In a laboratory, one can create potential differences of up to 32 megavolts. Let us imagine that we put an electron to an electric potential which makes it potential energy much less than -511 keV. How does such a strange electron behave?
An electron inside a charged metal sphere, or close to it
|
_____
/ \ metal sphere
--- | • e- |-----
\_______/
|
E symmetric electric field
If the electron is put inside a metal sphere, it will influence (polarize) charges into the sphere, such that its field will be centered at the center of the sphere. Then the inertia of the electron is only 511 keV inside the sphere. This may explain why experimenters have not noticed that the inertia of the electron depends on the potential. They should create the potential with static charges and remove all metal from the experiment setup.
Moving the electron tangentially close to the surface of the sphere induces an electric current to the surface, which creates a magnetic field B. This does increase the inertia, but then we interpret that the inertia is in the current.
An electron approaching a charged metal sphere
What about an electron approaching a charged metal sphere?
e- ● | ● e+ "mirror image"
metal wall
Close to a flat metal wall, the field of an electron, and the charges it displaced in the metal, will appear as a dipole field, because the field has to be normal at the wall. There is a "mirror image positron" on the other side of the wall.
If the metal wall is replaced by a metal sphere, then the charge which the electron repelled cannot escape to infinity. The repelled negative charge goes to the other side of the sphere.
We guess that the field of the electron and the metal will look like that the metal sphere "stole" some charge from the electron and put it to the center of the sphere.
The inertia of the electron in a tangential motion relative to the sphere may be unchanged, 511 keV?
e- ● --> v | × |
sphere
Let the electron approach a negatively charged metal sphere. Then there is energy shipping to the field of the sphere from the kinetic energy of the electron. This energy is shipped, on the average, to the center of the sphere? Does the energy shipping increase the inertia of the electron if it moves radially?
The kinetic energy is at the electron. The field which gains energy is centered on the center of the sphere. The energy flow is visible in the Poynting vector.
Can one "grab" the energy flow? Then the flow would contain some momentum which is visible to outside observers. The energy flow shows up in the gravity field as the mass-energy of the system moving to the right. It is plausible that one can "grab" it, and it does increase the inertia of the approaching electron. It may be possible to measure this effect.
Experimenters may have thought that the changes in the inertia are due to the kinetic energy of the electron, or due to currents in the metal. The changing inertia may have escaped their detection.
Conjecture. A charge q in a slow radial motion toward a charged metal sphere shows a larger inertia than in empty space.
Negative energy in an electron
If the metal sphere has a large positive charge, then an electron inside it has a negative total energy.
Our reasoning above says that the electron will still appear to have a positive inertia of 511 keV, because the field of the electron & the metal outside the sphere does not move if we move the electron inside.
This may explain why the spectrum of an atom does not change with the electric potential. If the potential is created with metal, the electric influence in the metal screens the change in the field?
A nucleus and an atom are not insulators?
Quarks move around inside a proton or a neutron, and protons and neutrons move around in a nucleus.
Electrons move around in the electron cloud of an atom. This suggests that the a nucleus or an atom may behave quite like metal, and does not increase the inertia of an electron in the electron cloud.
If we change the inertia of the electron in the x direction, but not in y or z, does that affect the spectrum of an atom? Yes
Above we speculate that the energy shipping close to a charged metal sphere changes the inertia of the electron in the radial direction. What would happen to the spectrum?
● proton
e- • --> v
There would be no change if the "spatial metric" would be squeezed to the radial direction. But it is unlikely that the electric field if the proton is squeezed. Thus, in the case of an atom, the inertia of an electron cannot be different to the radial direction.
The paradox of the energy flow of an electron circling a nucleus and a charge Q some distance away
--> E' field of electron
<----- E field of Q
● • e- ● Q charged
proton sphere
1 meter
Let us put the atom at a distance 1 meter from a charged metal sphere Q. The electron makes some 10¹⁶ rounds around the nucleus in a second, in which time light moves about 10⁻⁸ meters.
Any energy shipping in the field E + E' can only come over a very short distance.
For very slowly moving charges q outside the sphere, the Poynting vector ships the energy from far away: from roughly the distance of one meter.
How does the energy flow in the combined field E + E' when the electron does its loop around the proton?
Is there an electromagnetic wave which moves energy back and forth in the field E?
The wave generated by the electron has the radial field E' essentially zero at large distances because there are no longitudinal electromagnetic waves.
