The equivalence of magnetic and kinetic energy
An electric charge placed in vacuum produces an
electrostatic field that surrounds the charge. When an observer moves
relatively to a charge, the observed electrostatic field changes in time. The observer
moving relatively to the charge will, apart from measuring an altering
electrostatic field, measure a magnetic field due to the relative movement to
the charge. The presence of a magnetic field indicates magnetic energy.
For an observer, moving relatively to a mass,
the relative speed of the mass presents kinetic energy. Kinetic energy exists
only if there is relative movement.
Kinetic and magnetic energy are comparable in
the way that both forms of energy exist only when there is relative motion of
mass respectively electric charge to an observer.
An electron is a mass and charge that moves
relatively to an observer and therefore presents kinetic and magnetic energy.
We will consider only non-relativistic velocities, because relativistic conditions
unnecessarily complicate the situation without adding any additional insight.
The questions I want to be answered are:
How much energy does the magnetic field of a
moving charge present?
What is the relation between the magnetic and
kinetic energy of a charged mass?
The magnetic energy of a single moving charge
An electric current induces a magnetic field in
the surrounding space. The magnetic energy of an electric current is described
by the formula 
(Joule), where L is the magnetic induction coefficient
of the electric circuit and I the electric current.
The magnetic energy Wm of an electric
current tends to conserve the electric current. Only when there is electric or
magnetic resistance the current I will decline in time and the magnetic
energy Wm will be “lost” and the electric current I will
eventually disappear completely.
An electric current normally consists of an
infinite number of moving electrons. There are however no theoretical
objections to an electric current consisting of one single moving charge. In
the electron theory of the Dutch scientist H.A. Lorentz an electric current Ids
(amp.m) induces a magnetic field dH (ampere/m) at a distance R
(m) equal to:
Figure 16.
The magnetic field of a current IdS.
The total magnetic field an electric current
induces at P is the summation (integration) of all the magnetic
fields dH each moving individual electron in the electric
circuit induces at P.
Theoretically the current IdS can
exist of one moving charge Qe, because the total magnetic field H
at P is the summation (and by approximation, when there are infinite
electrons, the integration) of the magnetic field of all individual electrons
passing through the electric circuit at the same moment.
When the current IdS consists of
only one moving charge than:
IdS=QeVe [charge.m/sec] IVedt=QeVe Idt=Qe [charge]
In the case of a single moving charge Idt=Qe,
where Qe is the charge of that single electron. The current I is
no longer dividable, so IdS=QeVe is the differential limit
of an electric current.
When the electric current IdS is
presented by a single charge Qe, moving relatively to P(x,y,z)
with speed Ve, the magnetic field H at P(x,y,z),
due to current IdS=QeVe, is according to the electron
theory of Lorentz:
Let us consider a bulb shaped charge Qe
with radius Re. Because in nature energy always tries to minimize the
energy level, the charge Qe will be situated at the surface of the bulb
(Re). In the figure below, the situation is sketched, where charge Qe
is at rest and the movement of the charge is revealed by the relative speed V.
Figure 17. The magnetic field of a moving
charge.
When an observer moves relatively to Qe
with speed Ve and wants to determine the magnetic field Qe
is inducing in the surrounding space, the observer can choose any coordinate P(x,y,z),
compared to the position of charge Qe (0,0,0).
Because there is only one moving charge the
magnetic field H in P(x,y,z) is simply
determined by means of the electron theory of Lorentz:
and 
The energy density of the induced magnetic
field at P(x,y,z), in vacuum, is:
The magnetic energy dWm, for the
observer, in volume 
is:
Integrating for da and db gives:
This is the energy of the induced magnetic
field in the bulb shell
at a radius R from the center of the charge Qe
and a relative speed Ve.
When the radius of the charge Qe is Re
the total energy of the induced magnetic field surrounding Qe, becomes:
Wm is the magnetic
energy the relatively moving (Ve) bulb shaped charge (Qe) with
radius Re induces in the surrounding (vacuum) space of the observer.
We mentioned a current that consists of only
one electron. The above mentioned is however valid for any single relative
moving charged bulb. The single charge can be any (metallic) charged bulb. The
induced magnetic field B at R(x,y,z) can
therefore be verified in an experiment according to the equation:
Qe is then, in the
above equation, the total charge of the bulb, Ve the relative speed of
the charge to the magnetometer and R the distance to the center of the
charge.
