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We are told that magnetic force is a relativistic manifestation of the electric force.

When a charged particle is in the vicinity of a current carrying wire it will experience a force proportional to its charge and its velocity with respect to the wire.

This force can be explained by imagining the charge moving at the same speed as the electron flow in the wire. Due to special relativity the positive charges are more concentrated due to length dilation and hence there is a net electric force acting on the particle.

But surely when the particle is at rest with the wire, i.e. at rest with the positive charges in the wire the same argument can be made that the negative charge of the moving electron current, compressed due to length dilation, will produce a net force as well?

How come it’s zero?

Qmechanic
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adlibber
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2 Answers2

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For a moment, forget about relativity and motion, and just consider circuits at rest.

Wires are not things that just float around with charge and current, they are parts of circuits and the charge and current on them depends on the circuit. If you want a net charge on the wire then you give it a large voltage relative to ground and it will charge up according to its self capacitance. If you want a net current through it then you give it a voltage difference across it and it will conduct current according to its resistance.

So, going back to relativity, in the wire’s rest frame you set up the current and the charge on the wire. Both depend on the circuit that the wire is part of, and not on relativity. Once you have that information, then you can use relativity to determine the charge and current on the wire in any other frame. Relativity, by itself, is insufficient.

How come it’s zero?

The reason that the force is zero in the charge’s frame is because the wire is uncharged in that frame. The reason it is uncharged is because that was part of the problem setup in the beginning. That zero charge is what the circuit produced. It has nothing to do with relativity.

From that fact you can then transform to other frames and use relativity, but that fact comes from the circuit and not from relativity.

Dale
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With the help of Bob DuHamel of RSD academy the answer to this is to recognise that when electrons flow through a wire in one direction there is an equal and opposite flow of positive charge “holes” in the opposite direction as electrons move from copper atom to copper atom.

This means that from the perspective of a charge outside of the wire there is zero net charge due to special relativity.

When the charge starts moving wrt the wire the velocities of the “holes” and electrons relative to the moving charge are difference.

Hence the net electric force on the charge.

adlibber
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