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What happens when an electromagnetic wave strikes a perfect conductor at normal incidence? Is the wave transmitted or reflected through the conductor?

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Normal incidence means perpendicular to the surface, and let's assume that in your case you are talking about a perfect conductor, a metal, and to make you see easier what would happen, let's make this a mirror.

Now there is:

  1. reflection, when light is not transmitted through the metal,

  2. and there is refraction, when light is transmitted through the medium

  3. absorption and re-emission of light on the surface (with visible light it creates color for the object)

In your case, when the angle of incidence is perpendicular to the surface, the metal of the mirror, usually aluminum:

  1. will reflect (elastically scatter) all visible wavelength photons, and

  2. will not absorb or re-emit any visible wavelength photons at the surface

  3. will inelastically scatter some non-visible wavelength photons, like infrared, that heats up the metal, and these photons will transfer part of their energy to the vibrational motion of the molecules of the metal

In this case, no energy is transmitted through the conductor. This would mean refraction of light, and it is not possible with visible light with normal incidence angle.

http://www.qwed.eu/QuickWave/qwmodeller_help/microwave_course/1_2_2_normal_incidence_of_a_plane_wave_on_a_perfect_conductor_surface.htm

It is why metals do not have their own color, they appear gray, silver, because they reflect all visible light.

Please see here:

Why are most metals gray/silver?

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