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If you shine monochromatic light on a reflective metal surface, say polished silver, does the reflected light undergo a $180^{\circ}$ phase change like it does when it reflects off a medium with a higher refractive index?

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Yes, if the metal has infinite conductivity so the induced currents in it are in anti-phase with the electric field of the primary wave.

In practice, metals have finite conductivity, the oscillations of charges on the surface will be slightly off the anti-phase so the reflected wave will have phase shift lower than 180 degrees.

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Light is electro-magnetic wave and obeys same physical law.

Please see the following article about Yagi-Uda antenna where animated graphics explain how the antenna works.

https://en.wikipedia.org/wiki/Yagi%E2%80%93Uda_antenna

Very similar process takes with light and mirror -- the difference is only in wavelength.

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