The reality of light effectively
corresponds to a succession of events a photon travels the distance x over a time interval t which
are separated by invariant spacetime intervals
.
It should be noted that the expression under the square root sign cannot be negative because the
photon does not travel at superluminal velocity. In fact, for a photon traveling from point A to B the
expression above can be multiplied by t and t respectively so as to yield the following:






 


This establishes the light cone separating time- and spacelike intervals for both observers. We will now
no longer be worried about the relativity of the reference frame: we hope the reader got a sufficient
understanding of what is relative and absolute in the description of (physical) reality, which consists of
matter (charged particles) and light (fields and electromagnetic waves causing changes in those fields).
98
Jean Louis Van Belle, 26 December 2020
97
These two energy states are associated with the hyperfine splitting resulting from the two possible states of spin
in the presence of both nuclear as well as electron spin. Spin is measured in units of h: the nuclear spin of the
caesium-atom is 7/6 units of h, while the total electron spin is (because of the pairing of electrons and the
presence of one unpaired electron) is equal to h/2. Depending on the energy, nuclear and electron spin will either
have opposite or equal sign. Hence, the two energy states are associated with total spin value (nuclear and
electron) F = 7/6 1/2 = 3 or F = 7/6 + 1/2 = 4.
98
For a short (15 minutes) brief, we refer the reader to our YouTube video on reality, philosophy, and physics.
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