Why doesn't stationary charge feel force from a current carrying infinite wire?

Question

If a stationary charge is kept in front of current carrying wire , from charge's perspective the protons are at rest with it and electrons are moving , so shouldnt there be a lenght contraction in electrons and there density increase and stationary charge should feel a force , but it is seen that it doesnt?? please answer the question in stationary charge's frame of reference when electrons are flowing in the wire

Answer 1

You are correct that there a Lorentz contraction effect, but it is already taken into account in the balancing of the charge densities.

Let $\rho_+$ be the charge density of the protons in the wire, in the frame where they are at rest. This is also the reference frame in which the charge placed near the wire is not moving. Let $\rho_-$ be the charge density of the electrons in the wire in this same frame. We have $$ \rho_- = - \rho_+ $$ which results in a neutral wire. Now the quantity $\rho_-$ should not be mistaken for the density of electrons in their own rest frame. In their rest frame they are spread further apart and their density is consequently $\rho_- / \gamma$ where $\gamma$ is the Lorentz factor associated with their motion relative to the protons. There is indeed a Lorentz contraction on passing from electron rest frame to proton rest frame, and it just suffices to bring the density up to $\rho_-$ and then it balances the protons. Note, the densities are not balanced in the rest frame of the electrons.