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Consider a current carrying wire. There is a stationary charge $q$ at a distance $r$ from wire.

In lab frame when the electrons in wire are moving at $v_1$ velocity we say that there is some linear charge density of poisitve ions equal to $\lambda_1$, and the linear charge density of electrons with the affect of length contraction is equal to $-\lambda$.

The electrons in wire were moving and therefore length contracted. Their linear charge density become equal to charge density of positive ions due to length contraction. Therefore the net charge on wire was $0$ and no electric force applied on $q$.

Now again in lab frame, we increase the current in wire and the velocity of electrons becomes $v_2$. The electrons are now more length contracted and their absolute linear charge density should increase from $|-\lambda|$. The density of positive charges is same as before. It means there would be a net charge density on wire and therefore a force on stationary $q$.

But it doesn't happen. Why? According to length contraction, more the velocity of electrons, more the contraction of electrons but it doesn't seem to work in case of current carrying wires. Why?

James Webb
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But it doesn't happen. Why?

The electrons don’t have a fixed separation in their rest frame. So when you change the current you also change the separation in the electron’s rest frame. Changing the current in the wire is not a boost.

Dale
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