Radiation of an Accelerated Charge

International Journal of Electromagnetics ( IJEL ), Vol 1, No 1, August 2016
1
RADIATION OF AN ACCELERATED CHARGE
Enrique Cantera del Río
C/ Padre Benito Menni 6-2-E 47013 Valladolid,Spain
ABSTRACT
Rethinking the dynamics of an accelerated charge from classical concepts. From the idea that radiation
comes from kinetic energy and managing the problem of the auto-energy of a point charge, a system of
non-linear dynamic equations are found and results amenable to experimental verification. In developing, a
relationship between the principle of causality, which affects the direction of time, and the constancy of
mass appears. Another consequence are the fluctuations in the motion of particles, compatible with
Brownian motion and Heisenberg´s indeterminacy principle. The case of gravitational acceleration is also
analyzed, concluding that no electromagnetic radiation is possible and there is no electric field that can
produce a constant acceleration on a point charge. Thus the constant acceleration is an exclusive feature of
gravity.
KEYWORDS
Radiation reaction force, Abraham-Lorentz equation, Causality.
1. INTRODUCTION
The behaviour of an accelerated electric charge, regardless of the accelerating force, is a limit
problem in classical physics. The radiation of a system of charges is described by Poynting’s
theorem, which is a logical consequence of Maxwell’s equations. The key point is that the
Poynting vector, which appears in this theorem, is interpreted as a flow of radiant energy using
the principle of the conservation of energy in an electromagnetic system. From this perspective, it
seems that radiation, similar to potential energy, is a behaviour that is associated with a system of
charges and not with individual charges. In this sense, the literature discusses dipole radiation,
quadrupole radiation, etc[4]. However, in classical theory, it is immediately clear that the
radiation of a system of charges can be calculated if the motion of the charges is known because
that is sufficient to determine the fields that define the Poynting vector. There is a direct
relationship between the motion of a system of charges and the resulting radiation. H.A. Lorentz
went further and extended the result for an isolated charge that is accelerated by any force (a
magnetic field, gravity,etc.), independently of the existence of electromagnetic potential energy.
This author demonstrated that the field in the proximity of a charge with spherical symmetry
becomes distorted by the combined effects of the acceleration of the charge and the finite
propagation velocity of changes in the field. This distortion generates a net “self-force”[3] of the
field on the particle on its own source; the displacement of this force can represent, at least in
certain cases, the radiated electromagnetic energy. In this way, Lorentz did not attribute radiation
to the relative accelerations between the charges in the system, as would be expected if there was
a relationship with the potential energy, but to the acceleration of a charge with respect to any
inertial coordinate system. An accelerated point charge emits energy and impulses in the form of
radiation. The reason for this attribution is the presence of emitted energy in the form of
oscillations of the particle’s electromagnetic field in regions relatively far from the particle; these
make up the radiation field. An observer in the radiation field can relate this energy to an event
that occurred at a point occupied by the charge at an earlier time. This delay time corresponds to
the propagation velocity of an electromagnetic signal from the particle, and the event referred to
is a change in the velocity of the charged particle. In regard to the conservation of energy, the
radiated energy is directly extracted from the mechanical kinetic energy of the charged particle,
not from the potential energy of the electromagnetic system.

2
2. LARMOR’S FORMULA
Consider a spherical surface in a radiation field with an accelerated charge at the centre. The
amount of energy W that exits the surface in the form of electromagnetic radiation per unit of time
is given by Larmor’s formula,
[ ] )1.2()(
6
2
3
0
2
dta
c
q
dt
dW
τ
πε
−=
where q is the amount of electric charge, a is its acceleration and τd corresponds to the previously
mentioned delay. The cause of the radiated energy at time t is the charge acceleration at time t -
τd. Following these ideas, we can search for a force whose displacement generates this radiated
energy. If such a force exists and obeys Newton’s laws, it must satisfy
)2.2(amFFamF rexti =+⇒=∑
where the subscript “ext” indicates the resultant of the external forces, and the subscript “r”
indicates the Lorentz “self-force” between the particle and its own field. Assuming that the effect
of this force is to decrease the kinetic energy of the particle and emit it as radiation, we can write
[ ]∫∫ −=• )3.2()(
6
2
3
0
2
dtta
c
q
rdFr
πε
.
Note that we are now disregarding the delay. Newtonian forces are essentially local magnitudes
and are from the perspective of an observer local to the particle, not from an observer located in
the radiation field. In principle, this should not be a problem because we have seen that the energy
that arrives in the radiation field comes from the charge’s acceleration, and that energy should be,
at the corresponding time, as close to the charge as we want it to be.
( ) )4.2()(
66
)(
66 3
0
2
3
0
2
3
0
2
3
0
2
avd
c
q
rd
dt
ad
c
q
Fadvavd
c
q
vda
c
q
rdF rr •=•







−⇒•−•−=•−=•
πεπεπεπε
It turns out that the self-deceleration force derived by Abraham and Lorentz is the second term of
the previous parenthetical expression,
)5.2(
6 3
0
2
dt
ad
c
q
Fr
πε
=
According to our development, this force can be maintained when the velocity and the
acceleration are always perpendicular or when there is oscillatory motion; then, the integral
average of the second term in each period is zero,
)6.2(0
66 0
3
0
2
0
3
0
2
=•=•







−∫
TT
r av
c
q
rd
dt
ad
c
q
F
πεπε
If we accept all of this, then, Newton’s law becomes the following in the case of an accelerated
charge[1]:
)7.2(
6 3
0
2
am
dt
ad
c
q
Fext =+
πε
When the external force is oscillatory (the unforced case) we can write, in one dimension,
)8.2(
6 2
2
3
3
3
0
2
2
dt
xd
m
dt
xd
c
q
xm =+−
πε
ω ,

3
which is an equation that has been widely studied in the literature and has physically feasible
consequences. Mathematical appendix includes an explanation of this eq. according the theory we
will develop. However, in the case of a constant external force, the differential equation is simple,
and its solution is
)9.2(
6
; 3
0
2
0
/
0
0
mc
q
ea
m
F
a text
πε
ττ
=+= ,
which predicts a constant acceleration, a self-acceleration :a spontaneous exponential increase in
the acceleration. The discrepancy between this result and a real situation is radical. In charged
particle accelerators, where this result can be observed, nothing similar occurs. Have we obtained
a reduction to absurdity? Let us review the important points and try not to repeat the same
mistakes.
3. THE RADIATION COMES FROM THE KINETIC ENERGY OF THE CHARGE.
Imagine an object in motion in our coordinate system. We introduce into this object a certain
amount of electric charge that is small enough that the modification of its mass is negligible; we
place the object in an external magnetic field (that is exclusively magnetic). Due to the Lorentz
magnetic force, the object experiences a force normal to its trajectory, and due to the acceleration
generated, the charge starts to emit electromagnetic radiation. The radiation is emitted at the
expense of the object’s kinetic energy and ceases when the object is at rest in our coordinate
system. This phenomenon has been confirmed in certain particle accelerators (cyclotrons),and
therefore, there is experimental support for our assumption that kinetic energy can be completely
transformed into radiation.
In the previous section, we used Larmor’s formula but used instantaneous rather than delayed
acceleration to introduce a force associated with the radiation. In the following argument, we
reason about the hypothesis that the radiation comes from the kinetic energy without resorting to
the possibility that this force exists. Suppose a force is applied to a charge that is at rest. From the
very definition of acceleration (a), we know that the change in the velocity (v) of the particle at
time dt is dv=a*dt; however, if at that time, the charge emits a certain amount of energy while
keeping its mass m constant at the expense of its own kinetic energy Ec, we can no longer be
confident of the definition of acceleration. In fact, from the definition of acceleration, it follows
that dtamvdEdtavmvdvmdtavd c •=⇒•=•⇒= ; that is, for a given instantaneous acceleration,
the instantaneous variation in the kinetic energy, dEc, is completely determined if we suppose that
the mass ,m, of the particle is constant whether the particle is charged or not. If the radiation
implies consumption of kinetic energy , then dtamvdEc •≠ , which conflicts with the definition of
instantaneous acceleration. Evidently, this situation has led us into a very basic analytical error
because we can not disregard the definition of instantaneous acceleration, and therefore, we must
conclude that:
1. An analysis that uses Larmor’s formula must include the effect of the delayed
acceleration.
2. An analysis of the motion of an accelerated charge must consider the possibility of
fluctuations in the mechanical mass.
A non-inertial accelerated observer that perceives a charge at permanent rest and without mass
fluctuations does not observe a change in the kinetic energy, and therefore, according to our
hypothesis, does not observe the emission of radiation. If the laws of physics must be valid for
any observer, then the mass fluctuation may be a valid explanation of the radiation perceived by
any observer. Note that with this approach, the existence of a radiation force is still possible, but
such a force is a consequence of the mass fluctuations of the particle.

