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I know that the impact angle is known with the "traveling vector" k in respect of the normal vector, let's say z or (0, 0, 1).

Doing the math i know it is 45º, but why the reflected wave has the same angle? The EM wave impacts a plane.

Another question, what if, instead of a plane, the EM wave impacts a cylinder? will this condition of (Incident Angle = Reflected Angle) work?

WhiteGlove
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The Law of Reflection (i.e. "the angle of incidence is equal to the angle of reflection") can be derived using Fermat's principle of Least Time. The argument is detailed here: http://www.ysagade.nl/2015/05/22/sicm-fermat-optics/

The Law of Reflection is true for any type of surface, even rough ones.

probably_someone
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  • So the surface the EM wave is hitting it doesn't change the angle of the wave or matter at all(i mean, the form of the object being hit)? – WhiteGlove Jan 12 '18 at 19:03
  • Not quite sure what you just asked. Can you ask your question in a different form? – probably_someone Jan 12 '18 at 19:04
  • Imagine that, instead of a plane, we have a cylinder or a surface which is not entirely plane. Does it change the angle? It is always true the Fermat's principle? (Avoiding when the wave is totally transmitted in that case there is not reflected) – WhiteGlove Jan 12 '18 at 19:07
  • For a given incident direction, the angle of incidence depends on the geometry of the surface (more precisely, the angle of incidence is defined relative to the normal vector to the surface), but the angle of incidence is still always equal to the angle of reflection. – probably_someone Jan 12 '18 at 19:09