Final answer:
To answer if the gem could be a diamond and at what angle light would be polarized in water, Brewster's angle is used. For a diamond, the index of refraction suggests a Brewster's angle of about 67.5°, which is higher than the given 62.5°, hence the gem is not diamond. For water, the solution is undefined without knowing the gem's refractive index, but the angle would be less than 62.5° due to water's lower refractive index.
Step-by-step explanation:
To determine if the gemstone in the ring could be a diamond and at what angle light would be completely polarized when the gem is in water, we need to understand the concept of Brewster's angle, which is the angle of incidence at which light with a particular polarization is perfectly transmitted through a transparent dielectric surface, with no reflection. When light is incident at this angle, the reflected light is completely polarized perpendicular to the plane of incidence. According to Brewster's Law, Brewster's angle (θ_B) can be calculated using the index of refraction (n) of the material using the equation θ_B = arctan(n).
(a) For a diamond, the index of refraction is approximately 2.42. The Brewster's angle for diamond is given by θ_B = arctan(2.42), which equals approximately 67.5°. Since the given angle is 62.5°, which is less than the Brewster's angle for diamond, the gem in the question cannot be a diamond.
(b) When the gem is in water, the angle of polarization would be determined by the ratio of indices of refraction of water (n_water) and the gem. Assuming water has an index of refraction of 1.33, and using the information from (a) that the gem cannot be diamond, let's consider an arbitrary refractive index for the gem greater than water's. The new Brewster's angle would be calculated using n_gem / n_water. Since the problem does not specify the refractive index of the gem, we cannot give a definitive answer without additional information. However, we can say that the Brewster's angle will be lesser than 62.5° when the gem is placed in water due to the decrease in the refractive index difference between the two media.