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It is well known that electromagnetic force depends on frame.

I was reading a book, it says

If a charge is moving parallel to a current carrying wire then a magnetic force will be exerted on charge. But if we start moving with the charge at same velocity then it is in rest for the moving frame but it will experience the force again and since both frames( stationary one and the moving one) has no acceleration w.r.t each other so acceleration on the charge will be same in both frames but reason of this acceleration in moving frame as there is no magnetic field must be an electric field

My question is that it is well known that current carrying wire is neutral so how can there be an electric field in moving frame and if it is there then what is the origin of this field?

As it was marked as a possible duplicate, I want to clarify that I want an intuitive answer. There is written that a moving current carrying wire will appear as charge. How it is possible when wire is neutral? I am new to electrodynamics so sorry if i ask bad questions.

Emilio Pisanty
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its-him
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2 Answers2

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The short answer is length contraction. In the rest frame of the current-carrying wire, it appears neutral, but that is no longer the case if the wire is observed from a reference frame that is moving along the direction of the current, because

  • the wire contains components of different charges moving at different velocities,
  • each of those components will experience a different amount of length contraction, because the relative velocities to the new frame are different,
  • those length contractions will impact the apparent charge density as observed by the new frame of reference,
  • and therefore the total charge density will be nonzero, as observed in the new frame of reference.

That nonzero charge density will then generate an electric field, which will attract or repel the (formerly moving, now stationary) charge.

For a detailed exposition of this transformation, the go-to place is Ed Purcell's Electricity and Magnetism; for a condensed take, this video by Veritasium and MinutePhysics is a good introduction. For previous takes on this topic here on this site, see the search results here, and particularly How Special Relativity causes magnetism, current in wire + special relativity = magnetism, Special relativity and electromagnetism, Need clarity about relativity/magnetism explanation, How Special Relativity causes magnetism, Is magnetic field due to an electric current a relativistic effect?, and the many questions in their Linked and Related sidebars to the right.

Emilio Pisanty
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The important thing to realize is that the wire is only neutral in one reference frame. In other frames it is charged. This is easy to see the other way: if you have some charge density $\rho$ in a frame where it is at rest, then in a frame where it is moving there is clearly a current density $J$ also.

It turns out that it works the other way too although it isn’t as obvious. $\rho$ and $J$ have the same relationship as $t$ and $x$ in relativity. There is $\rho$ “dilation” and $J$ “contraction”.

Importantly, the quantity $\rho^2-J^2$ is invariant (in units where c=1). So as $J$ changes in different frames, $\rho$ must also change to keep it invariant. This leads to the neutral current-carrying wire in one frame being charged in any other frame.

Dale
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