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

[LCSphysicist]

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?

 

[TSny]

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.

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.]