To be able to relate the magnetic energy Wm
of the moving charge to the electrostatic energy of Qe, we have to
consider the potential electrostatic energy of a bulb (Re) shaped charge
Qe. The electrostatic energy of a charged (Qe) bulb (Re)
in vacuum is given by the formula:
For the observer, moving relative to charge Qe
with speed Ve, the total energy (Wt) the charge presents is the
sum of magnetic (Wm) and electrostatic energy (Wp):
Wt=Wm+Wp
Considering 
we derive:
Wt is the total energy
the moving charge presents to an observer: the electrostatic energy and the
dynamic energy. Considering the mass Mp the electrostatic energy Wp
presents:
Substituting the equation for the electrostatic
mass Mp with the formula for the total energy Wt of the moving
charge we derive:
The magnetic energy (Wm) of the moving charge,
expressed in the mass equivalence (Mp) of the electrostatic energy,
becomes:
This derived formula for the magnetic energy of
a moving charge is remarkable considering the kinetic energy (Wk) of a
“normal” mass Mp, moving with relative speed Ve, is:
The moving electron and magnetic energy
The classical radius or Compton radius (Rc)
of an electron is calculated by means of the Compton-equation:
meter
With Me the rest mass of the electron, Qe
the elementary charge of an electron.
The equation for the magnetic energy of a
charged (Qe) bulb (Rc) moving with a relative speed Ve,
is:
When substituting the mass equivalence, Me,
of the Compton-equation in the formula for the magnetic energy Wm and
considering 
, we
get:
So when we assume the electron has the radius Rc,
derived with the Compton-equation, the magnetic energy of the moving electron
presents energy equal to the kinetic energy of that same electron.
A moving “pure electrostatic mass” presents
magnetic energy according to
, while the magnetic energy
of a moving electron with the Compton-radius Rc has a magnetic energy of
.
What causes the difference?
Considering the Compton-equation 
and the electrostatic energy of a charged bulb 
, we observe that the difference between the formulas is a
factor two. The Compton-equation “stores” twice as much energy as a charged
bulb with the same charge and radius. This difference explains exactly the
difference between the equations for the magnetic energy 
, for the moving charged bulb, and 
for the moving electron with the Compton-radius.
The Compton-equation “stores” twice as much
energy as the formula for a charged bulb. We know that apart from a charge, the
electron also has a spin. The magnetic spin of the electron is not considered
in the Compton-equation.
In the previous chapter “The Electron” the
total energy of an electron Wt, at rest, is presented by the formula:
In this presentation of the total energy of an
electron at rest, half the energy is presented by electrostatic energy 
, consistent with the energy of a charged bulb, and the other
half of the energy by 
, the magnetic spin energy of the electron.
Calculating the radius with this equation, the
radius Re of an electron becomes 
meter: exact the
Compton radius.
The magnetic energy of a moving electron, when we
consider half the intrinsic energy of the electron at rest, is presented by
electrostatic energy and the other half by the magnetic spin energy, will be:
When half of the intrinsic energy of the mass
of the electron is presented by the electrostatic energy and the other half by
the magnetic spin energy, the calculated magnetic energy of the moving electron
is equal to the kinetic energy of that electron.
Because the kinetic energy of an electron is 
and at the same time the magnetic energy (Wm) is also 
, the kinetic energy of the electron must be the same energy
as the magnetic energy. Otherwise the conservation law for energy is violated
every time an electron is accelerated or slowed down.
The Electromagnetic Mass
In the chapter “The magnetic energy of a
single moving charge” we demonstrated, with the help of the Electron Theory
of Lorentz, that the magnetic energy (Wm) of a single moving bulb charge
is equal to:
, where Mp is the mass
equivalence of the electrostatic energy of the charged bulb according to 
.
In “The moving electron and magnetic energy”
we calculated theoretically the magnetic energy of the “classical electron”. We
derived the following equation for the magnetic energy of the moving electron:
Let us suggest that the electron, a charged
mass, can be presented by the energy/mass of a charged bulb and a, not yet
identified (mechanical) part of the mass.
The electrostatic mass of a charged bulb with
the Compton radius explains exactly half the energy/mass of the electron. The
magnetic energy of the moving charge equals the kinetic energy of the moving electron.
The question to be answered is: “What kind of energy/mass presents the other
half of the intrinsic energy of the electron?” We refer to the chapter “The
Electron”, where the magnetic energy of the spin of the electron is calculated
at exactly half the intrinsic energy of the electron.
