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The transformations in special relativity of current and charge density fields are given as follows: $\bar J' = \begin{matrix} \gamma(J_x - Vρ) \\ J_y \\ J_z \end{matrix}$

$ρ' =\gamma (ρ - {V J_x \over c^2}) $ with the relative motion happening in the x axis and J is the current density, V is the volume and ρ is the charge density.

The one term I cannot understand is that of the tittle of this question: It's the second term in the charge density in the moving reference frame, that is as you can see: $V J_x \over c^2 $

I believe I understand how a charge density gives a current in relative moving frame- it's doe to the relative velocity that has as a result the appearance of a new current.

Question: I cannot understand how a current density gives at the moving reference frame a "new" term of charge density. What is the different thing that the moving observer sees and makes him count this term?

Thank you.

Note. I have read the mathematical proof, but I don't understand what happens and this term makes an appearance.

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There is a nice thought experiment that explains this phenomenom, usually called the wire argument. I think you can find it in Feynman's lectures though I don't know who thought of it first.

The set up is the following: let's consider a wire with zero total charge but with a current going through. For simplicity's sake, let's assume that the charge carriers have the same absolute charge.

The fact that there is a global current implies that the charge carriers of different sign do not move at the same speed otherwise their contribution to the total current would cancel out. And indeed, usually in a wire, we can consider the positive charge carriers (the nuclei) to stay still and the negative charge carriers (that is the electrons) to be moving.

Now, let's change our frame of reference. The positive charge carriers now moving, their density is changed by length contractions. The same thing happens for the negative charge carriers but because they have a different speed, they get a different contraction. If the densities cancelled out before, they no longer can and now, you have a charge density in your new frame of reference.

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