What is the inertia of the electron like?
Let us imagine an analogous system in newtonian mechanics.
m spring M
• /\/\/\/\/\/\/\/\ ●
<-->
If we accelerate the small mass m very slowly we will observe that its inertia is m + M, because of the spring.
But if we move m back and forth very rapidly, its inertia appears as m.
This may explain the spectrum of atoms close to a charge Q. Since Q is much farther than 10⁻⁸ meters, the electron will not see much extra inertia to the radial direction, or to any direction. The combined field E + E' of the electron and Q will not have time to react to the rapid movements of the electron.
However, we have a quantum mechanical problem here. Let us have a hydrogen atom in the ground state. We bring it close to a charge Q. If the combined field of Q and the electron would "vibrate", that would steal energy from the atom. One cannot steal energy. This shows that there cannot be any visibly changing field of the electron outside the atom.
We have two solutions for the paradox:
1. for any rapidly oscillating electron, the far field with an external charge Q does not have time to adjust to the movements of the electron: Q cannot give much extra inertia to the electron;
2. an atom in a ground state, or in an almost stationary excited state, cannot possess an electric field which oscillates visibly outside the atom: the combined field of Q and the electron cannot give much inertia to the electron.
An electron close to a charge in an insulator
A macroscopic, charged insulator contains a huge number of electrons. Very little polarization makes the insulator to mimic the influence effects that an electron has on a metal object.
If we have a macroscopic charge Q, its electric field cannot exist inside metal, but can exist inside an insulator.
Can even very weak electric fields, like that of an electron, exist inside an insulator? Let us check the literature.
Wikipedia states that an insulator always contains a small number of mobile charge carriers. If we add a significant charge Q into an insulator, do mobile charge carriers still exist?
The literature says that static electric charges are typically in the nanocoulomb range. They do create significant electric potentials. Then mobile charge carriers presumably decide their location based on the potential. If we have an individual electron close to the insulator, it probably cannot affect much the location of charge carrierd.
We conclude that a charged insulator does not behave like metal when we move an electron close to it. It might be possible to measure the inertia change of an electron in the electric field of the insulator.
The negative energy of the electron close to a nucleus, or close to a charged insulator
What would happen if we have an electron at a potential where its total energy is negative, i.e., if its potential energy is < -511 keV?
The rod inside a cylinder example of our August 28, 2024 blog post is a way to study this question.
| | | | | | | E
+ ----------------------------------- cylinder
rod - -------- ---> v
+ ----------------------------------- cylinder
| | | | | | | E
-m "negative mass"
● M
The cylinder is attached to a table at a laboratory. The rod makes a "hole" into the energy density
1/2 ε₀ E²
of the electric field of the cylinder. Is it possible to "grab" the momentum if the hole moves?
Let us make the cylinder into a torus by attaching the ends of the cylinder. Let the rod be long. As the rod moves inside the torus, there probably is angular momentum present in the field E of the torus. One can probably "grab" the angular momentum through the gravity which the field E creates.
Let us put a mass M close to the torus. Let it grab some angular momentum from the rotating energy in the field E. M gains momentum and kinetic energy that way. This implies that the rotation in the field E must have contained kinetic energy.
Our reasoning suggests that, even though the total energy of the rod is negative, as it is located in a very low potential, the rod still has a positive inertia, and a moving rod stores kinetic energy into the rotating field E.
It is like moving a submerged balloon in a swimming pool. The water around the balloon gives inertia to the balloon.
Conclusions
Our analysis assumed that charges move relatively slowly. If an electron is close to a charged metal object, like inside a charged metal ball, the inertia of the electron may not increase at all, since the polarization of charge in metal screens changes in the combined field of the metal object and the electron. This may be the reason why experimenters have not noticed any change in the inertia of the electron.
Quantum phenomena probably screen the change of the inertia of an electron inside an atom.
For very rapid oscillations of an electron, the far field of the electron does not have time to adjust. This screens changes in the inertia in certain cases.
A proton, or a nucleus, may behave somewhat like metal, since there are moving charges in the quarks inside. This may explain why an electron inside a hydrogen atom seems to have a constant inertia.
If an electron is close to a charged electric insulator, we should see a change in the inertia of the electron, since substantial energy in the combined field of the insulator and the electron move around.
Experimenters should measure the inertia of an electron close to a charged insulator. If the inertia does not differ from a free electron, then the existence of the electric "field" as an entity comes into question.
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