Charge density seen from a moving reference frame S' (SR + EM)

AI Thread Summary
The discussion focuses on calculating charge density in a moving reference frame S' using electric field equations. One suggested method involves calculating the electric field in S' and applying the formula E = λ / (2 π ε l) to find λ. Another approach involves expressing λ in terms of the number of positive charges per unit length (n_p) and free electrons per unit length (n_e), utilizing length contraction to understand how these quantities transform between frames. Both methods are deemed valid for determining charge density. The conversation emphasizes the importance of understanding the transformations of charge densities in different reference frames.
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Homework Statement
a cable infinite and very thin that carries current and has charge density
null in the reference of fixed positive charges, in addition, consider a
charge particle q positioned at a distance l from the cable and traveling
in parallel direction to the cable with the same speed (relativistic) u
of the electrons in the cable.
Relevant Equations
.
There are some question involving the statement. One of them is about the charge density in S' frame. It asks to calc it.

I thought that i could calculate the electric field in the referencial frame S' and, then, use the formula
$$ E = \lambda / 2 \pi \epsilon l $$
In that way, i would obtain ##\lambda##. Is that a reasonable way to find the charge density? Is there another way?
 
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Herculi said:
I thought that i could calculate the electric field in the referencial frame S' and, then, use the formula $$ E = \lambda /( 2 \pi \epsilon l) $$
In that way, i would obtain ##\lambda##. Is that a reasonable way to find the charge density?
Yes.

Herculi said:
Is there another way?
You can express ##\lambda## in terms of ##n_p## and ##n_e##, where ##n_p## is the number of positive charges per unit length of the cable and ##n_e## is the number of free electrons per unit length. Use the idea of length contraction to see how ##n_p## and ##n_e## transform when switching from frame S to frame S'.

[Edited in order to improve notation.]
 
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