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When I hold a plastic transparent cup half filled with water and see it from above then its sides (up to which water is filled )appear shiny. Some of my fingertips (which are holding glass ) are wet so they are clearly visible through the shiny surface and other finger areas which are not wet are not visible. What is the reason behind it

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You are seeing total internal reflection: when light bounces against the side of the cup, due to the angle and water/air index of refraction diffrence there is no refracted ray that would leave the cup, so you get near-perfect reflection. But wet fingers makes the index of refraction on the other side equal, allowing light to pass through.

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