The movement of the test charge is explained solely through the Coulomb interaction. There is no magnetic field.
Suppose that we switch to a moving frame. Clocks which were placed at various spatial coordinates of the laboratory frame appear to go slower, and they are no longer in synchrony when viewed from the moving frame.
If an observer in the laboratory frame measures the test charge to move from a position x_1 to x_2, and measures the elapsed time as t, then the observer in the moving frame will measure a different time interval.
Furthermore, the measured time interval depends on the spatial distance x_2 - x_1. The speed of the charge appears to affect the force on the test charge, when the force is observed in the moving frame. The moving observer interprets that a magnetic field is affecting the path of the test charge. The force depends on the field strength and the speed of the test charge.
We see that in this simple case, we can say that the magnetic force is just an illusion which appears when the test charge path is Lorentz transformed over to a moving frame. A human has problems understanding the transformation and explains the surprising movement with an imagined magnetic force.
The simple model can explain the magnetic field which we see around a wire carrying an electric current.
http://www.feynmanlectures.caltech.edu/II_13.html
Richard P. Feynman calculated the relativistic effect of the electrons moving inside a wire, relative to the protons. He considers a negative test charge moving at the same (very slow) velocity v as the electrons inside the wire.
Feynman writes that there is a magnetic field which will cause the test charge to curve toward the wire. If we move to a frame moving along the test charge, then, he writes, the protons in the wire will appear length-contracted, and the negative test charge, in this moving frame, is pulled by their electric attraction.
Let the density of protons in the laboratory frame be P.
The density of electrons in the laboratory frame is γE, where γ > 1 (because of the length contraction) and E is the density of the electrons in the rest frame of the electrons. If there is no electric Coulomb force in the laboratory frame, then it must be that
P = γE.
Let us then switch to the moving frame. The density of electrons is E, and the density of protons in this frame is γP. The density of protons is γ^2 times the density of electrons. There, indeed, is a pulling electric force on the negative test charge.
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Feynman writes that, "of course", we cannot reduce magnetic fields solely to the Coulomb electric field and a Lorentz transformation. He does not specify the reason, though.
http://www.feynmanlectures.caltech.edu/II_13.html
Richard P. Feynman calculated the relativistic effect of the electrons moving inside a wire, relative to the protons. He considers a negative test charge moving at the same (very slow) velocity v as the electrons inside the wire.
Feynman writes that there is a magnetic field which will cause the test charge to curve toward the wire. If we move to a frame moving along the test charge, then, he writes, the protons in the wire will appear length-contracted, and the negative test charge, in this moving frame, is pulled by their electric attraction.
Let the density of protons in the laboratory frame be P.
The density of electrons in the laboratory frame is γE, where γ > 1 (because of the length contraction) and E is the density of the electrons in the rest frame of the electrons. If there is no electric Coulomb force in the laboratory frame, then it must be that
P = γE.
Let us then switch to the moving frame. The density of electrons is E, and the density of protons in this frame is γP. The density of protons is γ^2 times the density of electrons. There, indeed, is a pulling electric force on the negative test charge.
---
Feynman writes that, "of course", we cannot reduce magnetic fields solely to the Coulomb electric field and a Lorentz transformation. He does not specify the reason, though.
When we consider configurations like the radio transmitter of our previous blog post, it is not clear at all how we could get rid of the magnetic field.
If we move a charge back and forth, some of the spherical waves which are produced will reflect back. If we just believe in a Coulomb field which spreads at the speed of light out from the moving charge, what then causes the reflection? There has to be a wave equation acting, if there is a reflection. And it is hard to write a wave equation without a magnetic field.
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