Ashcroft and Mermin's Solid State Physics, chapter one, has an exercise about surface plasmons. They give the electric field normal to the surface of the metal as $E_z=Be^{iqx}e^{-Kz}$ How can this be? Wouldn't this imply a current going out of the metal? Shouldn't a negative charge accumulate at the boundary, cancelling the field?
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$\begingroup$ Why would it imply a current? $\endgroup$– G. SmithCommented Nov 22, 2018 at 5:39
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$\begingroup$ Well, this is inside the metal, so isn't the current equal to the conductivity times the electric field? $\endgroup$– Gene NadenCommented Nov 22, 2018 at 15:34
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$\begingroup$ I see. I thought you were talking about the field outside. $\endgroup$– G. SmithCommented Nov 22, 2018 at 17:32
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It is a characteristic of conductors at electrostatic equilibrium is that the electric field upon the surface of the conductor is directed entirely perpendicular to the surface.
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$\begingroup$ Oh, yeah... But the problem statement has components both normal and parallel to the surface. $\endgroup$ Commented Nov 22, 2018 at 15:15
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$\begingroup$ These are fields inside the metal, contrary to the usual treatment of conductors. It is a mystery to me. $\endgroup$ Commented Nov 22, 2018 at 15:36
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$\begingroup$ In your case, you should look at the boundary condition of Maxwell's equation in the medium from respective portions in Griffith's electrodynamic book. $\endgroup$ Commented Nov 22, 2018 at 16:49