4
4. POYNTING'S THEOREM FOR AN ACCELERATED CHARGE.
Following the observations above, consider a spherical surface centred on the charge at time dt.
The sphere’s radius (cτd), being c light’s velocity in vacuum and τd a time delay, is large enough
to suppose that it reaches the radiation field but small enough to use a linear approximation for
the delayed acceleration resulting from the charge’s instantaneous acceleration. Applying
Poynting’s theorem to this limited volume V yields
( ) )1.4(
2
1
2
22
∫∫∫∫ •×+•=





+− SdHEdVEvdVBdVE
dt
d
ρ
µ
ε .
If we consider volume divided into integration elements,dV, and bordered by a surface divided
into elements,dS, we have the following concepts:
1-The first term is the rate of change of the amount of electromagnetic energy contained in the
volume. In the case of a point charge accelerated by an external field, we can decompose the field
into the sum of the external field and the particle field as follows:
( ) ( )
∫∫∫∫∫∫
∫∫
•+•++++
=+++
dVBBdVEEdVBdVEdVBdVE
dVBBdVEE
parextparextparparextext
parextparext
µ
ε
µ
ε
µ
ε
µ
ε
1
2
1
22
1
2
)2.4(
2
1
2
2222
22
.
The summands that depend only on one field correspond to the external field’s own energy and
the charge’s electromagnetic mass in the volume to be integrated over. The summands that
depend on the scalar product of two different fields correspond to the interaction energy between
the particle and the external field.
2-The second term includes the charge density, the velocity and the electric field. The Lorentz
force on a point charge, allows this term to be written as the rate of change of the kinetic energy.
For a point charge distribution with mechanic impulse p and kinetic energy Ec
( ) )3.4(
2
dt
dE
dt
pd
vdV
dt
pd
v
dV
d
dV
dtdV
pd
vdVBvEvdVEv c
=•=







•=•=×+•=• ∫∫∫∫ ρρ .
3-The third term corresponds to the flow of electromagnetic energy per unit time that leaves the
integration surface. Suppose that no particles travel through this border; otherwise a term
associated with the flow of mechanical energy through the surface must be included. Under these
conditions, and in the case of an accelerated point charge, this term corresponds to Larmor’s
formula,
( ) [ ] )4.4(;)(
6
2
3
0
2
radiusspherecta
c
q
SdHE dd −=−=•×∫ ττ
πε
.
Summarizing the above into an equation with the assumption of a constant external field, we have
[ ] )5.4()(
62
1
2
1 2
3
0
2
22
dcparparparextparext ta
c
q
EdVBdVE
dt
d
dVBBdVEE
dt
d
τ
πεµ
ε
µ
ε −+





++=





•+•− ∫∫∫∫
In regard to the particle’s field, the theoretical results indicate that there are two component
fields[1]:
1-Ep
par, Bp
par:A quasi-stationary field that is the same as the field of a point charge that moves at a
constant velocity but that depends on the delayed velocity. The electric field lines go through the
charged particle.

5
2-Er
par, Br
par:A radiation field that is independent of the previous one. The lines of this field do
not go through the charged particle.
Consequently, we have
)2.6.4(
1
2
1
2
1
2
1
2
1
)1.6.4(
2222
222
2
222
2
∫∫∫∫∫
∫∫∫∫∫




 •+



+



=



 +=




 •+



+



=



 +=
dVBBdVBdVBdVBBdVB
dVEEdVEdVEdVEEdVE
r
par
p
par
r
par
p
par
r
par
p
parpar
r
par
p
par
r
par
p
par
r
par
p
parpar
µµµµµ
ε
εεεε
.
In The Feynman Lectures on Physics [3], the reader can find an explanation of the
electromagnetic mass, me, and see that it implies that particles are actually objects of a defined
size that can vary. Under these conditions, for velocities that are much lower than the speed of
light, we have
( ) ( ) .)7.4(
2
1
2
1
;
2
2222
∫∫ == dVBvmdVEcm p
pare
p
pare
µ
ε .
In principle, values of the electromagnetic mass are defined for stationary fields; however, we can
assume that the fields evolve relatively slowly so that a quasi-static approximation is applicable,
which enables us to express Poynting’s theorem as:
In this result, the left hand side of the equation includes a sum of terms associated with the
interaction of the external field with the particle’s own field and with the interaction between the
external field and the particle’s radiation field. On the right hand side of the equation, we have
terms that are associated with the kinetic energy, including the electromagnetic mass, and terms
associated with the radiation field. Among these, a term that represents the interaction between
the charge’s own field and the radiation field appears; in principle, this is the term that is related
to the self-deceleration force.
5. THE DYNAMICS OF AN ACCELERATED POINT CHARGE. CAUSALITY.
After the introduction of the electromagnetic mass, it seems necessary to attribute an internal
structure to that which we have considered a point particle. However, we introduced the kinetic
energy of a point particle, and logical coherence tells us that it should be possible to continue the
line of reasoning using this hypothesis. In this sense, we introduce the following relationships:
1-The variation of the energy of the radiation field inside the integration sphere: It seems evident
that this variation is calculated by taking the difference between the radiated energy at time dt
close to the charge, which we can associate with the instantaneous acceleration, and the energy
that is lost when it travels through the surface of radius cτd, which is associated with the delayed
acceleration, as follows:
[ ] [ ] )1.5()(
6
)(
622
2
3
0
2
2
3
0
222
d
r
par
r
par ta
c
q
ta
c
q
dVBdVE
dt
d
τ
πεπε
εε
−−=









+




∫∫
.
2-The variation in the energy in the particle’s own field, which is related to the electromagnetic
mass, inside the integration sphere: In the Berkeley Physics Course on Electromagnetism [5], the
reader can find an intuitive calculation of the energy variation of a particle’s own field in the
appendix on the radiation of an accelerated charge; that is, the field lines go through the charged
particle. According to this reference, we can write the following result, which is equivalent to the
above equation:
[ ]2
3
0
222
22
)(
6
1
222
1
)8.4(
11
d
r
par
p
par
r
par
p
par
r
par
r
parcee
r
parext
r
parext
p
parext
p
parext
ta
c
q
dVBBdVEE
dt
d
dVBdVE
dt
d
Evmcm
dt
d
dVBBdVEE
dt
d
dVBBdVEE
dt
d
τ
πεµ
ε
εε
µ
ε
µ
ε
−+









