A charge and two points of view

Consider a charge $-q$ (negative) that, seen from reference frame A (the laboratory), moves to the right with velocity $\vv{v}_0$ in the presence of a uniform magnetic field $\vv{B}$, pointing into the page. The Lorentz force pushes it downwards. But what does an observer moving together with the charge see?

On the left, the laboratory (frame A): the charge moves and the Lorentz force $\vv{F}_L$ pushes it downwards. On the right, the charge’s own frame (frame B): the charge is at rest and the field slides past it.

Let’s compare the two descriptions:

• In frame A: the Lorentz force $\vv{F}_L = -q\,\vv{v}_0\times\vv{B}$ is well defined and the charge accelerates downwards.
• In frame B (moving together with the charge): the charge is at rest, so $\vv{F}_L = -q\,\vv{0}\times\vv{B} = \vv{0}$. Yet, by the principle of Galilean relativity, the charge must still accelerate downwards: otherwise the two descriptions of the world would contradict each other.

Here is the paradox: in frame B the magnetic force is zero, but something must still accelerate the charge. Who does the work on the charge in frame B? The Lorentz force alone cannot answer this: a piece of physics is missing.

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Topics: Induzione elettromagnetica · Magnetismo Concepts: Forza di Lorentz

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