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Why does light from the sun gets a slight (horizontal) polarization after bouncing from the ground?

And same question for light which is scattered from the sky.

Thanks.

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The two effects have slightly different origins.

Polarization of reflected light is a function of the angle of incidence and the refractive indices of the two media on either side of the surface - as described in this reference. Specifically, there is an angle (called the Brewster angle) where the reflected light is entirely polarized with the electrical vector parallel to the reflecting surface. The exact mathematics was derived by Fresnel, and the equations are known as Fresnel equations. The derivation can be found at this link. Basically, it follows directly from the continuity conditions at the boundary - in particular, at the Brewster angle it is not possible for TE polarized light to refract into the medium since the electric vector would have to be along the direction of propagation (which is not possible for an EM wave). The actual math is not that hard - see the above link.

The polarization of light from the sky is a result of Rayleigh scattering, as I described in this earlier answer

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