There is no acceleration of the electron in the x direction in the case of a charged electric insulator cylinder
negatively charged insulator cylinder
========================== -
^ u W = F Δy
|
-v <--- • e-
|
v F Coulomb's force
^ ^
| |
E E'
^ y
|
------> x (laboratory frame)
Above, E is the electric field of the insulator cylinder and E' is the electric field of the electron. As the electron comes closer to the long cylinder, the electron does not acquire any momentum in the x direction.
The electron loses its kinetic energy as it comes closer to the cylinder. The inertia of the electron grows smaller. But the extra field energy in the field
E + E',
relative to the plain fields E and E', grows as much as the electron loses its kinetic energy. The combination of the electron plus the extra energy in the field retains its inertia. The electron does not acquire any acceleration in the x direction.
The magnetic acceleration in the comoving frame of the electron: the Lorentz magnetic force formula fails for an electric insulator
negatively charged insulator cylinder
========================= ---> v
○ ○ ○ ○ B field lines point
out of the screen
^ u
|
• e-
^ y
|
------> x' (comoving x')
In the comoving frame of the electron, the electron sees a magnetic field B which should accelerate it to the negative x' direction.
But this is inconsistent with the fact that the electron has no x acceleration in the laboratory frame.
What is going on? We proved that the Lorentz force formula is wrong for an electric insulator which is charged. The charges in the cylinder cannot move. It is an electric insulator.
A symmetric charged metal wire: the Lorentz magnetic force formula holds
midpoint wire electrons
===========|=========== -
^ u
|
-v <---- • e- test electron
| | | | | | | E''
^ y
|
-----> x (laboratory frame)
If the charged cylinder is a metal wire, then conducting electrons can freely move inside it. Let the test electron be close to the wire, at the midpoint of the wire. The test electron repels the electrons in the metal. The combined electric field E'' of the system will be roughly constant far away. There is no stronger field at the location of the electron.
Now the electric field does not increase the inertia of the electron in the x direction. The inertia of the electron decreases as it approaches the charged wire. The electron accelerates to the left.
We see that the Lorentz force formula about B holds if the charged cylinder is an electric conductor. The Lorentz force takes into account both Coulomb's force and the backreaction of the electrons in the wire.
If the electron is not approaching the wire at the midpoint, then the situation is more complicated since the extra field energy is stored, on the average, at the midpoint. What happens then?
A single electron inside an electron gas cloud: the inertia is a constant me?
Let us look at the electron gas inside a block of metal. The thermal velocity of electrons is roughly 100 km/s.
If we move one electron e- along a long wire at a speed v, then to keep the field approximately constant, it has to be compensated for by moving a cloud of electron gas of N electrons to the opposite direction at the speed
v / N.
There N is on the order of 10²³. The energy needed for the movement is
~ 1/2 N me v² / N²
= 1/2 me v² * 1 / N,
where me is the rest mass of the electron.
We conclude that the inertia of the single electron e- does not grow appreciably from the effect on the electron gas.
The conclusion: when an electron e- is very close to a metal wire, the "evening out" of the electric field done by repulsed electrons in the wire does not increase the inertia of the electron.
Ultimately, this effect should be measured, since it is a quantum mechanical process, and surprises can occur. An Internet search only returns one experiment where the inertia seemed to change. Apparently, in most cases the inertia change is negligible.
Also, the effect of the electrons in the gas moving at 100 km/s should be studied. The hypothetical electron which we "move", actually moves already at 100 km/s. The other electrons, which it repels, move at 100 km/s, too.
The electric influence of an electron close to a wire: a deficit of electrons worth e+
If we add a single new electron e- into an electron cloud, e- "makes room" for itself by repelling the amount e- of charge around it. There is a deficit of charge equivalent to e+ close to the electron.
deficit of electrons
+ + + + + + +
============|============ wire
midpoint length L
• e-
What about a metal wire and a single electron e- close to it? Let us assume that the wire is very long and the electron e- is at the midpoint.
There is an excess of electrons now at the ends of the wire. Does the excess have any effect? The excess probably is ~ 1/2 e-. The charge density of the excess is
~ e- / L.
The electric field of the excess close to the center of the wire is negligible if L is very long.