 •+



 •+









+



+





++
=





•+•−





•+•−
∫∫∫∫
∫∫∫∫

6
[ ] [ ] )2.5()(
6
)(
62
1 2
3
0
2
2
3
0
2
22
dee ta
c
q
ta
c
q
vmcm
dt
d
τ
πεπε
−−=





+ .
Substituting this into the equation we derived using Poynting’s theorem, we obtain
[ ] [ ] [ ] 









 •+



 •+







 −−++
=





•+•−





•+•−
∫∫
∫∫∫∫
dVBBdVEE
dt
d
tatata
c
q
dt
dE
dVBBdVEE
dt
d
dVBBdVEE
dt
d
r
par
p
par
r
par
p
par
c
r
parext
r
parext
p
parext
p
parext
µ
ετ
πε
µ
ε
µ
ε
1
)()()(
6
)3.5(
11
222
3
0
2
.
A first approximation of the acceleration terms is
[ ] [ ] [ ] [ ] )4.5(2)()()()(
2222
dd
dt
ad
atatatata ττ •+≈



 −−+ ,
but note that this first approximation is the same as the approximation of the square of the
acceleration not delayed but advanced.
[ ] [ ] )5.5(2)()(
22
dd
dt
ad
atata ττ •+≈+ .
This situation can affect the principle of causality because even to a first approximation, we
expect results that are consistent with causality. When introducing, as a first approximation in the
principle of energy conservation, the value of a parameter at a future time, the evolution of the
physical system depends on the future conditions under which, in accordance with the principle of
causality, we suppose that cause always precede effects. To avoid this situation, Poynting’s
equation must provide us with an alternative that removes the “non-causal” term in the first
approximation. We can see that the integral associated with the interaction between the radiation
field and the particle’s own field can do this. If we assume that the interaction between the
external field and the radiation field is also negligible, or we include it in the corresponding term
(the first eq.), decoupling Poynting’s equation generates two other equations. According to our
approach, we consider the parameter τd approximately constant so that
[ ]
[ ] )7.5(0)(
6
1
)6.5()(
6
1
2
3
0
2
2
3
0
2
=







+



 •+



 •
+=





•+•−
∫∫
∫∫
d
r
par
p
par
r
par
p
par
c
p
parext
p
parext
ta
c
q
dVBBdVEE
dt
d
ta
c
q
dt
dE
dVBBdVEE
dt
d
τ
πεµ
ε
πεµ
ε
.
It is straightforward to interpret eq. (5.6) according to the principle of conservation of energy: at
time dt, the first term corresponds to the energy transferred by the external force, and the second
is the sum of the kinetic energy variations and the radiated energy. Equation (5.7) corresponds to
a coupling between the charge and the radiation emitted by it so that the energy associated with
the coupling between the radiation field and the particle’s own field is directly related to the
radiation process of the particle. To calculate this integral directly, it is necessary to know the
internal structure of the particle; however, the development that we have followed raises the need
for this coupling integral to have a determined and non-negligible value in the first
approximation. Formally, the previous result corresponds to a constant of motion with dimensions
of energy or mass. In the line of this study, we interpret this relationship as a way of
compensating so that the mass of the particle remains constant. We can see that the causality
condition lead us to the mass constancy.
If by following classical mechanics, as in the second eq.(5.7), we introduce a force f for this
coupling, we can write
)8.5(0
3 3
0
2
=•+• da
dt
ad
c
q
vf τ
πε
.

7
When the particle is at rest at time t-τd, we have aτd≈v, and we can estimate the coupling force as
)9.5(
3 3
0
2
dt
ad
c
q
f
πε
−= .
Aside from a factor of ½ and the minus sign, this force is similar to the one given in equation
(2.5).
For an inertial observer at instantaneous rest with respect to the accelerated charge, the radiation
emitted by the charge at that time is emitted with symmetry such that its mechanical impulse is
null [2]. When the relativistic transformation of the mechanical impulse is applied, we obtain
)11.5(
1
0
2
2
2
c
v
c
v
p
Pl
−
∆−∆
==∆
ε ,
and therefore, for any other inertial observer, ∆p=v∆ε/c2
, where p and ε are the mechanical
impulse and the energy of the emitted radiation, respectively, and v is the particle’s velocity. If we
assume that the radiation comes from the particle’s kinetic energy, we can then take [∆p, ∆ε] as
components of the particle’s energy-impulse. If we divide (5.11) by the corresponding time, ∆t,
and we include all of these in the equation (5.9) for the radiation force and use the Larmor’s
fórmula, we obtain in the limit
)12.5(0
3
2
23
0
2
2
=+⇒−===
dt
ad
a
c
v
dt
ad
c
q
dt
d
c
v
dt
pd
f
πε
ε .
Note that according to the fundamental dynamic equation, a force that is proportional to the
velocity is a sign of a variation in the mass,
)13.5()(
dt
vd
m
dt
dm
vvm
dt
d
dt
pd
+==
From this perspective, we see that the effect of the radiation coupling force is to restore the lost
mass. Taking the scalar product of eq. (5.12) and the acceleration, dv/dt, we obtain
( ) )14.5(00
0
2
222
2
2
=














+








⇒=•





+• ∫
t
vd
c
a
dt
vd
dt
d
dt
vd
v
c
a
dt
vd
dt
vd .
Note the similarity of this equation with eq. (5.7). This similarity suggests that eq. (5.13) is only
applicable when the external fields are constant. It is evident that if the radiation phenomenon did
not occur, the acceleration of the particle in a constant electric field would be constant, a=qE/m,
neglecting relativistic effects. Therefore, in the case of a constant electric field, the only cause of
variations in the acceleration is the emission of radiation. If the field is not constant, it is evident
that the particle’s acceleration can vary in response to the external field and that the radiation is
only responsible for variations in the difference between the real acceleration of the particle and
the acceleration caused by the external field. Therefore, we generalize eq. (5.13) as follows:
[ ] )15.5(0
2
=





+





×+− v
c
a
BvE
m
q
a
dt
d
extext
From eq. (5.15)1
for constant fields, we can determine the following integrals immediately:
1
The final form of this equation will be shown in the section entitled Quantum interpretation.

8
[ ] ( )
[ ] [ ] ( ) )17.5(
2
00
)16.5(
2
exp
2
1
ln0
1/
0
22
2
2
0
2
0
2
0
2
2
0
2
22
22222
vBa
m
q
B
v
m
q
av
dt
d
Bav
m
q
dt
ad
vvv
c
a
Bav
m
q
dt
ad
v
c
vv
aavv
ca
a
dt
dv
ca
dtda
dt
dv
c
a
dt
da
va
c
a
Ba
m
q
dt
ad
a
extextextext
ext
•=







+×⇒=××−×⇒=×





+××−×







 −
−=⇒−−=





⇒=+⇒





+⇒=•





+







×−•
According to eq. (5.17), when B=0 or B and v are perpendicular, there is a constant vector that is
perpendicular to the instantaneous velocity, and therefore, the motion develops on a plane. If B
and v are not perpendicular, then, the motion cannot be considered planar. Continuing the
derivation using the result given in eq. (5.16), we obtain
)18.5(0
1
0
22
22
222
2
22222
=+





