Faraday's resolution

The paradox of the charge seen from two frames dissolves once we admit a new ingredient. In frame B the charge is at rest and the magnetic field moves past it: that $\vv{B}$ which changes in time (at the point where the charge sits) generates an induced electric field. It is this electric field, not the Lorentz force, that does work on the charge in system B.

Principle — The induced electric field

A magnetic field that varies in time generates an induced electric field $\vv{E}$, endowed with two new properties compared with electrostatic fields:

1. It is not conservative: it admits no potential.
2. Its field lines are closed.

The intuition is that this induced field $\vv{E}$ does not resemble the electrostatic ones met so far at all. It does not arise from stationary charges, but from a changing $\vv{B}$; and its lines, instead of starting on positive charges and ending on negative ones, close on themselves, like the lines of $\vv{B}$. Precisely because the lines are closed, the circulation of $\vv{E}$ along a closed path can be non-zero: this is the gateway to Faraday’s law.

The two descriptions of the paradox become consistent again: in frame A it is the magnetic force that pushes the charge; in frame B it is the electric field induced by the changing $\vv{B}$. The same acceleration, two different names depending on the frame: this is the first hint of the deep unity between the electric field and the magnetic field.

Links

Topics: Electromagnetic induction Concepts: Faraday-Neumann-Lenz law · Electric field

Related exercises: Worked exercise — loop entering and leaving a field · Problem — Ranking the EMF of four square loops · Problem — Loop entering a field (qualitative)