• c test charge
C ● --> ----------------------
charge field of C
The test charge sees the field of C as if C would have been moving at a constant velocity. The velocity in this example is 0. This is the usual assumption about retardation.
In our blog we assume that there is extra inertia associated with c when it moves in the field of C.
Hypothesis. The test charge c feels the extra inertia as if C would have been moving at a constant velocity.
How to prevent a perpetuum mobile?
Retardation seems to open the doors for a perpetuum mobile.
+ ● ---> <--- ● +
Suppose that we have two charges with the same sign. We suddenly move them closer to each other. Each charge sees the "old" field of each other, which is weaker than the field would be if it would be updated infinitely fast.
The force when we move the charges is less than the force which will repel them later? Do we have a perpetuum mobile?
No. The fields of each charge "bend" when we move them. We have to do work to cause the bending. We cannot recover all that work after the process is completed. That spoils the perpetuum mobile.
/ \ field lines = "wires"
/ \
+ ● ● +
This can be understood through a wire model of the fields. The repulsion between the charges come from stresses of the wires. We can cheat with the repulsion for a moment by moving the charges quickly, but we have to pay the price of bending the wires. No extra energy can be gained.
A test charge c close to a large stationary charge C; a mass-energy shipment
This is a common configuration. The test charge feels a force field which is static. The Coulomb force is equivalent to the one which we could calculate from the energy of the combined field of c and C.
If we think in terms of the field energy, the static field magically "knows" how much the field energy would change if we moved c.
Our hypothesis above claims that the static field magically knows the inertia of such a movement.
more field energy
• + c -->
less
● + C
more
We have marked in the configuration how a test charge modifies the field energy density of the large charge C. Suppose that we suddenly move c right. The energy flows in the field. How could the field know beforehand how much energy flow there will be, and be able to tell to c how much inertia there should be?
The test charge c changes the energy density by
~ E E',
where E is the field of C and E' is the much weaker field of c.
The test charge c has to emit a "mass-energy shipment" to the the field, so that the energy can be shipped around the right distances.
A mass-energy shipment might be defined:
s W,
where s is the distance and W is the amount of energy. Alternatively, as
s m,
where m is the mass shipped.
Conservation of the center of mass is the associated property. A shipment must be balanced by an opposite shipment.
Is there some analogy in mechanics which could simulate a shipment in the electric field? Any electromagnetic wave which c creates is very weak, since the field of c is very weak. The process can only leak minimal energy and momentum into space.
Suppose that we have a tapering rope which becomes very narrow far away.
^ movement
|
####====----- . . . . rope
It might be that if we input a shipment to the heavy end of the rope, that makes the entire rope to move a fixed distance up in the diagram. That is, the heavy end "feels" the entire inertia of the rope immediately.
1. Only minimal energy can escape as radiation when we move the test charge c.
2. The common field of C and c cannot remain oscillating. It becomes static very quickly.
Items 1 and 2 suggest that the field does behave like the tapering rope. The field can relay shipments but it cannot oscillate.
Compare the tapering rope model to a simple rod:
^
|
================ rod
If we move the end of the rod suddenly upward, we do not feel the entire inertia of the rod immediately. The rod will start to oscillate. But such oscillation has not been observed in static electric fields. This has to be a wrong model.
Why does the test charge feel the field energy instantaneously? The Coulomb force
Let us ask the question: why does the test charge c in the field of a large charge C instantaneously "see" how much the combined field energy will change when we move c? That is, how can the Coulomb force predict the required energy?
We may appeal to the same argument as for the extra inertia. Let us move c quickly closer to C. If we could input less energy than is required to update the entire field, then there would be oscillation. But such oscillation has not been observed experimentally.
We must input the right amount of energy when we move c. That energy is then shipped to the appropriate locations in the combined electric field.
What would be a mechanical analogue for this energy flow process? For inertia, it was the tapering rope.
●/\/\/\/\/\/\●
If we push the ball on the left quickly, the spring carries the energy to the right end. This mechanism might be analogous to the combined electric field of c and C. The energy is shipped as "pressure waves".
Conclusions
We presented a hypothesis that a test charge immediately feels fully the extra inertia caused by an external electric field.
We argued that the hypothesis is reasonable, and might be the only way that extra inertia can work. Other mechanisms would imply that the static electric field of the charges could oscillate, but such oscillation has never been observed.
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