+⇒=





+
dt
da
dt
dv
cdt
vd
c
a
dt
ad
dt
dv
c
a
dt
da
which tells us that, if they exist, the extreme values of the square of the acceleration and the
square of the velocity are related, i.e., a maximum of a2
corresponds to a minimum of v2
and vice
versa. However, note that eq. (5.18) does not establish the existence of these extremes.Note that
according to our line of argument, in principle, eq. (5.6) for the conservation of energy is only
valid when the external fields are constant. This suggests approximating the power transferred to
the charge as
)19.5(
66
2
3
0
2
2
3
0
2
a
c
q
vama
c
q
dt
dE
vEq c
ext
πεπε
+•=+=•
Recalling that the kinetic energy is proportional to v2
, we see that the kinetic energy minimum in
eq. (5.19) must result in a non-zero velocity so that the previous expression does not result in a
kinetic energy minimum of zero. We can imagine a positive charge that initially moves against an
electric field. It is evident that the deceleration due to the field ensures that the kinetic energy
reaches a minimum at some time, and if the motion is rectilinear, it is not easy to understand why
the kinetic energy cannot become zero at some time. In this way, we have not been able to give a
concrete solution to eq. (5.6) by following a classical description of the dynamics of an
accelerated charge. However, eq. (5.15) does allow a dynamic solution in a simple case, and the
results can be contrasted with eq. (5.19), as we will see in the following section.
6. RECTILINEAR MOTION OF A POINT CHARGE IN A CONSTANT ELECTRIC
FIELD. BROWNIAN MOTION?
In this case and for a positive charge, the velocity, the acceleration and the electric field maintain
the same constant direction so that eq. (5.15) can be written in terms of modules, and because the
acceleration does not cancel out, we can divide the expression and integrate it as follows:
)1.6(
1
0
1
222
qE
m
c
x
ac
v
dt
da
a
−=+−⇒=+
where x is the position of the particle in the direction of motion ;initially, x=0 and the acceleration
is Newtonian. From the previous equation we obtain
)2.6(1ln
2211
2
2
2
0
2
22






+=−⇒
+
=⇒
+
= ∫∫ mc
qEx
c
vv
dx
mc
qEx
qE
dxma
mc
qEx
qE
ma
Approximating the logarithm of a small displacement with respect to x=0 or for E close to zero,
we obtain the expression
)3.6(....
3
1
2
1
2
1
2
1 3
42
33
2
2
22
2
0
2
+







−







+−≈ x
cm
Eq
x
mc
Eq
mvmvqEx ,

9
where we immediately recognise the form of eq. (5.19) with the potential and kinetic energies,
and by cancelling terms, the energy emitted in the radiation is
)4.6(
6
....
3
1
2
1 2
3
0
2
3
42
33
2
2
22
ta
c
q
x
cm
Eq
x
mc
Eq
πε
≈+







−






 ,
and the acceleration, a, has an average value. At the limit in which the electric field tends to zero,
we assume that the average acceleration also tends to the Newtonian value: as E→0,a→qE/m.
Therefore, we consider the following limit
)5.6(
3
1
3
6
....
3
1
2
1
0
2
2
2
2
0
3
0
22
3
42
33
2
2
22
0 t
mc
q
x
tq
mcx
t
c
aq
x
cm
Eq
x
mc
Eq
LimE
πε
πε
πε
≈⇒==
+







−







→
Note the reader that the above limit, if any, should be independent of how it reaches him : taking
the field E in the +x direction or the opposite -x. This consideration suggests that the most
appropriate is that the value found x2
corresponds to a mean square value of a particle that can
move with equal probability in both directions +x and -x. Thus equation (6.5) is similar to the
brownian motion of a particle with a constant average kinetic energy. Therefore, even though it
performs an intrinsic behaviour, a free charged particle appears as if it were moving in a viscous
medium and affected by random impacts.
7. QUANTUM INTERPRETATION.
Equation (5.15) is not physically satisfactory because the mechanical impulse of the radiation
should be proportional to the charge, q, as is the case of the radiated energy in Larmor’s formula.
We generalize the equation by introducing a dimensionless factor β as follows:
[ ] )1.7(0
2
=





+





×+− v
c
a
BvE
m
q
a
dt
d
extext β ,
which is the dynamic equation for an accelerated charge. The mathematical consequences of this
formula are the same as those of eq. (5.15) with the change c2
→β-1
c2
. In the limiting case in
which a constant electric field tends toward zero in eq. (6.5), the result becomes
)2.7(
2
exp 2
2
0
2
0 






 −
−=
c
vv
aa β
In the limiting case in which a constant electric field tends toward zero in eq. (6.5), the result
becomes
)3.7(
3
1
1
3
6
....
3
1
2
1
0
2
2
12
2
0
3
0
22
3
422
33
2
21
22
0 t
mc
q
x
tq
mcx
t
c
aq
x
cm
Eq
x
cm
Eq
LimE
πεββ
πε
πε
ββ
≈⇒==
+







−







−
−−
→
.
When we derive the result, we obtain
)4.7(
6
1
0
2
c
q
xmv
πεβ
≈
Interpreting this equation according to Heisenberg's Uncertainty Principle reveals that the
following constant is an admissible value for β:

10
)5.7(;
4 0
2
h
h
≈∆∆=⇒= xpxmv
c
q
πε
β ,
where ħ is Planck’s constant. In addition, the reader can observe that for the case of the
elementary charge(e) β is the fine-structure constant, α, which yields
)6.7(
44
2
0
2
2
0
2
α
πεπε
β Z
c
e
Z
c
q
===
hh
,
where Z is the number of elementary charges, e, contained in the charge, q. Consequently, in eq.
(5.15), the square of the charge is a factor of the radiation impulse term. Therefore, if we have
two charges with the same velocity and kinematic acceleration, the one with more charge emits
more radiation, which is in agreement with Larmor’s formula.
8. ANALYTICAL COMPATIBILITY OF THE EQUATIONS.
Generalizing eq.s (5.19) and (7.1) for a force f(t), we obtain the following system of equations:
( )
mc
q
v
c
a
m
f
a
dt
d
amvfam
3
0
2
0
2
2
0
6
;
)2.8(0
)1.8(0
πε
τ
β
τ
=









=





+







−
=+•− .
If we derive the first energy equation with respect to time and replace the first term of the second
equation in the result, we obtain
( ) ( )
)3.8(21
02102
02
2
02
2
0
dt
ad
m
c
v
amf
dt
ad
mf
c
v
ama
dt
ad
amfamafam
dt
d
v
τβ
τβτ
+







−=
⇒=








+−







−•⇒=•+−•+−•
.
It is clear that this last equation is not compatible with the result given in eq. (5.16) when f is a
constant force. If the accelerated particle starts at rest, its velocity increases over time, and
according to eq. (5.16), its acceleration decreases over time and the derivative of the acceleration
is both negative and decreasing over time. In contrast, in the previous expression, if the velocity
increases over time and the acceleration decreases over time, the derivative of the acceleration
can only increase over time. Therefore, eq.s (8.1) and (8.2) are incompatible. A possible solution
is that this incompatibility is due to relativistic effects that have not been considered in the energy
equation although the dynamic equation already includes them. Let us derive a first relativistic
correction to the kinetic energy
( )( ) )4.8(
2
3
1
2
3
1
2
1 2
2
22
2
2
2








+•≈•=⇒××−







+≈⇒







+≈
c
v
avm
dt
pd
v
dt
dE
vav
c
m
c
v
am
dt
pd
c
v
vmp c ,
and for the power radiated, we have
( ) )5.8(
1
31
2
2
2
2
2
0 





×−







+≈ av
c
a
c
v
m
dt
dW
τ .
Therefore, the system of equations is now
( ) ( )









=





+







−
=





×−







++•





+−
)7.8(0
)6.8(0
1
31
2
3
2
2
2
2
2
2
02
2
v
c
a
m
f
a
dt
d
av
c
a
c
v
mva
c
v
mfam
β
τ

