UPDATE August 25, 2024: Where is the kinetic energy of the "extra energy" W stored? It may be stored by the field of q being squeezed horizontally by length contraction. More field lines of q become vertical, and strengthen the vertical field of the cylinder.
But what if q moves close to the speed of light? Then almost all field lines of q are vertical. How can we store ever more energy as q approaches the speed of light?
field lines
| | |
q -------- ---> v
| | |
Also: we can make the charge q rod-shaped. Then its field points vertically, and length contraction has no effect on W.
----
The kinetic energy of moving field energy
\ | /
\ | / -
-----|------------------------------
P, v' <-- Q ● m, q • ---> v, p
-----|------------------------------
M / | \ -
/ | \ --> pW
"extra" field energy W
moves with the charge q
Let us have a long uniform cylinder with a uniform negative charge. We put a small negative charge q inside the cylinder. We assume that we have to do a lot of work when we put q inside. Let the work be
W = N m c²,
where m is the mass of q, and N could be 10. Let the mass-energy of the cylinder plus Q be
M = N² / 2 * m.
If we move the charge q inside the tube, we expect most of the inertia of q to come from moving the extra field energy W in the combined field of q and the tube.
We attach a small negative charge Q inside the cylinder, close to q. The charge Q pushes q to the right. We ignore all other forces, except the repulsion between Q and q.
The charge q is accelerated to the right. Let it move relative to the laboratory at a speed v. Let the cylinder move at a speed v'.
The Poynting vector
S = 1 / μ₀ * E × B
is aware that the extra field energy W moves along with the electron to the right.
The electromagnetic action knows about the momentum of W, but is not aware of a possible kinetic energy of W. If the charge q moves a distance s to the right, all the released field energy must go to the kinetic energy of q and its own field. None to W. The released energy is
p² / (2 m) = 1/2 m v².
The total momentum to the right is
(N + 1) m v.
Momentum conservation requires that the momentum of the cylinder to the left is
P = -(N + 1) m v.
The kinetic energy of the cylinder is then
P² / (N² m) ≈ m v².
This is nonsensical. The cylinder is much heavier than the charge q. The repulsion between Q and q did much less work to move the cylinder to the left. But the kinetic energy in the formula is twice the kinetic energy of q.
We assume that the cylinder and its field behaves approximately like any ordinary object of a mass M. This is plausible because q does not alter the field of the cylinder that much.
The paradox of a positron in a very low potential
In laboratory, it is possible to produce voltage differences of up to 32 MV.
Let us assume that we put a positron inside a sphere which has a very large negative electric charge. Then the positron may be in a potential much less than the 511 keV mass of the positron.
Does the positron then have a negative energy? Does it have a negative inertia?
A possible solution: even though the positron formally would have a negative inertia, the electromagnetic action may still make it to move in external fields in a reasonable way, as it would possess a positive inertia. We have to study this question.
Conclusions
No one in the literature claims that Maxwell's equations, or the associated action, would be perfect. There are several long-standing open problems in electromagnetism. The best known one is the renormalization or regulation of the electron field energy. The 4/3 problem is another one.
The question where is the energy of a field located, and how does the energy move, is open for most field theories. But we would need that knowledge, in order to determine the gravity field of another field (or the gravity of the gravity field itself).
Does our thought experiment above prove that Maxwell's equations or the action are flawed? We do not know. Maybe we overlooked something, which would save energy conservation and momentum conservation. We will analyze this further. In this blog we have been very interested in the inertia of energy flowing in a field. The example above suggests that we have to include the inertia of the extra field energy W to the inertia of the charge q. But what happens if W is negative?
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