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Say an electron outside a wire is moving with the same velocity as the electrons inside, then due to Lorentz contraction, an electric field would be generated in the outside electron's frame of reference and hence the electron will move towards the wire, and the positive charges moving in this frame of reference with the same velocity of the electrons but in the opposite direction will create a magnetic field that wouldn't affect the electron that is not moving in this frame of reference. Now, if a grounded shield is placed around this wire, the electric field would not penetrate it and therefore there should be no attraction but then why is the electron attracted to the wire in spite of this shield?

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  • $\begingroup$ You asked the same question six days ago; you should be editing the original question, rather than reposting. $\endgroup$ – Daniel Griscom Nov 9 '15 at 21:02
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Something to remember is that the potential, $\Phi$, is not a Lorentz invariant. Rather, it transforms as the zeroth component of a four vector, $(\Phi, \vec{A})$. In the presence of a nonzero vector potential, then, potential will change under a Lorentz transformation.

So what's happening here? In the rest frame of the wire, the wire has no net charge, the covering has no net charge, and the electron experiences a magnetic force. In the rest frame of an electron, the wire is charged, the cover is still uncharged, and the electron feels an electric force. However, the cover is no longer grounded. In fact, whatever you attached the cover to in order to ground it now has nonzero potential, as does the cover itself. This is because $\Phi$ is not conserved under Lorentz transformations.

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The electron "sees" the magnetic field created by the current inside the wire. The deflection is due to this magnetic field . It is not the electric field!

How is the electric field of the wire is looking like? The electrons in the wire are accelerated along the electric field, so it spreads along the wire.

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If electric fields can't pass through a grounded shield, how can a magnetic force be generated on moving charges outside a wire?

Magnetic fields can pass through shields. They are not stopped by Faraday cages.

"Electrostatic shielding is the phenomenon that is observed when a Faraday cage operates to block the effects of an electric field. Such a cage can block the effects of an external field on its internal contents, or the effects of an internal field on the outside environment." Source

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