The magnetic spin energy is responsible for and
explains the physics why the charge of an electron is confined. The expanding
force of the charged bulb of the electron is compensated by means of the
contracting force of the spinning magnetic field surrounding the electron. The
charge of the electron is trapped.
The above presented E&M physics for the
electron is however inconsistent with the QM perspectives and therefore must be
addressed. We refer to the QM perspectives in “Lectures on Physics” part II
chapter 28 “The Electromagnetic Mass” by Feynman. In this chapter the
electromagnetic mass of the electron is derived by means of the momentum
density g (Equation 27.21 Lectures on Physics Volume II, Feynman).
g=e0
ExB
According to the QM approach in (28-2) the
magnitude of vector g is:
because the momentum density vector is directed
obliquely toward the line of motion
(*).
Furthermore, and I quote (28-2): “The fields
are symmetric about the line of motion, so when we integrate over space, the
transverse components will sum to zero, giving a resultant momentum parallel to
v. The component of g in this direction is g sin(q), which we must integrate
all over space.” (**)
In the above argumentation (*) and (**) vector p
is thought to be partly compensated by the opposite vector p.
Vector summation is allowed in static situations, where the vectors for example
express the magnitude and direction of a static force. p is
however a dynamic vector, presenting the impulse of the moving mass/energy
density at a certain point.
(For * and ** see Lectures on Physics Volume
II, Feynman)
The total momentum p, according
to the QM approach is then:
where 
is the volume element. The integration over
all space gives:
(Equation 28.3 Feynman).
This equation, expressed in the symbols used in
this article, gives the impulse:
The calculated electromagnetic mass
according to equation 28.3 is 
,
which is 4/3 the mass equivalence of the electric field or 2/3 the mass of the
electron Me.
Although the QM approach differs from the
approach in this article, the outcome should be consistent with each other.
Because there is no consistency between the outcome of both approaches there
must be an omission. In the QM approach the momentum density magnitude of
vector g is diminished with the factor 
,
because of the previous mentioned and marked arguments (*) and (**).
To comprehend the effect of the correction of
the magnitude of the momentum density g with the factor 
we
have to consider the QM equation for the total momentum; the integration of the
momentum density g over space to p, in more detail
(Feynman 28-2):
The motion of the charge v, in
the above equation, is independent of any variable in the equation, so we are
allowed to abstract v out of the integral.
Because 
presents
the mass density (kg/m3) of the energy of the electrostatic field E
and 
is
the volume element, the integration of the formula presents the calculation of
the mass equivalent of the electrostatic field surrounding the charge. The
corrections mentioned in (*) and (**) do not alter the implied physics of the
integration. By integrating the momentum density g around the charge
all over space, the physical interpretation of the integration is the
calculation of the mass of the electrostatic field surrounding the charge. A
mass is a scalar and therefore the correction of the magnitude of the momentum
density vector with factor 
will
be an omission, by which the conservation law for energy is violated. Part of
the mass of the electrostatic field is unjustly ignored.
The magnitude of g presents the
magnitude of the mass density impulse of the moving electrostatic field E
in the direction of v. The direction of vector g
does not present the direction of the movement of the electric field and
therefore not the direction of the impulse. Compensation of the magnitude of g
with factor 
violates
the energy conservation law because part of the mass/energy of the
electrostatic field is then ignored.
The momentum density g at P as a
result of the moving electrostatic field/energy E(r) with v
at r should be:
g=Mr.v
where 
presents the mass density (kg/m3)
of the electrostatic field at r.
Integration of g over space
gives:
Equation (28.3 Feynman) is now:
This equation, expressed in the symbols used in
this book, is:
or 
or
(Me is the mass of the electron and Mp
is the mass equivalence of the electrostatic field).
The corrected total momentum of the moving
charge p, according to the QM approach, is now completely
consistent with the derived equations in this chapter. The correctly derived electromagnetic
mass melec, according to the QM approach, equals now the mass of the electron Me.
Therefore the dynamic energy of the electron with the QM approach is also be
fully magnetic.
Discussion
How is it be possible that the
magnetic energy of an electron is the kinetic energy as well?
The answer to that question is that both forms
of energy are different presentations of the same “dynamic” energy.
If an electron moves and collides with another
particle the change in kinetic energy is transferred from one particle to the
other. The kinetic energy of the electron changes, because the electron moves
now with a different speed. The magnetic energy also has to change, because the
charge of the electron now also moves with a changed velocity.
The same argument is valid when an electron
(electric current) loses magnetic energy through magnetic induction. The
electrons slows down and loses kinetic energy.
