Download the coulomb force is not valid for a moving test charge

REFORMULATION OF COULOMB AND NEWTON FORCES, EXERTED
BY A SOURCE CHARGE, ON A MOVING TEST CHARGE
PART I: COULOMB FORCE EXERTED BY A SOURCE CHARGE,
ON A MOVING TEST CHARGE
Tolga Yarman
Okan University, Akfirat, Istanbul, Turkey
([email protected])
ABSTRACT
Over the years, we were all taught, and did teach that Coulomb Force exerted by a source charge at rest on a test
charge, is the same, whether the test charge is at rest, or in motion. Previously we have shown that the spatial
behavior of Coulomb Force, or that of Newton Force, is a must imposed by the Special Theory of Relativity,
though for static (electric or gravitational) charges, exclusively. In this article we will show that, to presume the
validity of the Classical Coulomb Force for a moving test charge, violates the relativistic law of conservation of
energy. Accordingly, we provide the correct expression for Coulomb Force exerted by a source charge, at rest,
on a moving test charge. It is that, the Classical Coulomb Force in the latter case, is reduced by the inverse of the
Lorentz dilation factor coming into play, or the same, the source charge appears to exert a greater acceleration on
the moving test charge, if viewed classically. This approach, facilitates considerably, the formulation of the
relativistic quantum mechanics. As regards to Newton Force, it allows us to reach the end results of the General
Theory of Relativity, though through a totally different set up than that of Einstein, and incomparable ease, and
thereby, the quantization of gravitation, just like the electric field, or in fact any field the object at hand interacts
with.
1. INTRODUCTION
The Classical Coulomb Force is assumed to act, between a source charge at rest, and a test
charge, either at rest or in motion.1
Let us define the following quantities.
Q :
q :
r :
FCC :
k :
intensity of the source charge, assumed at rest, throughout
intensity of the test charge
distance between the test charge and the source charge, assumed to be at the center of
the coordinate system
strength of Coulomb Force reigning between Q (at rest) and q (either at rest or in
motion)
proportionality constant, being unity in the CGS unit system.
Thus, as usual (in CGS unit system), 2
Qq
FCC  2
[Eq.(1) of Part I of the precedent article] .
r
(Classical Coulomb Force reigning between the
two charges, no matter whether q is at rest, or in motion)
1
In the precedent article we have shown that, the above equation is well valid, for a test charge
at rest, and this result is a must imposed by the Special Theory of Relativity (STR).
The finding in question will constitute the basis of the derivation we will now present, as
regards to a moving test charge (the source charge being assumed at rest throughout).
The conclusion we will derive is that, the above relationship is incorrect, if q is in motion.
And this is implied by the relativistic law of conservation of energy.
In other words, assuming that the above relationship is valid, were q is in motion, constitutes a
clear violation of the relativistic law of conservation of energy.
We exclude (for simplicity, but without any loss of generality), the possibility of creation of
charges on the way, which is for instance the case of a charged particle, suddenly entering in
an accelerator.
2. COULOMB FORCE EXERTED BY A SOURCE CHARGE AT REST, ON A
MOVING TEST CHARGE
Herein we will derive an expression about Coulomb Force reigning between a source charge
at rest, and a moving test charge.
We start with the following postulate, essentially, in perfect match with the “relativistic law
of conservation of energy”, thus embodying, the mass & energy equivalence of the STR.
Postulate: The mass of an object at rest, bound gravitationally, electrically, or else, amounts
to less than its rest mass measured in empty space, the difference being, as much a
a mass deficit, equivalent to the binding energy vis-à-vis the field of concern.
For simplicity we consider two charges bound to each other, at rest, one of them being
infinitely more massive than the other.
In order to calculate the binding energy coming into play, for electrically bound particles, we
will make use of the Coulomb Force we have derived, with respect to static charges, solely
based on relativistic considerations.2
Thus we know that Coulomb Force, in this case, is rigorous.
Hence, the framework we consider, fundamentally lies on the STR, more profoundly on the
Galilean Principle of Relativity.
Based on this framework, below, we will derive the strength of Coulomb Force exerted by a
source charge at rest, on a moving test charge.
To concretize our approach, we would like to consider, the electron of charge e, in motion
around a nucleus of charge Ze (Z, being the number of protons residing in the nucleus). Ze
and e, together are assumed to constitute a closed system.
2
The Equation of Motion
Suppose that the electron is cruising in a full “electric isolation”, with a uniform translational
velocity. So does a nucleus of charge Ze. Then (based on the STR) we would be certain that,
the electron in its own frame of reference, all the way through, preserves its identity, defined
at infinity. So will also do, the nucleus in consideration. If now, we remove the previous
electric isolations, the electron and the nucleus of charge Ze, because of the electric attraction
force, they mutually create, will sense each other, and shall start getting accelerated toward
each other. The “extra kinetic energy” they would acquire, as well as the energy they would
radiate through this process, shall be supplied by the (closed) system made of the two. (In
order to assure an easy wording, for the time being, we will neglect the energy emitted
throughout by radiation, without though any loss of generality.) The total energy of the
electron and the nucleus of charge Ze [i.e. (the sum of their relativistic masses) x c 02 ], through
the motion, must remain constant, and equal to the equivalent of the sum of their initial
relativistic masses. (Otherwise, the relativistic law of conservation of energy would be
broken.)
Since we assume that the nucleus of charge Ze is infinitely more massive than the electron,
owing to the law of conservation of linear momentum, it will stay in place, all the way
through. We will elaborate on this point, soon. Let us suppose for simplicity that, Ze and e
start, far away from each other, at rest; then, their initial masses (more precisely, relativistic
energies), are essentially equal to their rest masses (more precisely, rest relativistic energies).
If after a while, the accelerating electron, hurts an obstacle and looses all the kinetic energy, it
would have acquired through the attraction process; it must concurrently dump a portion of its
rest mass, and this, as much as the amount of the kinetic energy it would have piled up, on the
way.*
Let us evaluate the “rest mass deficit” in question. Thus, suppose that the nucleus of charge
Ze and the electron are held with the aid of convenient obstacles, transparent to electric
interaction, at a distance R 0 from each other.† We now propose to carry the electron, very
slowly, away from Ze. The reason we anticipate to achieve this task “very slowly”, is to avoid
the complication that would result from “the radiation of an accelerated charge”. Note
however that our anticipation will not lead in any way to a lack of thoroughness. Then, the
static binding energy E SB (R 0 ) we have to furnish to the electron, in order to bring it from R0
to infinity, based on the work we have to achieve against Coulomb Force is (in CGS units
system), as usual,
2
 Ze
Ze 2
.
(1)
E SB (R 0 )  
dr

