● Q
• q ----------->
o/
|
/\
observer
Let the observer lower the negative test charge q close to Q. The inertia of q is reduced by
2 |U| / c²,
where U is the potential energy of q in the field of Q?
This does not make sense. There is energy flowing in the common field of Q and q. A better hypothesis is the following.
A new conjecture of the inertia of a test charge q inside the electric field of another charge Q
Corrected conjecture about inertia inside an electric field. If Q and q have the same sign, and the potential of q is U, then the inertia of q in a radial movement relative to Q is
2 U / c²
larger than in empty space. In a tangential movement it is U / c² larger. However, if the field of Q is canceled by a charge -Q close to it, then there is no excess inertia. If Q and q have a different sign, then the excess inertia is
2 |U| / c²
in a radial movement and |U| / c² in a tangential movement.
Thus, an opposite electric field can reduce the extra inertia which would come from an electric field. This makes a lot of sense. The new formulation of the inertia rule says that the energy in the combined field of q, Q, and Q- is not "private". Also, the new formulation says that any interaction increases the inertia of q, or keeps it constant. This sounds good.
Assumption about paradoxical momentum exchange. When q and Q have the same sign, Q is static in the frame, and q gains more inertia I in the electric field of Q, that paradoxically speeds up the velocity of q by a factor
1 + I / m,
where I is the gained inertia and m is the inertial mass of q. If Q and q have different signs, then there is a slowdown of the velocity of q by a factor
1 - I / m.
The slowdown sounds reasonable. If Q is sitting still, and q gains more inertia from Q, we expect the velocity of q to slow down. The paradoxical assumption is required to make the magnetic field to guide q to the right direction when Q and q have the same sign. We have to think about this. Can we find a reasonable explanation for the momentum exchange in the paradoxical case?
How "private" is the Coulomb interaction?
v <-- • e-
=============== wire
^ V
|
• -----> V'
q = e-
Let q approach with the velocity V. The magnetic force accelerates q sideways. To explain the magnetic force, we first have to calculate the "inertia force" for the field of the electrons in the wire and then subtract the inertia force of the protons in the wire.
The inertia force is "private" in the movement of the electrons. We cannot first sum the electric fields of the electrons and the protons, since there would be no electric field at all. But at the end of the calculation we are allowed to subtract the inertia in the field of the protons from the inertia in the field of the electrons. In this phase, the interaction is "public".
The privacy definition is equivalent to traditional electromagnetism. A "moving" electric field induces a magnetic field. One can subtract electric fields, but that does not cancel the magnetic field.
The problem of the 1-2-3 wire: the electric field
On October 17, 2023 we tried to figure out the field of a mass flow of this form:
1 3
\ / ^ v
\ / /
\__________/ • e-
2
^ V
|
• q = e-
m
Let us analyze the analogous electric wire. The mass flow is replaced with an electron flow. The parts 1 and 3 point directly at q.
In our November 14, 2023 blog post we assume that v² is very small, and we can ignore it. Then γ = 1, and the corrections come from the product v • V in the Lorentz transformation formulae above, and from the inertial mass of the test charge q:
m + 1/2 (V ± v)².
Let us analyze the part 1 of the wire. The electrons there are moving toward q at a velocity v, while the protons stand still. In the comoving frame of the electrons, q is moving at a larger velocity,
V + v.
The Coulomb force Fc accelerates q in the comoving frame. Switch to the laboratory frame: the acceleration has a correction from the different inertial mass, as well as from the acceleration of the moving charge q.
We proved in the previous blog post that the Biot-Savart law holds for 1: there is no force on q. The Coulomb part in the laboratory frame is:
ac = (Fc (1 - v • V / c²) - V (Fc • v / c²))
/ (m + 1/2 m V² / c²),
and the "inertial force" part:
ai = (-V (Fc • V / c²) + V (Fc • v / c²)
+ v (Fc • V / c²))
/ (m + 1/2 m V² / c²).
In the Coulomb part, "1" is canceled by the Coulomb force from the protons. The acceleration term
-Fc (v • V / c²)
points away from the part 1. The term
-V (Fc • v / c²)
points directly down. We can say that the Coulomb force of approaching electrons pushes q down, relative to the attraction of the protons. This sounds reasonable.
The first term in the inertia force is canceled by the protons. The second term cancels the third term of the Coulomb force. The third term cancels the second term of the Coulomb force. The inertia force pulls q up. This is a result of the paradoxical momentum exchange. Without the paradox, q would be pushed down.
The magnetic effect by the part 1 on q is zero. Is this a coincidence, or somehow "required" about a magnetic effect?
In this analysis, we analyzed the field of the electrons "privately". Only at the end we subtract the effect of the protons.
If we would do a "public" analysis, then we would conclude that there is no electric field in the laboratory frame, and consequently, there are no inertia effects whatsoever: no magnetic force.
However, the "public" method fails spectacularly for the part 2: it predicts that there is no magnetic field, which contradicts experiments.
We conclude that the "private" analysis is the correct one in the case of the electric field.
The 1-2-3 mass flow in gravity
In gravity, the analysis of the part 1 of the flow is different from an electric field. A major difference is that the gravity charge of the test mass depends on the velocity of the test mass, while the electric charge of q stays the same.
Another difference is that gravity modifies the spatial metric.
1 3
\ / ^ v
\ dm / / mass flow
\__________/
2
r = distance (dm, m)
^ V
|
• m
Let us compare the gravity of the mass flow to the case where the mass would be static. This is analogous to the electric version where the protons were static and the electrons moved.
Rather than doing the calculation here, let us write a separate blog post where we derive the "Biot-Savart law" for gravity.
Conclusions
We will continue from where we were left in this blog post, and we will derive the "Biot-Savart law" for gravity in a new blog post in December 2023. We will adapt the calculation of November 14, 2023 to include the main difference between gravity and the Coulomb field: the "gravity charge" of a test mass is m / sqrt(1 - V² / c²).
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