When we consider the formulas for magnetic
energy of an electric current and the kinetic energy of a mass we find there
are similarities. The magnetic energy of a current is 
[Joule],
while the kinetic energy of a moving mass is 
[Joule].
Consider an electric current, where the
electrons move twice as fast than the induced magnetic energy Wm will be
four times as large and so is the kinetic energy of the moving electrons in the
current. Both formulas, for the magnetic and kinetic energy, are consistent
with the presentation of equivalence for kinetic and magnetic energy.
The logical consequence of the equivalence for
magnetic and kinetic energy is, that every mass that moves possesses kinetic
energy, also possesses magnetic energy!
The moving proton and magnetic energy
The proton is, like the electron, a single charged
(+Qe) particle. The observable difference between the proton and
electron is the opposite sign of the charge and the difference in mass. In
chapter “The Proton and Neutron” we see that the intrinsic energy of the mass
of a proton can be expressed by the equation:
Where the mass of a proton Mp is
accounted by the electrostatic energy for half the mass and for the other half
by magnetic spin energy. Rp is the radius of the bulb shaped proton.
The magnetic energy of the induced magnetic
field (Wm) by the moving proton will accordingly be:
The moving hydrogen atom and neutron
The proton is a positive charged particle. The
moving proton must posses magnetic energy. But if kinetic energy is the same as
magnetic energy, any moving mass must posses magnetic energy!
The hydrogen atom possesses no electric field
outside the radius of the molecule. All electrostatic energy and therefore all
magnetic/kinetic energy of the moving hydrogen atom concentrate in the atom,
between electron and proton.
The charge of a separated moving electron and
proton is not shielded, as it is in the hydrogen atom. The separated electron
and proton therefore have a much larger range in which the electric fields are
present and can interact with other charged particles when there is relative
movement. The electrostatic field in the hydrogen atom, and therefore the
kinetic/magnetic energy, is contained in the space between the proton and the
electron.
For the moving hydrogen atom the same arguments
as for the moving electron and proton are valid. The kinetic energy of the
hydrogen atom is the induced magnetic energy by the moving electrostatic field
between proton and electron. Outside the hydrogen atom there is no electrostatic
field or dielectric displacement, so outside the atom there is no magnetic
field or energy. The magnetic/kinetic energy is confined to the area of the
electric field, between proton and electron, in the hydrogen atom.
When a proton and an electron fuse to a
neutron, during the fusion process the potential electrostatic energy of proton
and electron is transferred to kinetic energy (magnetic energy). Although
fused, the positive and negative charges of proton and electron still oscillate
in the neutron, so the electrostatic field between both charges still exists,
only now concentrated and therefore confined in the neutron.
The neutron does not posses an electrostatic
field we can observe, because the oscillation frequency of the neutron is far
too high (approx. 2.10^26 Hz chapter “The Proton and Neutron”) to be detected.
Not being detectable does not mean that the equivalence of magnetic and kinetic
energy for a neutron would not be valid.
Comment added November 2007:
When J.J.
Thomson (1881) derived that the EM-theory could not explain the electromagnetic
mass of an electron, during which he brutally violated the energy conservation
law, no one questioned the correctness of his conclusions. His false analysis
was completely copied by QM and served them as proof that the EM-theory was inadequate to describe elementary
particles. Feynman copied Thomson’s mistake in his Lectures on Physics to which
article I refer above.
Why did
QM-physicists never question Thomson’s fundamental false derivation?
The answer to
this question is to my knowledge the relentless faith of physicists in Maxwell’
equations. Maxwell’s equations describe
the interrelationship between the electric field,
the magnetic field,
electric charge,
and electric current. Although Maxwell’s equations
are mathematically correct, apparently no physicist ever verified whether these
equations are also valid in physical sense!
Apparently
QM-physics from the end of the 19th century until today is nothing
more than math. As long as derived mathematical equations look nice and seem
intuitive not to violate fundamental physics laws they are accepted as
physically correct. At least one can say that QM handles “theories” over the
last century in an improper scientific way.
Nowadays many
(top-) scientists know that QM incorrectly disqualified the EM-theory as being
inadequate to describe elementary particles. Actually by now they know the
QM-approach is the fundamentally flawed theory.
Knowing this
these scientists should, if they were scientists living up to the moral and
ethical standards of their profession, address the mistakes. They decide however
that their personal interest is of more importance than the moral and ethical
standards of their profession.