11
By deriving the first equation and using the second equation, we obtain
( )
( )
( )
0
1
6312
2
3
1
3
2
3
)8.8(0
6
312
2
3
1
3
2
3
2
2
2
2
022
2
02
2
22
2
2
202
2
02
2
22
2
2
2
=




















×−
+






















××
+







++−













−++••+








•
⇒=
•
+•







++•








−







++•








•++−
v
c
av
c
a
m
c
dt
ad
vv
dt
ad
c
v
mf
c
v
amvav
c
m
a
dt
ad
c
vm
v
a
c
av
m
dt
ad
a
c
v
maf
c
v
amvaav
c
m
dt
ad
c
vm
va
c
m
ττβ
ττ
β
Disregarding terms that contain v2
/c2
(or that are lower by a factor of sine or cosine), as a first
approximation, we obtain
)9.8(26 02
2
0
dt
ad
mv
c
a
mamf ττ ++≈
Note that the first two summands of the second term can be interpreted as the derivative of the
mechanical impulse (p=mv) with respect to time, which includes an increase in the mechanical
mass. Removing the second summand of the right hand side, based on eq. (8.7), we obtain
)10.8(
3
12
66
2 0
00
0
dt
ad
mam
dt
fd
f
m
f
a
dt
dm
dt
ad
mamf 





−+≈−⇒







−−+≈
β
τ
β
τ
β
τ
τ .
We can integrate this result by multiplying by an exponential factor, exp{(t-τ)/τ1}, taking τ1
conveniently and integrating by parts as follows
If, for low velocities, we assume that the force, the mass and the acceleration are invariant with
respect to Galilean transformations, then each of eq.s (8.6,8.7),separately, turn out not to be
Galilean invariant. Note that the fundamental law of dynamics for a particle with variable mass
would be also not a Galilean invariant law if it only includes the particle’s mechanical mass;
however, if the entire system is considered, including the split matter that moves with velocity vs
(5.13) becomes
( ) )12.8(
dt
vd
m
dt
dm
vvF s +−= ,
which is the correct form for the fundamental law of dynamics and a Galilean invariant law due to
the appearance of a relative velocity v-vs. The same occurs with eq. (8.11) or eq.s (8.6,8.7) as a
system, which is a Galilean invariant law.
{ }
{ } { }
)11.8(exp
3
)(
exp
3
)(exp)(exp)(
exp
2
exp)(
exp2exp)(
1
exp)(expexp)(
1
exp)(exp)(
exp2expexpexpexp)(
6
1
111
11
0
1
1
0
111
1
1111
1
1
1
0
1
1
11
1
1
0
1
∫
∫∫
∞
∞∞∞





 −
+=
⇒




 −
=−=




 −
−=












 −
−





 −
=












 −
−
⇒




 −
+













 −
+













 −
−




 −
=













 −
+













 −
−




 −
⇒




 −
+




 −
−




 −
=




 −
−




 −
⇒=
t
ttt
d
t
d
adm
amtf
d
t
d
adm
amtf
t
famd
t
fam
d
d
t
d
admt
fam
d
d
t
d
ad
m
t
a
t
a
d
d
m
t
am
t
f
t
f
d
dt
f
t
dt
ad
m
t
d
ad
m
t
am
t
d
fdt
f
τ
τ
τ
τ
β
τ
τ
τ
τ
β
τ
τ
ττ
τ
τ
τ
τ
τ
τ
ττ
τ
τ
τ
τ
τ
τ
τ
τ
τ
τ
τ
τ
ττ
τ
τ
τ
τ
τ
τ
τ
τ
τ
ττ
τ
τ
τ
τ
τ
τ
τ
τ
τ
τ
τ
τ
τ
τ
τ
τ
τ
τ
τ
τ
τ
τ
τ
β
τ
τ

12
9. FLUCTUATIONS.
As shown, eq. 8.11 is contradictory. We see that the corresponding integral times τ extends to
greater than the observation time t, which would be a problem in order to maintain the principle
of causality in which this work is based. On the other hand the integral value is determined, at a
time dt, from the equation; assuming known mass m, the force f and acceleration a at that instant.
However the characteristic time τ1 of eq. 8.11 equals to 4ћ/mc2
, into the quantum physics domain
; and we must abandon the idea of knowing accurately the relation a(t), f(t) at this time scale.
Based on the relationships of energy-time uncertainty of quantum physics, the concept of present
can not be reduced to a geometric point in the coordinate timeline, but we must accept an
amplitude or uncertainty in this concept. In this context we can interpret that the limits of
integration in 8.11 correspond to this amplitude over time. On time scales far above ћ/mc2
seconds have a net fluctuation of acceleration: ∆a, and linear dynamic eq. 8.11 combines the
external force, the average acceleration and the fluctuation
)1.9(exp
3
;)(
1
∫
∞





 −
≅∆∆+=
t
d
t
d
ad
aamamtf τ
τ
τ
τ
β
Now we review the development done from the beginning. The hypothesis that radiation comes
from a particle’s kinetic energy leads us to consider the fluctuation of the mass of a radiating
particle. The particle loses and regains mass. If the mass is lost at a rate that is higher than the
recovery rate, then the difference is related to the kinetic energy that contributes to the radiation.
However, if the rate of mass loss is different than the recovery rate, then the process cannot be
continuous in time because the particle’s existence would be compromised if it were. Therefore,
the radiation emission process must fluctuate; that is, it must be discontinuous in time for the
particle to have a margin of recovery. In regard to the radiation, the classical model of a point
charge assumes that the particle has infinite electromagnetic energy and so infinite mass. In this
model, a continuous loss of mass should not be a problem because the mass of the particle
remains infinite; however, there is an evident mechanical contradiction because an infinite mass
cannot be accelerated by a finite external force. In contrast, we have the Lorentz force, in which
the particle’s mechanical mass is clearly finite. For consistency, we must study the case in which
the radiation is emitted by a particle of finite mass, and this leads us to consider a discontinuous
radiation process when the charged particle must maintain a stable mass. Under these conditions,
the Lorentz force and Larmor´s formula can only be statistical averages in reality. For both the
kinetic energy and the radiated energy of a particle with constant mass, there is a relationship with
the mechanical impulse; however, a direct transfer from the kinetic energy and impulse to
radiation is not possible, as can easily be observed in the following reductio ad absurdum where
the subscript 1 refers to a particle of constant mass and the subscript 2 refers to the radiation.
)2.9(,;; 12
2
1221122211 E
c
v
EPPEEE
c
v
PPvE ∆=∆⇒∆−=∆∆−=∆∆=∆∆•=∆
According to our hypothesis, the problem is that the radiation is associated with a fluctuation in
the particle’s mass that, on average, must keep the particle mass constant. When we introduce a
small fluctuation of the mass, δm, into the relativistic energy U , we obtain
( ) ( ) ( )( ) )3.9(22 422224222242222
mcmcpmcUmcUmcmmccpcmmcpU δδδ +≈+−⇒++≈++= .
For velocities that are much lower than the speed of light, it is valid to suppose U+mc2
≈2mc2
, and
therefore,U-mc2
≈p2
/2m+ẟmc2
; if we compare this result with eq. (5.19), we obtain qEx=U-mc2
,
and for the mass fluctuation,
)4.9(
6
2
3
0
2
2
ta
c
q
mc δ
πε
δ ≈ .