R0 r 2
R0
(energy necessary to bring the electron at rest, at
a distance R 0 from the fixed nucleus of charge Z, to infinity)
*
The author is honored to have discussed with Professor R. Feynman, the roots of the idea presented herein,
and to have been encouraged by him in developing this idea, through a Fulbright visiting stay at California
Institute of Technology, in 1984.
†
One may wonder how: Thus consider for instance the molecule, H2O. It can be visualized as a dipole made of
the charges, on one side, -2e located near the Oxygen Atom, attracting the Hydrogen Atoms’ electrons, and on
the other hand +2e located in between the Hydrogen Atoms giving away their electrons. The dipole’s charges
ares then well held at a given distance from each other, once the molecule is formed.
3
Through the process in question, this much of energy must have been transferred to the
system made of the nucleus of charge Ze and the electron, originally located at a distance R 0
from each other. The nucleus of charge Ze, was supposed to be held at rest, since it is much
more massive then e, and it will not be altered through the process in question; thus, it is the
electron that should have acquired the energy of concern.‡ (A detailed mathematical analysis
on the subject is provided in Appendix A.)
Henceforth, based on the relativistic mass & energy equivalence, i.e.
E (energy released, or piled up)
2
= m (magnitude of the algebraic increase in the rest mass) x c 0 ,
(2)
the mass of the electron at a distance R 0 from the nucleus of charge Ze, should increase as
much as E SB ( R ) / c 02 , when carried, to infinity.
Thus, in mathematical words, and via Eq.(2), the mass m e (R 0 ) of the electron at R 0 ,
becomes
Ze 2
,
m e (R 0 )  m e 0 
R 0 c 02
(3)
(mass of the electron at rest, at a distance R 0 from the nucleus
of charge Ze , as referred to the mass of the free electron )
where m e 0 is the mass of the electron at infinity.
Eq.(3) is the mathematical expression of the above postulate. In fact, again, this is nothing
else, but the relativistic law of conservation of energy, written via taking into account the
STR. Accordingly, we cannot say that the nucleus and the electron are the same, after we have
added to, or retrieved from the system made of the two, a given amount of energy.
The greater Z, the larger is the energy extracted from the system made of the nucleus and the
electron; thus the worse will be the harm caused in essentially the electron’s internal
dynamics (the nucleus, being assumed untouched, chiefly for big Z’s). The bound electron’s
internal dynamics in fact weakens as much as the static binding energy coming in to play (still
assuming that the nucleus remains untouched). Consequently the electron’s proper mass
defined at infinity, must decrease just as much through the binding (see Appendix A).
Let us rewrite Eq.(3) at the location R 0  r0 , as
‡
If indeed, now the other way around, the electron practically at infinity, is set free in comparison with a fixed
nucleus (not really at infinity, where the force exerted by the proton on it is null, but at a very big distance
where the force strength in question, is very small, without however being null), then the electron will start
falling onto the nucleus. It will acquire kinetic energy on the way, and will eventually radiate. Ultimately it
will come to hurt the obstacle at the distance R 0 from the proton, and will in here dump the kinetic energy
gained on the way. The relativistic law of conservation of energy requires that the kinetic energy, the electron
will transfer to the obstacle, together with the energy, it would have lost on the way via radiation, should in
total, come to amount to the RHS of Eq.(1), i.e. Ze 2 / R 0 . Thus, based on the relativistic mass &energy
equivalence, the electron must have lost that much of energy (divided by c 02 ). Or the same, the electron’s mass
