# Special relativity and electromagnetic fields

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.

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

• Does it mean that the electric field in the frame of the point charge cannot be explained by Faraday’s law in this example? Apr 21, 2020 at 13:56