13
Evidently, this mass fluctuation, which also appears in eq. (8.1), grows over time, and if it were
not emitted in the form of radiation, it would necessarily affect the particle’s stability. Therefore,
the dynamics of the particle includes phases in which it accumulates mass and phases in which it
releases the excess mass as radiation. Equation (8.9) indicates an effective increase of mass and
then τ1 is a characteristic time of the accumulation phase. Equation (2.7) and Larmor´s formula
indicates that τ0 is a characteristic time of the release phase. In a constant electric field, the
accumulation phases provoke decreases in the acceleration by increasing the mass, and the release
phases provoke increases in the acceleration by decreasing the mass. Thus the mass fluctuation
entails a fluctuation of acceleration. If the accumulation phases are longer than the release phases
on average, then the average effect of the process is similar to that of a friction force that tends to
decrease the particle’s velocity and acceleration. Equation (7.1) provides a way of measuring this
phenomenon by relating variations in the acceleration to the amount of radiation emitted. In this
context, eq. (5.19) must be interpreted as an average over the appropriate time interval. In this
way, we can understand the apparent discrepancy between eq.s (8.1) and (8.11) when the
acceleration is constant : If there are no variations in the acceleration, then there are no mass
fluctuations, and no radiation is emitted. This outlook is stable if τ1>> τ0 ; in appendix 12.5 we
can see a limit in this inequality and a possible relation with phenomena in Solar Corona.
The concept of mass fluctuation is similar to what some authors call Schott energy or
acceleracion energy:
“Changes in the acceleration energy correspond to a reversible form of emission or absorption of
field energy, which never gets very far from the electron.”[9]
10. RADIATION AND ACCELERATION IN A GRAVITATIONAL FIELD.
In the last section we found that If there are no variations in the acceleration, then there are no
mass fluctuations, and no radiation is emitted. If the particle’s acceleration is entirely due to a
gravitational field, then fluctuations in its mass cannot affect its acceleration. This follows from
the equivalence principle of general relativity, and therefore, a charge that is accelerated by a
gravitational field does not experience the friction due to radiation; that is, the radiation will not
cause variations in the particle’s acceleration. Therefore, for an accelerating charged particle, we
draw the following conclusions:
1-If the acceleration is due to a gravitational field, then the charge does not emit radiation.
2-The charge emits radiation only if its acceleration is due to an external electromagnetic field or,
in general, a force that acts on the charge.
Equation (5.19) relates the kinetic energy and the radiated energy, but we have seen that there is
no lineal direct physical relationship between these two energy forms. We can think that the
average needed to arrive at eq. (5.19) goes through a non-linear and irreversible process. One
sign of this non-linear process is that eq. (7.1) contains Planck’s constant and eq. (5.19) does not.
This non-linear process confronts the classical idea of energy as the displacement of forces and is
related to the principle of causality, as we saw in the section on the dynamics of an accelerated
charge.
There is a well-known image in which an observer inside a box accelerated by an external motor,
assumed to be unobservable , is unable to distinguish between the box’s real physical state and a
state of rest in a uniform gravitational field. The equivalence principle, which is sometimes
explained using this image, assumes a local physical symmetry between a gravitational observer
and a different accelerated observer.
Therefore, we can imagine, for example, that by experimenting, the observer arrives at the
following physical law: for any physical object to remain unaccelerated, a force of magnitude mg,
where m is the object’s inertial mass and g is the inertial acceleration, must act on it. Invoking the

14
symmetry of the equivalence principle shows that this is also true for an observer at rest on the
earth’s surface when g is interpreted as the intensity of the gravitational field and m is interpreted
as the mass or gravitational load. The relevance of the symmetry is that the inertial mass and the
gravitational mass are always equal to each other, which has been accepted without explanation
since Newton’s time. Therefore, the acceleration due to gravity is the same regardless of the mass
and the physico-chemical nature of the object; these facts have been known since Galileo’s time.
However, there are physical phenomena that seem not to obey the symmetry between gravity and
acceleration. One in particular is the radiation emitted by an accelerated charge, which is known
in classical electromagnetic theory. In this case, we begin with a charged object that is at rest.
According to the theory the charge in the accelerated box emits radiation. However, the symmetry
leads us to suppose that a charged object that is at rest on the earth’s surface emits radiation. It
seems that this does not occur, based on our experience on earth and knowledge obtained from
astronomical data. Also, the origin of the radiated energy should be justified. The gravitational
observer clearly does not appreciate the origin of this possible energy, and the accelerated
observer would conclude that the source or origin of the radiated energy is not inside the box;
instead, it comes from the external object that causes the box to accelerate. That is, this observer
would give a non-local explanation for the radiation process.
The symmetry that we refer to is restricted to constant accelerations and gravitational fields. If we
apply the dynamic equation, eq. (7.1), to a charge that moves with constant acceleration a0 in an
external electric field, E, we obtain
)1.10(2
020
2
02
2
020 ra
cq
m
EErda
cq
m
Edva
c
E
m
q
a
dt
d βββ
+=⇒=⇒−=





− .
In contrast, if we apply the equation of the conservation of energy, eq. (5.19), to the field, we
obtain
( ) )2.10(02
00
2
0200 =+•−•− amvra
c
mvEqam τ
β .
When we derive this expression twice, we see that the result only makes sense if the
acceleration,a0, is zero, which indicates that there are no charges that move with constant
acceleration due to the effect of an electric field.
Imagine a classical pendulum with a charge on its end oscillating under the effects of string
tension and gravity. According to the quantum theory of harmonic oscillators, states that cancel
out the emitted radiation are possible. The tension is related to the chemical bonds that are
between the molecules of two immediately adjacent parts of the string, and therefore, it is a force
of quantum nature rather than an electromagnetic force.
These ideas allow us to think that there is no radiation emitted for an accelerated observer or for a
gravitational observer, and thus, the symmetry is maintained.
11. CONCLUSIONS.
The classical electromagnetism has not correctly included the role of mechanical mass; and this is
the origin of the problem on the radiation of an accelerated charge. In electromagnetism you can
easily describe a point charge in arbitrary motion; but a point charge, according with Einstein´s
mass-energy equivalence, has an infinite mass, and therefore can not be accelerated. A movement
that is arbitrary and non-accelerated is absurd. A charge distribution is unstable if only electrical
forces act and the best example is nuclear forces, but what is the corresponding force on the
electron? This paper suggests that the mechanical mass of a charged particle fluctuates, so they
are acting both tendencies: instability and recovery in a compensatory process. This leads us to
the acceleration in Larmor’s formula should be relative to an external electric and/or magnetic
system. We have seen that this compensation is a non-linear process that depends on Planck's

15
constant and, in this sense, it is of quantum nature. However we have also seen that
electromagnetism is not totally alien to this compensatory mechanism and can describe at the
same time a stable mass of the particle and physical causality. A sign of this aspect can be seen in
Lenz's law: the currents associated with induced electromotive forces in a conductor by altering
the external magnetic flux, generate magnetic fields which, in turn, tend to cancel external
magnetic flux disturbance; and therefore the electromagnetic system reacts trying to restore
equilibrium conditions. The stability we have found is similar to thermodynamic equilibrium
because of the competition between opposing tendencies; this balance is conditional and may
change or break for charges and accelerations sufficiently high, conditions that can happen in the
solar corona.
12. APPENDIX.
12.1. Constant Electric Field
Equation (7.1) can be represented using the intrinsic coordinate system associated with the
particle’s trajectory. If the unit vectors are T(tangent), N(normal) and B(binormal), then applying
the Frenet formulas results in
)1.1.12(
1
0
232
2
222
Tv
c
a
BT
v
N
v
dt
d
N
v
dt
dv
T
dt
vd
Tv
c
a
N
v
T
dt
dv
dt
d