must have increased by this amount, as it is carried from R 0 to infinity (c.q.f.d.).
4
m e (r0 )  m e 0 c 02 (r0 ) ,
(mass of the bound electron, at rest)
(4)
where (r0 ) is
(r0 )  1 
Ze 2
;
r0 m e 0 c 02
(5)
To many of us, Eq.(4) is new. It is even unacceptable. The useful simplification of visualizing
the electron as a point–like particle, unfortunately still remained as an oversimplification, and
deprived many of us from conceiving an internal dynamics to be associated with the electron,
with a given energy. This was unfortunately a big danger, for it led to the bias of considering a
universal electron rest mass, no matter where the electron is, no matter what it does. To the
author the rest mass of the electron is nothing, but its internal energy, say the internal energy
of the charge of the electron, holding this charge together. Just like two parallel currents
attract each other, a given internal motion of the electron’s charge, may be holding it, in one
piece, away from otherwise a fatal dissociation, as many anticipated. We hate to speculate; the
essential though, is that it is pointless to conceive any particle as a point-like particle,
including the electron. No matter how tiny it is, the electron must still have an internal
dynamics, thus an internal energy. Such an energy is no doubt subject to a change. The
electron’s internal energy can decrease, just like it does when bound to a positive charge. An
excited electron would not weigh as much as an electron on the ground level in an hydrogen
atom; it must weigh more when excited. The hydrogen atom does not, in the first place, weigh
as much as the proton and the electron weighed separately from each other. It seems to have
been silly to have considered thus far, that eggs are not a bit altered in an omelette. The
electron is altered when bound.
Thus the so called proper mass, can only be defined for an object free of any field it may
otherwise interact with.
Thus we come to the definition of a rest mass of an object interacting with a given field
strength (and not just a universal proper mass). Here how ever, we assume that the interacting
particles do no lose their original identities.
Now suppose that the electron is engaged in a given motion around the nucleus of charge Ze;
the motion in question can be conceived as made of two steps:
i) Bring the electron quasistatically, from infinity to the given location r0, on its orbit
around Ze, but keep it still, at rest. (As assumed the proton, through this process,
remains in place.)
ii) Then deliver to the electron, at the given location, its orbital motion.
Note that, we are dealing with a strictly closed world made of just the nucleus in
consideration, and the electron, and while achieving the two steps in question, we exclude any
interference, one may be carried to think of, with that world.
5
Thus, the first step yields a decrease in the mass of m e 0 as delineated by Eqs. (4) and (5).§
The second step yields the Lorentz dilation of the rest mass m e (r0 ) , at the location r0 , so that
the overall mass m e (r0 ) , or the same, the overall relativistic energy of the electron on the
given orbit, becomes
m e (r0 )c 02  m e (r0 )c 02
1
1
2
0
2
0
 m e 0 c 02
Ze 2
r0 m e 0 c 02
v
v 02
1
1 2
c
c0
(overall relativistic energy, or the same, total energy
of the bound electron on the given orbit)
;
(6)
v 0 is the magnitude of the tangential velocity of the object at r0 .
The overall relativistic energy of the electron in orbit [i.e. m e (r0 )c 02 ] must remain constant,
so that for the motion of the object in a given orbit, one finally has
Ze 2
1
r0 m e 0 c 02
2
2
.
(7)
m e (r0 )c 0  m e 0 c 0
 Constant
2
v
1  20
c0
(total energy written by the author, for the
electron in motion around the nucleus)
This relationship is in fact the integral form of our general equation of motion (given below).
One can notice that Eq.(7) is different from what one would write classically, i.e.




