Polarizing windows, filters, etc. are often used to reduce the amount of light that enters the lens of a camera or into a room or a car. A library atrium has an overhead skylight that lets in too much light during the day which heats up the interior of the library far too much. The building engineer installs new double paned polarizing sky lights to reduce the intensity. If sunlight, which is unpolarized, has an average intensity of 1366 W/m2 what angle should the polarizing axis of the second pane of the window make with the polarizing axis of the first pane of the window in order to reduce the intensity of the sunlight to 33% of the original value

 θ = 35.7º

Explanation:

For this exercise we must use the law of Malus

          I = I₀ cos² θ

where the angle is between the two polarizers.

When the unpolarized light from the sun reaches the first polarizer, only the light polarized in the direction manages to be transmitted, which is why

          I₁ = I₀ /2

this light reaches the second polarizer

           I₂ = I₁ cos² θ

           I₂ = I₀/2    cos² θ

           cos² θ = 2 I₂ / Io

indicate that the transmitted light is 33% = 0.33 I₀

          cos² θ = 2 0.33

          cos θ = √0.66

          θ = cos⁻¹ 0.8124

          θ = 35.7º

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