On the other hand, the deficit of electrons close to the center of the wire does have a significant effect on the electric field close to the center.
Conjecture. The electron e- repels the electrons in a long wire in such a way that the deficit of electrons close to e- is equivalent to e+.
If the electron e- moves along the wire, it carries a fuzzy "positron e+" along with itself inside the wire.
An electron moving along a negatively charged metal wire: a more detailed analysis
deficit of electrons
"e+"
v <--- + + +++ + +
====================== - wire
negative charge
^ u R = distance(e-, wire)
|
v <--- • e-
| | | | | | E
^ y
|
------> x
The "electric influence" of the electron e- repels electrons in the wire. A "positron e+" travels along with the electron.
The velocity component u takes the electron closer to the wire, and the velocity component u slows down. The inertia of the electron e- is reduced. But how does the inertia of the field, or the electrons in the wire, behave?
The combined electric field E of the wire and the electron e- at distances r > R from the wire has the lowest energy if E is approximately uniform. The field strength
~ 1 / r
and the energy density
~ 1 / r².
The circumference at a distance r from the wire is ~ r. Thus the total field energy is the integral of
~ 1 / r² * r = 1 / r,
which diverges.
The field E is the sum of the field of the electron e-, the "positron e+", and a uniform charge density in the wire.
Most of the "interaction energy" of these fields is at distances r > R?
====================
deficit of
field energy E
v <--- • e-
In the diagram we have marked the volume where the field E has less energy than it would have if the electron e- were not present. Does this "bubble"add much to the inertia of the electron e-?
As the electron moves to the left, field energy must flow to the right, so that the bubble moves with the electron. Does this energy flow possess much kinetic energy?
It is the Poynting vector
1 / μ₀ * E × B
which describes the energy flow. The flow takes energy from the left to the right along quite a straight path. Is there a lot of kinetic energy in this flow? Can we "grab" the flow?
The kinetic energy of flowing liquid
Suppose that we have a metal object M submerged into a water pool. The mass M of the object adds inertia to the object. The water in the pool starts to move if we move M. How much inertia does water add to M?
This is a complicated question.
In the electron gas cloud section above, we simply guessed that the extra inertia is negligible. The total kinetic energy of the system gives us a hunch of what the inertia might be, because
1/2 M' v²
is the energy, where M' is the inertia of the object and v is its velocity.
If M causes a very large mass of water to flow very slowly, then the kinetic energy of that very large mass is negligible. If M is streamlined, the flow of water around it may have very little kinetic energy?
Then we can produce a significant momentum into the liquid, even though we only consume very little energy. How is that possible?
m m
■■|■■
■■|■■
|
|
| -->
lever
The streamlined object may act like a "lever" which gives two large masses m a significant momentum, even though the energy spent is very little. In the lever configuration, there will be momentum both to the left and to the right.
The bubble of a lower field energy density has a constant effect on the inertia of the electron e-?
In the diagram in a preceding section, the electron drags along with it a bubble of a lower field energy density.
Can we harvest energy from the movement of the bubble, and harvest it separately from the movement of the electron?
We can "grab" the bubble at least through gravitation. The bubble is equivalent to a block of negative mass moving superposed to a constant density mass distribution. The flowing liquid example proved that the energy associated with such a movement can be negligible.
----> x ================= - wire
| ^
v r | E bubble
| E'
v
• e-
R = distance(e-, wire)
Let us calculate how much energy is missing from the bubble in the case where there is an electron close to a negatively charged wire.
The field of the wire is
-E₀ / r.
The missing energy density is
~ E E'
in the bubble. Close to the electron,
E' ~ 1 / (R - r)²,
and E is roughly constant. The integration volume is ~ (R - r)² dr. The integral is
~ R - r.
Close to the wire, E ~ 1 / r, E' is roughly constant, and the integration volume is ~ r dr. The integral is
~ r.
Let us calculate a very rough approximate value of the missing energy in the bubble. Let
E = E₀ / r.
The electron is at a distance R. The electron field at the midpoint between the wire and the electron is
E' = 1 / (4 π ε₀) * e- / (R - r)².
The missing energy density at the midpoint is
ε₀ E E' ≈ ε₀ E₀ * 2 / R
* 1 / (4 π ε₀) * e- * 4 / R²
= 2 / π * E₀ e- / R³.