2

 Ze 2
m0c0
m e (r0 )c 02  m 0 c 02  
 m 0 c 02  
 Constant : Wrong !
2
r


v0
0
 1 2

c0


(total energy one would write classically, for
the electron in motion around the nucleus)
(8)
What is “wrong” with this latter equation?
What is essentially wrong with it, is that, as silly as it may look, on the whole it delineates a
violation of the “relativistic law of conservation of energy”. Although [when compared with
Eq.(7)] this seems trivial, it is unfortunately overlooked through very many decades. The
restitution of the mistake in question, evidently, is to alter so very many related derivations.
This may be unfortunate, but that is the way it is. We elaborate on this below.
§
The fact that the electron is brought to the location in consideration, quasistatically, provides us with the
facility of not having to deal with the “radiation problem”, that would arise otherwise.
6
Eq.(8) assumes that the overall relativistic energy, or in short, the “total energy” is composed
of the “initial relativistic rest energy”, the “relativistic kinetic energy” and the “potential
energy”. And what is really wrong with this? To compose the total energy, that way, is what
we all learned at already the high school level, and that is what most of us kept on teaching.
For one thing, in this presentation we do not make use of, or refer to the “concept of potential
energy”. We only consider the mere “concept of energy”, more precisely the “concept of
relativistic energy”. Thus, the energy of the statically bound electron, is decreased as much as
Ze 2 / r0 [cf. the first step we have considered, in writing Eq.(7), i.e. Eqs. (4) and (5)], and it is
the “remaining electron’s relativistic energy of the electron”, which is dilated by the Lorentz
factor (and not the electron’s rest mass at infinity), while we deliver to it, its motion on the
orbit [cf. the second step we have considered, in writing Eq.(7)].
We can provide another way of looking at the traditional mistake we unveil, and it is the
following. Eq.(8), tacitly assumes that Coulomb force holds, between a “static source
charge” (here the nucleus), and a “moving test charge” (here the electron) (cf. Appendix B).
This is too, something we all have learned, and we all taught. And what reasons do we have to
do so? Seemingly, none! Nowhere we know of, it is not even specified whether we postulate
that Coulomb Force is still valid for a moving test charge, just like this, or we borrow this
assumption from a straight empirical evidence. There are books which mention the latter
argument, but this remains as only a hope, and we show herein that it is in effect, a false hope.
What though we have assumed, to develop the present theory, is just “Coulomb’s Law,
reigning in between two static charges, exclusively”; this, in fact, as we have shown, turns to
be a straight requirement imposed by the STR, and we did not even borrow it, from Coulomb
or direct experimental results.
“Coulomb’s Law reigning in between, only two static charges”, as stated, does not, in any
way, tell us how the law of force will look, if one of the charges moves, and clearly, what we
used to believe “true”, is not; Coulomb’s Law does not hold in between the nucleus (assumed
at rest, throughout) and the moving electron (the way it holds, between “the nucleus and the
electron, both at rest”). This is the crucial point. In other words, as detailed in Appendix B,
Eq.(8) would be valid, if Coulomb’s Law were to hold between the nucleus and the moving
electron, just the way it does for the nucleus and the electron, but both at rest. How ever, it
does not; thus Eq.(8) is only approximate.
One way or the other, it is that, the rest mass of the bound electron, is not the same as the rest
mass of the free electron, and as trivial as it may look at this stage, this is what essentially had
been overlooked throughout the passed century, and unfortunately (as elaborated above) the
eggs in an omelette were assumed to be same as the uncooked eggs!
How Coulomb’s Law must be written for the nucleus and the moving electron, will soon be
specified. Thus, although Eq.(7) looks straightforward, to our recollection, it happens to be
new. The way we write it, induces the need of elaborating on the concept of “field”; this will
be undertaken below.
We show elsewhere that Eq.(7) furthermore constitutes, the basis of a relativistic quantum
mechanical description, well equivalent to that of Dirac, if geared alike, yet established in an
incomparably easier way.3,4 We have to stress that the present approach is in full harmony
with all the existing quantum electrodynamical data.
7
The differentiation of the above equation leads to
v 02
1 2
c0
dv
Ze 2
.

