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Reasons for the existence of a magnetic force can be found in the rest frame of a moving electron next to a current carrying wire as explained in Chapter 5.9 "Interaction between a moving charge and other moving charges" of Purcell: Electricity And Magnetism.

However, I am looking for an explanation in the lab frame, since an explanation should not depend on observations in a specific reference frame. An explanation should be possible in any reference frame, hence also in the lab frame, where two current carrying wires attract each other. I would like to understand how moving charges can create a magnetic force there.

Frank Breitling
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Let us say we construct a clock that consists of two very large freely moving positively charged plates, with large inertia.

This clock reads zero when the plates are stacked together. The plates accumulate momentum at constant rate. We read the clock by measuring this accumulated momentum.

Okay now we send this clock to a trip to Alpha Centauri and back at constant speed 0.87 c.

Let's say Alpha Centauri is in the up-direction, while plates repel each other in the left-right-direction.

The trip takes 9.74 years earth time. But the clock that took the trip proceeded only 4.85 years.

Compared to a charged-plates-clock that stayed on earth the charged-plates-clock that made the trip accumulated half the time. And half the momentum.

The force on the plates in the lab (earth) frame was the change of momentum of the plates divided by 9.74 years. (I am talking about the clock that made the trip)

The force on the plates in the rest frame of the center of mass of the charged plates was the change of momentum of the plates divided by 4.85 years. (I am talking about the clock that made the trip)

The reduction of the force in the lab frame is the magnetic force.

stuffu
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