The area illuminated at the base of the hemisphere can be found using the concept of critical angle and total internal reflection.
The critical angle (θc) for the interface between air and glass can be calculated using the formula:
sin(θc) = 1 / μ
Given that μ = √2, we can calculate θc:
sin(θc) = 1 / √2
θc = sin^(-1)(1/√2)
θc = π/4
The light beam will be totally internally reflected if the incident angle is greater than the critical angle. Since the glass hemisphere is of radius R and the light beam has a diameter of √2R, we can determine the angle (φ) that the light beam makes with the normal to the surface as it strikes the hemisphere as:
sin(φ) = R / √2R
φ = sin^(-1)(1/√2)
φ = π/4
This is the angle of incidence and because the angle of incidence is equal to the angle of reflection the angle of reflection = π/4 . With this, the cone formed by this angle of reflection is obtained and is used to calculate the illuminated area.
The illuminated area is the basis reverse projection of the cone and can be given by the formula:
Area = π * (R/ tan(π/4))^2
Area = π * (R/ 1)^2
Area = π * R^2
So the correct option is:
c. πR²/8