 v0 0
2
2
dr0
m 0 r0
Ze
1
r0 m 0 c 02
(9)
[differential form of Eq.(16), equivalent
to the equation of motion]
One can transform Eq.(9) into a vector equation; the RHS, is accordingly transformed into the
acceleration (vector) of the electron on the orbit.
Thus, recalling that the LHS of Eq.(10), i.e. m e (r0 )c 02 , is constant, one can write

d v (t )
v 02 r 0
Ze 2
1

 m e (r0 ) 0 0 ;
2
2
dt 0
r0
c 0 r0
(10)
[vectorial equation written based on Eq.(9, or the same,
equation of motion written by the author, via the energy conservation
law, extended to cover the relativistic,” mass – energy equivalence”]
here, r 0 is the “radial vector” of magnitude r0 , directed outward, and v 0 is the “velocity
vector” of the electron, at time t 0 ; recall that m e (r0 ) remains constant throughout [cf.
(Eq.(7)] .
For a small Z, thus a small v0, the orbit would be as customary elliptical; otherwise it is open;
in other words, the perihelion of it shall precess throughout the motion. Eq.(10) is anyway the
same relationship as that proposed by Bohr, except that Coulomb Force intensity is now
decreased by the factor 1  v02 / c02 , similarly to what empirically, but approximately proposed
by Weber, by the end of nineteen century.5,6,7,8 [We would like to racall that Weber by trial
and error, introduced the coefficient 1  v 02 /( 2c 02 ) , instead of 1  v02 / c02 .]
We would like to state, our finding as a theorem.
Theorem: Coulomb Force exerted by a source charge Ze, on a test charge e, moving at r0 ,
with the velocity v 0 , is not Ze 2 / r02 , but (Ze 2 / r02 ) 1  v 02 / c 02 ; thus in relation to
its accustomed form, it is decreased by the factor 1  v02 / c02 .
One other, also useful way of interpreting this outcome, is that to the classically adopted
intensity of Coulomb Force, now corresponds a greater equivalent acceleration. And
accordingly we may state, this is exactly what yields the precession of the given orbit.
Recall that what we do, is in no way in conflict with quantum mechanics. Quite on the
contrary, we show elsewhere that through our approach, one can land at the de Broglie
relationship, which is the basis of quantum mechanics.9
8
At this stage, it seems useful to draw the following table displaying the differences between
our approach and the standard approach.
Table 1 Differences Between the “Standard Approach” and “Present Approach”,
Based on the Electron Bound to the Proton (assumed at rest throughout)
Standard Approach
Present Approach
Ze 2
r02
The same.
Force Between the
Proton and the
Electron,
Altogether at Rest
Total Energy of the
Statically Bound
Electron
Total
Dynamic Energy of
the Electron
Mathematical
Expression of the
Total Dynamic
Energy of the
Electron
Force Between the
Proton and the
Moving Electron
m(r 0 )c 02  m 0 c 02 
Ze
r0
2
Rest Energy
+ Potential Energy
+ Kinetic Energy
m  (r0 )c 02 
2
0 0
m c
1
Ze 2
r02
v
c
2
0
2
0

Ze 2
r0
The same; but classically the mass of
the bound electron is not considered
to be altered; it is the overall “field
energy” which is believed to
decrease, as much as the “potential
energy”, coming into play.
The concept of “potential energy”,
as considered classically, is
misleading.
1
m  (r0 )c 02  m 0 c 02
Ze 2
r0 m 0 c 02
1
v 02
c 02
v 02
Ze 2
1