−=





+−+







++⇒=





+







+ βτ
ρρρρ
β
ρ
,
where ρ is the curvature and τ is here the torsion of trajectory. The equality of the vectors requires
the torsion to be null, and therefore, the particle’s trajectory is planar. In the component N,
)2.1.12(0 3
2
kv
v
dt
dv
dt
dv
=⇒=







+ ρ
ρρ
,
where k is a constant; therefore, the curvature is proportional to the cube of the velocity.
Combining this result with eq. (5.16) and using the acceleration components, we obtain
( ) ( ) ( )
( ) )3.1.12(1/exp
2
1/exp
2
1/exp
1
/exp
22222
2
22222
2
22222
2
22
222
2
2
2
2
−−±=
⇒−−±=⇒−−=











⇒−−=







−=





cvvkA
kdt
dv
cvvkA
dt
dvk
cvvkA
dt
dv
kv
vk
cvA
v
a
dt
dv
β
βββ
ρ
This expression is in principle directly integrable, and v2
(t) can be obtained. As a result, we see
that there is a case in which the velocity decreases with time, as it should for a particle with an
initial velocity that opposes the electric field, and cases in which the velocity increases with time.
As we see in Fig. 1 there is a lower bound on the velocity module of approximately 1/kA. In this
way, the solution explicitly eliminates velocities of zero, which is consistent with the problem
about kinetic energy discussed in eq. (5.19).
Fig. 1 :dv2
/dt versus velocity modulus

16
Note the reader that zero velocity in this case would mean repose regarding the fountains of the
external electrostatic field.
12.2. Constant Magnetic Field.
Using the equation of energy, eq.(5.19),and the result given in eq.(5.16), we obtain
( ) ( ) ( )
( ) ( )
)1.2.12(
6
;21ln
21exp2exp
2)(expexp
2
1
2
exp;0
3
0
2
0
2
002
2
2
0
2
2
002
2
0
2
2
2
00
0
2
0
2
2
2
2
00
2
0
2
0
2
2
2
0
2
2
2
00
2
2
0
2
20
2
0
mc
q
ta
c
c
vv
ta
c
vv
c
tavv
c
c
tavdvv
c
vv
c
am
dt
dv
mvv
c
aaam
dt
dE
v
v
v
v
c
πε
ττ
β
β
τ
ββ
τ
β
β
τ
ββ
τ
β
τ
=





−+=
⇒−=−





−⇒−=





−
⇒−=





−⇒





−−−=⇒





−−==+ ∫
under normal conditions the argument of the logarithm is very close to 1 and can approximate
)2.2.12(2 2
00
2
0
2
tavv τ−≈
12.3. Deflection Of Charged Particle By Coulomb Centre Of Force :
Bremsstrahlung.
Rutherford´s famous experiment on atomic structure was based on the deflection of charged
particles by centres of force associated with the atomic nuclei in an ultrathin layer of gold.
Evidently, the acceleration generated by the centre of force on the particle causes radiation to be
emitted, which should produce an increase in the deflection of a trajectory that we can consider
approximately hyperbolic, in accord with the Newtonian formalism of forces. If we assume that
the accelerated particle’s motion is planar, we can use polar coordinates in the complex plane to
write eq. (7.1) as follows:
)1.3.12(33233
;;2;
2
22
3
2
2
3
3
2
2
32
22
3
3
2
222
22
2
2






+−=








−





−++++−





−
⇒−=





−=
















++





−=





+=
dt
d
ir
dt
dr
a
cdt
d
r
A
dt
d
r
dt
rd
dt
d
dt
d
r
dt
d
dt
dr
i
dt
dr
r
A
dt
d
dt
d
r
dt
d
dt
dr
dt
rd
va
cm
f
a
dt
d
e
r
A
m
f
e
dt
d
r
dt
d
dt
dr
i
dt
d
r
dt
rd
ae
dt
d
ir
dt
dr
v iii
θβθθθθθθθθ
βθθθθ θθθ
Let us take the real component of the previous equation and approximate a constant angular
momentum, as in the Newtonian model, as follows:
)2.3.12(;;02
2
3
3 2
24
2
23
22
3
3
dt
d
mrLe
r
A
m
f
a
dt
dr
r
A
cdt
dr
r
A
dt
d
dt
d
r
dt
d
dt
dr
dt
rd i θβθθ θ
==≈=++





−





− ,
where we also approximate the square of the acceleration in the radiation term using its
Newtonian value. Removing the derivative of the angle from the angular momentum L, we obtain
)3.3.12(0
3
3
1
02
3 23
2
2
2
2
2
2
34
2
2
2
2
3
3
=








−







−−⇒=+







−+ −−−−
Arr
c
A
m
L
dt
rd
dt
d
dt
dr
Ar
dt
dr
r
c
A
m
L
dt
rd ββ .
Applying the initial conditions for r very far from the centre of force and multiplying by dr/dt, we
continue the integration process as follows:

17
)4.3.12(22
3
0
32
1
2
1
0
3
3
1
0
3
3
1
12
2
2
2
22
12
2
2
2
22
23
2
2
2
2
2
2
23
2
2
2
2
2
2
BArr
c
A
m
L
dt
dr
Arr
c
A
m
L
dt
dr
dt
d
dt
dr
Ar
dt
dr
r
c
A
m
L
dt
dr
dt
rd
Arr
c
A
m
L
dt
rd
=+







−+





⇒=








+







−+





⇒=−







−−⇒=−







−−
−−
−−−−−−
β
βββ
Removing the variable dt from the angular momentum, L, we obtain
)5.3.12(
1
3
1
1
2
22
3 222
222
4
2
12
2
2
2
22
2
Bm
r
Am
rLc
mA
d
dr
rm
L
BArr
c
A
m
L
mr
L
d
dr
=+
















−+





⇒=+







−+




 −− β
θ
β
θ
.
The reader can check the similarity of this result with the Newtonian case in [6]; it allows us to
write the following after solving for A and β and relating B to the energy E:
)6.3.12(
4
1
3
1
1
2 0
2
2
22432
4
2
E
r
eZZ
rL
ZZ
d
dr
rm
L rmrm
=±






















−+





πε
α
θ
h ,
where α is the fine-structure constant, ħ is Planck’s constant, Z is the number of elementary
charges, the subscript m refers to the moving charge and the subscript r refers to the charge that is
at rest. The positive sign represents repulsion and the negative sign represents attraction between
the particle and the centre of force. As in [6], we find the corresponding Binet’s differential
equation, which is
)7.3.12(
1
;0
43
1 2
0
22243
2
2
r
u
L
meZZ
u
L
ZZ
d
ud rmrm
==±














−+
πε
α
θ
h ,
and the corresponding solution for the trajectory is
)8.3.12(
43
1cos
4
21
2
0
222432
2
0
2
2
L
meZZ
L
ZZ
L
meZZ
L
mE
r
rmrmrm
πε
α
θ
πε
m
h














−


















+=
,
with a negative sign for repulsion and a positive sign for attraction between the centre of force
and the particle. If we call the trajectory’s Newtonian deflection ∆N, we see right away that the
effect of the radiation is an increase in the deflection,∆,
)9.3.12(
3
1
3
1;)2/tan(
2243
22432






−
∆
=∆⇒∆=





−∆=∆
∞
L
ZZL
ZZ
LP
meZZ
rm
N
N
rmrm
N
h
h
α
α
,
where P∞ is the modulus of the linear momentum very far from the centre of force. According to
this result, the corrective term is only appreciable in practice when the angular momentum, L, is
on the order of Planck’s constant,
)10.3.12(pxvmbL ∆∆≈≈= h ,
where b is the straight arm of the angular momentum or the distance between the centre of force
and the asymptote. The previous relationship indicates that the moving particle passes very close
to the “fixed” centre of force, at a distance comparable to the De Broglie wavelength of the
moving particle. Therefore, at this level, the phenomenon must be studied using quantum
mechanics as an individual photon emission process. Tipical data on the alpha particles in
Rutherford’s experiment are as follows: mass = 6.64424×10−27
kg and initial velocity = 2×107
m/s. These data lead to b≈8×10-16
metres. In the case of an alpha particle aimed directly towards a
gold nucleus (which, therefore, has L=0), the maximum approach, r, can be calculated using the
conservation of energy; the result is
)11.3.12(
42
1
0
2
2
r
eZZ
mv rm
πε
= ,
which assumes that r≈2.7×10−14
metres, which is very far from the previous value of b; therefore,
the corrections presented here are negligible in the case of Rutherford’s experiment.

