Why doesn't a current carrying wire exert any force on a stationary charge?

Question

By the idea in Section 13-6 of this chapter I came to up with an absurd question. Actually I understand completely the contents in the above text. The main objective of the above mentioned section is to explain that electric field and magnetic field are relative to one another and depend on the reference frame of the observer. As explained, in the reference frame S' in which the test charge is stationary and wire is moving past it, we see that due to length contraction (as due to the Special Relativity) the charge densities of positive charge and negative charge tranform thus resulting in a net charge density which is not zero which then results in Coulombic Forces between the wire and test charge. I am not diving in the details as all of it is explained very well in the above texts.
So, the question is very simple, if a test charge $q$ is kept near a wire without any current through it, we see no force acting in the scene as all the charges are distributed uniformly throughout the wire. But, when the negative charges start moving in the wire, which constitute current, the length of these should contract, due to the length contraction, thus resulting in a net negative charge density in the wire. Then why is it that we don't see any electrostatic forces between the test charge and the wire?

Note: In the given question, suggested as duplicate, it is already assumed that there is no force on a charge around a current carrying wire.

The current carrying wire doesn't apply any magnetic force on nearby charge q ( positive stationary charge) because it has 0 velocity in lab frame. We found that there is no force on q by wire.

But I am asking the reason for that. Also I have supplied arguments from my side that why (according to me) the charge should experience force. All I am asking is the correction to my wrong idea about the theory.

Answer 1

I have same question while reading the Feynman Lectures recently. Here is my thought.

Let's say that there are two reference frame S and S'. The wire is at rest in S. The moving conduction electrons which constitute current are at rest in the S'. And S' has speed $u$ relative to S in the x-axis direction, which means that the moving conduction electrons have speed $u$ relative to the wire.

The key point is the conduction electrons are initially stationary in S and are suddenly accelerated when an electric field appears in the wire.

Let's focus on S first. Before the acceleration, assume two conduction electrons b and c are separated by $l$ in the wire. Their coordinates are ${x_b}$ and ${x_c }$ respectively, the y and z coordinates are omitted for convenience. At the time $t$, an electric field is suddenly established in the wire, and the two electrons begin to accelerate. So in frame S, the two events b and c start to accelerate are $(x_b,t)$, $(x_c ,t)$ respectively, and $x_b+l=x_c $.

Let's switch the perspective to frame S'. The x coordinate where b starts to accelerate is: $$x'_b = \frac{x_b-ut} {\sqrt{1-u^2/c^2}}$$ The x coordinate where c starts to accelerate is: $$x'_c = \frac{x_c-ut} {\sqrt{1-u^2/c^2}} = x'_b+\frac{l} {\sqrt{1-u^2/c^2}}$$ In addition, in frame S', the time when b and c start to accelerate is different.The time b starts to accelerate is: $$t'_b = \frac{t-ux_b/c^2} {\sqrt{1-u^2/c^2}}$$ The time c starts to accelerate is: $$t'_c = \frac{t-ux_c/c^2} {\sqrt{1-u^2/c^2}} = t'_b-\frac{ul} {\sqrt{1-u^2/c^2}}$$ If we assume that the acceleration process of conduction electrons is completed almost instantly. In S' the events that b and c complete acceleration are $(x'_b,t'_b)$, $(x'_c ,t'_c)$, and the electrons come to rest after the acceleration in S'. So the distance between b and c in S' is: $$x'_c-x'_b = \frac{l} {\sqrt{1-u^2/c^2}}$$ Thus, the distance between b and c in S is: $$\frac{l} {\sqrt{1-u^2/c^2}}\cdot\sqrt{1-u^2/c^2} = l$$ The negative charge density won't change in the wire. In summary, the length contraction occurs when switching reference frames, not when the electrons accelerate.