8. A thin beam of light falls normally on a plane-parallel glass plate. Behind the plate, at some distance from it, stands an ideal mirror (its reflection coefficient is equal to one). The plane of the mirror is parallel to the plate. It is known that the intensity of the beam that has passed through this system is 16 times less than the intensity of the incident beam. The reflection coefficient at the glass-air boundary is considered constant regardless of the direction of the beam. Neglect absorption and scattering of light in air and glass. Find the reflection coefficient at the glass-air boundary under these conditions. (10 points)
Answer: 0.5

## Solution.

Let $k$ be the reflection coefficient, then we get $I_{1}=I_{0}(1-k)$ (2 points). Similarly, $I_{3}=I_{2}=I_{1}(1-k)=I_{0}(1-k)^{2}$ (2 points). As a result: $I_{k}=I_{0}(1-k)^{4}$ (2 points). According to the condition $I_{0}=16 \cdot I_{k}=16 \cdot I_{0}(1-k)^{4}$ (2 points). In the end, we get $k=0.5$ (2 points).