r02
c 02
9
Discussion, About the Concept of Field
The setup of Eqs. (7) and (8) is clearly not achieved by just customarily used concepts, and
mainly, the regular concept of “field”. This concept, to us, is nothing else, but a
“mathematical convenience”. Indeed, the intensity of it, cannot be measured, unless, one
makes use of a “test charge”. Consequently, it is not the “field intensity” that one would
measure, but the strength of the “force” developed by the present static charges, on a test
charge. Therefore, along what we did, we should stress that, the “relativistic energy”
delineated by two electric charges (contrary to the general wisdom) is, not materialized by the
surrounding space, but by the changeable “internal dynamics” of the charges of concern.
What is essential, is the “Coulomb Force, reigning in between two static charges”. This is, as
discussed above, essential in two ways: i) the electric charges, as imposed by the Galilean
Principle of Relativity (which is in fact the underlying principle of the STR) are Lorentz
invariant, ii) the 1/distance2 dependency of Coulomb Force between two static charges, is
imposed by the STR.
What is believed so far is that Coulomb Force holds, if the “source charge” is static,
regardless whether the “test charge” is, at rest or in motion. However, we discover that, this
is not so; if the test charge is in motion, then Coulomb Force strength is altered, by the factor
1  v 02 / c 02 [cf. Eq.(10)]. This occurrence drives us to consider the electron (contrary to what
has been so far done), not in an extreme simplistic way. That is, we sympathize by the fact
that, the electron is generally considered as a “point-like particle”. It must be obvious though,
as tiny as it may be, the electron cannot be reduced to a “point”, given that a point cannot be a
“material being”. Thus, it is futile to consider the electron as a point-like particle. Doing so,
furthermore embodies a severe danger, which seems indeed to have led all of us to various
known problems, to be overcome by normalizations and renormalizations, and so forth. The
electron must embody an “internal dynamics”, just like any other particle. Perhaps its
“mass”, as elaborated above, is simply the “internal energy” of its “electric property”,
which we call “electric charge”. This internal energy, is thus, to be associated with (how ever
it may be), the internal dynamics delineated by the electric charge.
When the electron is bound, say, to a proton, its internal dynamics is then (as a requirement of
the relativistic law of conservation of energy), slown down, as much as the binding energy
coming into play, assuming for convenience that the proton (being much more massive than
the electron), is not affected by the process of binding (cf. Appendix A). Our claim regarding
the weakening of the internal dynamics of the bound electron can be checked right away
through the “reverse process”. Suppose then, we propose to bring, the bound electron, back
to infinity. Accordingly, we have to furnish to it, an amount of energy equal to its binding
energy (still supposing that, moving away the electron, would not disturb, the proton). The
two particles, forming a “closed system”; furnishing energy to the electron, owing to the
relativistic law of conservation of energy, will increase the internal energy, thus the rest mass
of the latter. In other words, when entirely detached from the interaction domain, with the
proton; the electron’s rest mass would then get increased as much as the energy we would
have furnished to it, i.e. by an amount equal to its original binding energy. Hence, the free
electron is not anymore the previous bound electron, or vice versa, the bound electron is not
anymore the same as the free electron.
10
The bound muon decay rate retardation10 may be considered as an experimental proof of our
prediction.
3. CONCLUSION
Over the years, we were all taught, and did teach that Coulomb Force exerted by a source
charge, at rest on a test charge, is the same, whether the test charge is at rest, or in motion. In
a previous work we have shown that Classical Coulomb Force or Newton Force is imposed by
the STR, though for static (electric or gravitational) charges, exclusively. A priori we have no
sign that either force will work, for a moving test charge (the source charge being throughout
at rest). In this article we have shown that, to presume the validity of Coulomb Force for a
moving test charge, violates the law of conservation of energy. Accordingly, we have
provided the correct expression for Coulomb Force exerted by a source charge, at rest, on a
moving test charge. It is that, the usual Coulomb Force in the latter case, is altered by the
inverse of the Lorentz coefficient coming into play. We will do the same separately, for the
Newton Force.
APPENDIX A
WHY ENERGY SHOULD BE RETRIEVED FROM THE TINY OBJECT
BOUND TO AN INFINITELY MORE MASSIVE CELESTIAL BODY?
Suppose indeed we set the very tiny object of mass m and the very massive object (MO)
(originally assumed at rest), simultaneously free, in the reference system of the distant
observer. Because of the attraction force, they will get accelerated toward each other, and at a
given distance from each other, they will come to acquire the velocities v and v MO (supposed,
without any loss of generality, far below that of the speed of light), thus the kinetic energies
**
M v 2MO / 2 and m  v 2 / 2 . Because of the conservation of the linear momentum, we have
M v MO  m v .
(A-1)
This makes that the fraction of the kinetic energy of
into play, turns out to be
m
m v 2
M
2

