On December 17, 2023 we derived the Biot-Savart law of electromagnetism from the Coulomb force and various assumptions about inertia. We wrote in the blog post that these assumptions are ad hoc – but is that really the case?
1. Assumption about the Coulomb force
This assumption is about how the test charge q "sees" the relative positions of the protons and electrons in the wire.
There is no clear reason why we should use the frame of the protons (= the laboratory frame). It is not "special" in any way.
The only frame "special" for q is the comoving frame of q itself. We conclude that this assumption is not ad hoc, but the only natural assumption.
2. Assumption of the extra inertia of a test charge q inside an electric field of another charge Q
We have discussed this many times in this blog. If q moves tangentially relative to Q, then q "carries" along it field energy to another location. We assume that the energy associated is the absolute value of the potential |V| between q and Q, and it moves along with q. The energy |V| adds to the inertia of q. This is a natural assumption.
If q moves radially relative to Q, then in addition to the moving of the field energy |V| around, there is also a "flow" of energy E to q, or from q, because of the Coulomb force F between q and Q. A good guess for the inertia effect of this energy flow is that it ships energy from Q to q, or vice versa.
The energy flow E doubles the inertia effect of |V| in a radial movement of q.
3. Assumption of the extra inertia of q when there is a current in a wire
Let us have q close to a wire element where an electric current is flowing. The protons and electrons essentially cancel out the electric fields of each other. Because of the canceling effect, we expect that the "flow" of energy E in the preceding section is negligible and it does not cause inertia effects.
However, the potential energy |V| of q in the field of the wire electrons can be assumed to "move" along with the electrons, while the potential energy |-V| of q in the field of the protons does not move.
When q approaches the wire element, it "picks" up more of the moving potential energy |V|. Therefore, q acquires a momentum p.
These assumptions are quite natural.
4. Assumption of a paradoxical momentum exchange
This assumption is not very natural. If an electron q is close to a wire where electrons are charge carriers, then things happen as if the charge carriers would have a positive charge and move to the opposite direction to the electrons. That is, they would be like "holes" in a semiconductor.
Can we find an intuitive explanation for this?
Let us consider the following configuration:
wire
v <-- ● ●
e- proton
^ V
|
• q = e-
The proton pulls q and gives it energy "which does not move" in the diagram. The test charge q puts that energy to the field between q and e-. The energy in that field "moves" to the left. To conserve momentum, the test charge q must itself start to move to the right. This explains away the "paradoxical" part!
Conclusions
We were able to show that the assumptions in our derivation of the Biot-Savart law are not totally ad hoc. We found intuitive grounds for all of them.
This result may help us to understand magnetic gravity. We have been wondering what is the gravitomagnetic field of a mass flow like.
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