The integration volume at the midpoint is something like
1/4 R² dr.
The value of r varies 0 ... R. We conclude that the missing energy in the bubble is on the order of
1 / (2 π) * E₀
electron volts. This can be compared to the mass of the electron, 511 keV.
In a typical experiment with a charged wire, the field strength at the distance of 1 meter might be at most 100 kV/m. If it affects the inertia of the electron, the effect is somehing like 15 keV, or 3%. The effect does not depend on the distance of the electron. Thus, the effect does not appear as a magnetic effect when the electron approaches the wire.
Conjecture 1. The bubble of missing lower field energy does not have much effect on magnetism.
The electric field at distances larger than R from the wire is roughly uniform, and does not add much inertia to the electron if it moves along the wire.
Conjecture 2. The inertia of the electron in a parallel movement with the wire does not depend appreciably with its distance R from the wire.
If the electron e- approaches the wire at a velocity u, it loses kinetic energy in its velocity component u. Conjecture 2 implies that its inertia is less as it comes closer. But the momentum is constant parallel to the wire. The electron e- accelerates in the direction parallel to the wire. This is seen as a magnetic "force".
Conjecture 3. An electron approaching a charged wire feels a magnetic "force" which is described by Maxwell's equations.
Poincaré stresses
Do Poincaré stresses have any signicance in the case of the charged electric insulator or a wire?
Poincaré stresses counteract the pressure components of the Maxwell stress tensor of the field. In many cases, Poincaré stresses cancel the effects of pressures on the momentum of the system.
Our analyses in the preceding sections have ignored pressures altogether. We have only looked at energy flows. We can justify our choice by claiming that Poincaré stresses cancel all effects of pressures.
But this should be studied in more detail.
Oppositely charged cylinders which are electric insulators: the Lorentz magnetic force formula holds
In the previous sections we studied the magnetic "force" felt by an electron approaching a charged metal wire. The "force" in that case did not change the momentum of the electron, and, therefore, is not a true force. The acceleration happened because the inertia of the electron decreased.
Normally, a magnetic field is produced by an uncharged wire inside which there is a flow of electrons. Let us first treat a simpler case where the electric current is produced by a moving electric insulator. Then there is no electric influence on the charges in the insulator.
charged electric
insulator cylinders
v <--- ---------------------------- + protons
============== - electrons
v <--- ---------------------------- + protons
○ ○ ○ ○ B
attraction
^ F
| |
v F'
repulsion
^ u
|
v <--- • e-
----> x
The outer cylinder is positively charged and is comoving with the electron e-. The inner cylinder is negatively charged and is static. The charges cancel each other out, so that the electric field is zero outside the cylinders. The magnetic field B is non-zero.
The static positive charge attracts the electron (the force F). The moving negative charge repulses (the force F').
In our September 18, 2024 blog post we were able to derive magnetism by assuming that the energy given to e- by F is "moving at a velocity v" to the left, and the energy taken by F' is "static".
Let us analyze this from the aspect of fields.
^ W' energy packet
|
^ u
|
v <--- • e-
<- p' surplus momentum
----> x
In the first section of this blog post, the electron e- loses some of its "u-kinetic" energy
1/2 me u²
to Coulomb's force. Let us denote this energy packet by W'. The packet does not "move" in the x direction. The electron would retain the x momentum in the u-kinetic energy; let us denote it by
p' = W' / c² * v.
But the electron e- has to give p' to the extra field energy in the combined field of the cylinder and e-, and must drag this energy along with itself.
Let us then add the positively charged cylinder to the configuration. One may imagine that the energy packet W' now does not come from the u-kinetic energy of e-, but it comes from the attractive force between e- and the positive cylinder. The packet travels at the velocity v to the left. The momentum p' is now held by e-, and a "magnetic force" accelerates e- to the left.
Is this explanation plausible? Why the surplus momentum p' is not absorbed by the cylinders? Why is it held by the electron e-?
Conclusions
Let us close this long blog post. We will treat later the case where an electron approaches a moving positively charged cylinder.
We would need empirical measurements of the electron behavior close to moving charged insulators. It is difficult to determine the correct model using pure reasoning alone.
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