1
2
2
M v MO m v
M  m

2
2
;
out of the total kinetic energy coming
(A-2)
likewise, the fraction of the kinetic energy of M out of the total kinetic energy in question,
becomes
m
0
M  m
**
(c.q.f.d.).
(A-3)
Note that, having the relativistic law of conservation of energy, along with Newton’s law or Coulomb’s law,
one is able to derive the law of momentum conservation. So this law may very well not be assumed.
11
The same philosophy well applies if there are more than two particles getting bound, since in
this case, it appears sufficient to handle the problem in the frame of the center of mass of the
system (at rest, throughout). At a first glance, in effect it may seem that our result depends on
the history of the recombining particles; this is not correct, for it does not in the frame of the
reference of the center of mass. Indeed, through say the recombination of a proton and an
electron (of initially random kinetic energies), yielding a hydrogen atom, the extra energy the
system would acquire, in the laboratory frame of reference, exhibits itself, as the translational
energy of the center of mass, or a rotational energy around the center of mass, or else. It is the
mass deficit, different elements of the system displays, after the system as a whole comes to a
rest, that we must be accounted for, and the resulting picture is well free of the history of the
recombining particles.
The essential idea is anyway that the overall rest mass of the bound particles, is less than the
total rest mass of these particles when weighed separately from each other. The difference is
as much as the total binding energy coming into play.
APPENDIX B
COULOMB FORCE WOULD HOLD, FOR A MOVING TEST CHARGE, ONLY IF
THE TOTAL ENERGY IS SET EQUAL TO THE SUM OF THE REST ENERGY
INCREASED BY THE LORENTZ FACTOR AND THE POTENTIAL ENERGY,
WHICH HOW EVER, CONSTITUTES A VILOATION OF THE LAW OF
CONSERVATION OF ENERGY
Coulomb Force would hold, for a source charge at rest, and a moving test charge, only if
Eq.(8) of the text were valid. What we know, though, as explained through the text, is that
Coulomb Force holds, for static charges exclusively. Based on this, we are not allowed to
speculate that Coulomb Force shall also be valid, for the pair of, a source charge at rest, and a
moving test charge.. What we shall show herein, is that (the wrong) Eq.(8), is equivalent to
the case in which Coulomb Force reigns between a static source charge and a moving test
charge. Thus consider the differential form of Eq.(8):
2
v0
c 02
dv 0
2 1
m 0 c 02
1
v 02
c 02
v 02

Ze 2
dr0 .
r02
(B-1)
c 02
[differential form of Eq.(8)]
Let us assume for simplicity, though without any loss of generality, that the electron
delineates a free fall toward the proton assumed at the origin. In this case one can write
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v0  
dr0
;
dt 0
(B-2)
Here the negative sign arises from the fact that, through the free fall, dr0 is negative (whereas
v 0 is positive).
One can then, via Eq.(B-1), write
m
dv0 Ze 2  v02 
 2 1  2  ,
dt 0
r0  c0 
(B-3)
where
m
m0c02
v2
1  20
c0
(B-4)
.
Eq.(B-3) is well equivalent to
d(mv 0 ) Ze 2
 2 .
dt 0
r0
(B-5)
Indeed
d(mv 0 )
d( v0 )
d ( m)
d ( v0 )
m
m
 v0
m
 v0 0
dt 0
dt 0
dt 0
dt 0
dt 0
v0dv0
c02
1
2
0
2
0
v
c
 v
1 
 c
2
0
2
0




Ze 2
r02
;
(B-6)
thus
m
d(v0 )  v02 
v2 dv
Ze 2  v2 
1  2   m 20 0  2 1  20 
dt 0  c0 
c0 dt 0
r0  c0 
,
(B-7)
leading well to Eq.(B-3) (c.q.f.d.).
This latter equation can as well be written in vector form, i.e. nothing else but the classical
electric equation of motion:
d(m v0 )
Ze 2 r 0
.
 2
dt 0
r0 r0
(B-8)
[the classical electric equation of motion
obtained from the “wrong” Eq.(18]
Note that the negative sign arises, because the vector quantities r 0 and d v0 are in opposite
directions, supposing that as usual r 0 is directed outward.
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In short, Eq.(B-8) (the classical electric equation of motion), is equivalent to Eq.(8) (since we
have started up with this latter equation to get it). Therefore, it is as wrong.. The correct
equation, is Eq.(10). The conclusion is that, quite on the contrary to the general wisdom,
Coulomb’s Law of Force does not hold for the pair of source charge at rest and test charge in
motion; it only holds for electric charges at rest.
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1
2
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4
5
6
7
8
9
10
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