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I am an amateur reading the book "The Theoretical Minimum: Special Relativity and Classical Field Theory" by Leonard Susskind. In the lecture about Maxwell's equations,an example similar to the following is given:

Suppose there is a stationary uniform magnetic field with only one component $B_z$ pointing out of the page, and a *positive point charge $q$ is moving with constant velocity $v$ to the right in the magnetic field.

By the Lorentz force law, there should be a downward force $q v \times B$ on the charge made by the magnetic field. In the frame of the point charge, however, the same downward force on the charge is driven by an electric field $E$.

My question is: If we know nothing about special relativity, can this electric field be explained by Faraday's law:i.e, $$\nabla \times E = -\frac{\partial B}{\partial t}$$

Although the magnetic field is moving to the left in the charge's frame, as it is uniform, I cannot see why there is a change of magnetic field. So $\frac{\partial B}{\partial t}$ should be zero. I cannot find a clear explanation in the book.

* In the book, it is a vertical wire instead of a point charge.

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2 Answers 2

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When you change reference frames you change both the fields and also the sources. In particular what is a pure current density in one frame becomes both a current density and a charge density in another frame.

Usually a uniform magnetic field is produced by a solenoid. In the rest frame of the solenoid it is uncharged with current circulating around the circumference. In a frame where the solenoid is moving one side will have a positive charge density and the opposite side will have a negative charge density. This will produce a uniform E field inside the solenoid which explains the force on the charge in that frame.

Of course, Faraday's law still holds in the moving frame, so at the solenoid where the B field goes from 0 outside the solenoid to non-zero inside the solenoid as the solenoid moves you do get a non-zero $\frac{\partial}{\partial t} B$ at that boundary. The full explanation of the field requires both applying Faraday's law at that boundary and also correctly accounting for the non-zero charge density of the solenoid. Faraday’s law by itself is insufficient

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  • $\begingroup$ Does it mean that the electric field in the frame of the point charge cannot be explained by Faraday’s law in this example? $\endgroup$
    – gnowme
    Commented Apr 21, 2020 at 13:56
  • $\begingroup$ My apologies, you are right, I had not addressed Faraday's law in my answer even though it was a part of your question. I have added a paragraph including that. $\endgroup$
    – Dale
    Commented Apr 21, 2020 at 15:24
  • $\begingroup$ Thanks, Dale. But I don't understand why we need to presume a model, e.g. solenoid, for the magnetic field. If there is a true uniform magnetic field, and we know its magnitude and direction, as well as the velocity relative to the charge, as described in my example, is it possible to calculate the electric field at the charge purely from the Faraday's law? The electric field should be consistent with qv X B. Note that the charge is entirely inside the magnetic field. It is not moving from outside to inside, so I don't understand why there is a change of B field. $\endgroup$
    – gnowme
    Commented Apr 22, 2020 at 1:19
  • $\begingroup$ @gnowme I don’t think that you can avoid specifying the sources if you wish to avoid relativity. With relativity we know how to transform the fields directly, but if you want to avoid relativity then you are left with transforming the sources and then calculating the fields. If you are not allowed to directly transform the fields and are not allowed to know the sources then I don’t think that you can conclude anything. That is not a reflection of a problem with physics, just that you have artificially rejected all the options. You need to allow the sources if you want to avoid relativity $\endgroup$
    – Dale
    Commented Apr 22, 2020 at 2:54
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    $\begingroup$ I think I sort of get it now. The charge density in the solenoid should be given by the effect of length contraction explained here . But I still found it very amazing that regardless of the source (should it be solenoid or permanent magnet) that gives rise to the magnetic field, the "elements" all work together to produce the same electric field perceived in the rest frame of the moving charge. It is almost magical. $\endgroup$
    – gnowme
    Commented Apr 23, 2020 at 15:23
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Yes. We knew before 1905 that electricity and magnetism are related. And a little secret special relativity doesnt explain how permanent magnets work.

There is a change because the density of the magnetic field changes over the wire.

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