On December 17, 2023 we claimed that we can derive the Biot-Savart law from Coulomb's force and a few additional assumptions. But the assumptions were dubious. Daniel Zile and James Overduin in their 2014 paper touched the same problem.
The derivation by Zile and Overduin of Biot-Savart from Coulomb and special relativity
The Wikipedia article contains a link to the following paper:
Daniel Zile and James Overduin (2014) derive Biot-Savart from Coulomb's force and a special relativistic transformation of forces in a moving frame. Their derivation uses a "mass transformation". The transformation seems to be the total energy of a moving point particle in another frame.
Their momentum transformation is the 4-momentum transformation of a particle to a new frame.
(Jim Branson, 2012)
Above, β = v / c. The 4-momentum transformation of Zile and Overduin contains a typo in the following line:
where mu probably means E / c² in the notation of the authors. It should be
v E / c².
The letter v is missing.
Q ●
^ u
|
• ----> V
q
^ y
|
------> x
Let us analyze what the authors calculate in the above simple configuration. Let Q be positive and q be negative. As q approaches Q, the particle q gains energy from the field of Q, and the velocity V slows down since the x momentum stays the same.
Let v = V at the time t = 0. We switch to a frame which moves at a velocity v to the right. In that frame, it appears that a force is accelerating q to the left because V is slowing down. We can interpret that in the moving frame, Q has a magnetic field B, and the Lorentz force
q v × B
is accelerating q to the left. This is the "force transformation" of Zile and Overduin.
This phenomenon is the inertia effect which we observed in our blog several years ago. It can explain B, starting from the Coulomb attraction between q and Q. Zile and Overduin made the same observation in 2014, as we made a few years later in this blog.
The error in the Zile and Overduin argument
In a wire carrying an electric current, the protons essentially cancel Coulomb's force which moving electrons exert on a test charge q near the wire.
The argument of Zile and Overduin rests on the fact that the total energy E of q increases (or decreases) as q approaches the wire. This does not happen.
How to correct the argument of Zile and Overduin?
We must take into account the momentum of the energy W which the test charge q gains in the field of Q.
Let q be a positron. As q approaches the electrons in a wire, it receives a "moving" packet of energy W in the field of the electrons. The packet contains momentum, besides energy.
The positron q must use the energy W when it climbs in the potential of the protons of the wire. But the positron cannot give up the momentum that it got in the packet W. The positron will steer to the direction where the electrons are moving.
Conclusions
Our proof of the Biot-Savart law on December 17, 2023 contained assumptions which are dubious. It is better to use the argument that the energy packet W which a test charge q receives in the field of a moving charge Q, also contains momentum. We will analyze this in a future blog post.
On August 28, 2024 we observed that the electromagnetic action seems to be ignorant of the kinetic energy of two overlapping electric fields. Our analysis of Biot-Savart above depends on the energy packet W possessing momentum. It is another subtle effect of overlapping fields. The electromagnetic action does handle it correctly by introducing the magnetic field B.
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