Final answer:
Using Malus's law, the fraction of the initial intensity of unpolarized light transmitted after passing through two polarizers at 45 degrees to each other is 25%.
Step-by-step explanation:
When unpolarized light passes through two polarizers with their polarization axes at a 45-degree angle to each other, the fraction of the incident light intensity that is transmitted can be found using Malus's law.
The law states that the intensity of polarized light after passing through a polarizer is I = I_0 * cos2(θ), where I_0 is the initial intensity and θ is the angle between the light's polarization direction and the axis of the polarizer. When light passes through the first polarizer, it's polarized and its intensity is reduced by half (50%).
After passing through the second polarizer set at 45 degrees, the intensity is reduced by another factor of cos2(45°) = 1/2.
Therefore, the fraction of the initial intensity that is transmitted is 1/2 * 1/2 = 1/4, or 25%