18
12.4. Simple Oscillator.
For a charged particle oscillating along the x axis subjected to a force F=-kx, we can present eq.
(7.1) as
( ) )1.4.12(; 22
2
2
m
k
va
c
xa
dt
d
=−=+ ω
β
ω .
This equation can be easily integrated as follows:
)2.4.12()(tan
)(arctan
1
0
0
2
0
2
0
2
2
2
2








−−=
⇒−−=








⇒−=








+
⇒−=+ ∫∫
xx
c
c
a
xxa
c
c
dx
a
c
da
va
c
v
dt
da
x
x
a
βω
β
ω
ω
ω
β
β
ω
ω
ω
β
β
ω
where x0 is a point of null acceleration for the particle. We can see the physical meaning of x0 by
Fig. 2, that depicts two Cartesian axes x-y and a charged particle moving to the right (dashed
arrow) on the x axis. The two arms of the particle radiation are also depicted; and, with
continuous arrow, the forces acting on the particle. The solid arrow to the right corresponds to the
force F = -kx. In the path of the particle between the ends [-A, A] will be an average reaction
force (arrow to the left) associated with radiation in a half cycle. For small values of the
argument of the tangent function in (12.4.2) we have -ω2
x(t)+ ω2
x0 ≈a so that x0 appears
related to the average radiation force in a half cycle. This point x0 also affects energy as equation
(5.19) tell us in this case
)3.4.12()(tan 2
0
2
0 avxxx
c
c
τω
βω
β
ω
=








−








−
and to a first approximation for a small argument of the tangent, we have
)4.4.12(2
00
2
avx τω −≈
This equation can not be verified, instantly, for a classical oscillatory movement (low radiation),
as it does not reflect that an increase in the acceleration module entails a fall in module speed.
Assuming valid eq. 12.4.4 on average, the energy conservation needs x0v product is always
negative. Changing the sign of the velocity at the end of each half cycle involves a change in the
position x0 to either side of the origin of coordinates and (12.4.4) qualitatively predicts the
behaviour shown in the drawing. Therefore the value x0 must be considered variable over time:
x0(t), similar to a square wave and the average eqs. (12.4.2;12.4.4) that mentioned above refers to
each half cycle where both, speed direction and point x0, are held constant. In the half-cycle
change we must assume a discontinuity, changing the initial condition x0.
We can calculate the dynamic equation using the following approaches:
A-A
Fig. 2 :A point charge oscilating
x
y

19
1-Simple harmonic motion of the particle:
)5.4.12()cos(;)(;)cos(;)( 32
tA
dt
da
tAsenatAvtAsenx ωωωωωωω −≈−≈≈≈
2-Simple harmonic motion of point x0 and average value of eq. (12.4.4)
)6.4.12(;)cos( 0000 AAtAx ωτω ≈−≈
this last approach is equivalent to the first component of Fourier series corresponding to the
square wave x0(t). We develop the dynamic equation as follows





≈++
≈+−
⇒−≈≈−≈
≈+−
)8.4.12(0
)7.4.12(0
)cos()(
)()(:.
2
0
2
2
0
0
2
00
2
0
2
0
22
xva
x
dt
da
a
v
dt
da
tAtx
atxtxeqDynamic
ωτω
ωτ
τωτωωτωω
ωω
these two equations are equivalent in our approach and we can see for the first (12.4.7) the shape
of the classical equation (2.8).
We can imagine the general motion of a charge affected by a force as a set of half cycles with
change of initial conditions at the ends. In each half cycle conservation of momentum and energy
is verified on average. If the time duration of these half cycles is small enough, at a microscopic
scale, the particle appears to move by shocks, although on average according to the direction and
intensity established by the external force. In my opinion Heisenberg's relations reflect the way
nature creates these half cycles.
12.5. Solar Corona.
The corona is the outermost layer of the solar atmosphere; it contains intense magnetic fields and
plasmas that contain highly ionized atoms, such as iron atoms that lack 19 electrons [7]. The
radiation emitted from the corona generally has high frequencies; ultraviolet light and X-rays
predominate in its spectrum. All this leads researchers to assign a temperature of millions of
degrees to the solar corona, which leaves us perplexed if we recall that the temperature of the
sun’s surface is approximately 5.000 celsius. One would expect a progressive decrease in the
temperature as the distance increases, as is the case with a bonfire.
In our approach, we can see expression τ1=6τ0/β (see eq. 8.11) as the relation among time of
accumulation phase (τ1) and release phase (τ0) and it should be τ1>>τ0; inequality that has a clear
limit
)1.5.12(281376
6
1
66
201 ≈×≈<<⇒>>=⇒>>
ααβ
ττ Z
Z
the presence of highly charged ions means a value of Z elevated, and this may affect to the time
balance among the phases in the radiation dynamic. Also, the presence of an intense magnetic
field and magnetic reconnection phenomena[8] favour an increase in acceleration and then
radiation braking. The emitted radiation can also increase the ionization of the particle or of other
particles from the inner layers of the solar atmosphere (spicules). In this way, the presence of
intense magnetic fields in the corona can provoke a combination of high-frequency radiation and
particle velocities that are relatively low for the expected temperature. The classical explanation
of the Fraunhofer lines in the solar light spectrum implies that there are radiation-absorbing cold
gases in the external layers of the sun.

20
REFERENCES
[1] Landau L D and Lifshitz E M (1975) The Classical Theory of Fields (Oxford: Elsevier)
[2] Bredow M M, Rumyantsev V V and Toptygin I N (1985) Classical Electrodynamics (Moscow,
Nauka).
[3] Feynman R, Leighton R and Sands M (1965) :The Feynman Lectures on Physics. Vol 2.
Reading,MA,Adisson-Wesley
[4] Wangsness,R.K., (1979),Electromagnetic Fields, John Wiley and Sons.
[5] Purcell E. M. and Morin J. (2013), Electricity and Magnetism (Berkeley Physics Course, Vol. 2),
Cambridge University Press.
[6] Goldstein H. (1980) Classical Mechanics (2nd ed.). Addison-wesley.
[7] About the Solar Corona: http://en.wikipedia.org/wiki/Corona,
https://www.youtube.com/watch?v=Hg5tla7s-ys,
https://www.youtube.com/watch?v=HFT7ATLQQx8
[8] About Magnetic Reconnection: nasa.gov May-12-2016: NASA Directly Observes Fundamental
Process of Nature for 1st Time Science Journal DOI: 10.1126/science.aaf9987
[9] Dirac, P. A. M. (1938).Classical theory of radiating electrons, Proc. Roy. Soc. London A167, 148
AUTHOR
Enrique Cantera del Río is Graduate in Physical Sciencies and Technical Engineer in Telecommunications
by Valladolid University (Spain).

Radiation of an Accelerated Charge

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