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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.

Dale
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gnowme
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2 Answers2

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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

Dale
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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.