(II) A piece of material, suspected of being a stolen diamond (n = 2.42) , is submerged in oil of refractive index 1.43 and illuminated by unpolarized light. It is found that the reflected light is completely polarized at an angle of 56°. Is it diamond? Explain.

Verified step by step guidance

1. Determine the condition for complete polarization of reflected light. This occurs at Brewster's angle, where the reflected and refracted rays are perpendicular to each other. The relationship is given by the equation: $n{t}_m=tan(\theta$_B), where $n{t}_m$ is the ratio of the refractive indices of the two media, and $\theta$_B is Brewster's angle.
2. Identify the refractive indices of the two media. The refractive index of the diamond (suspected material) is $2.42$, and the refractive index of the oil is $1.43$. The given Brewster's angle is $56°$.
3. Use the equation for Brewster's angle to calculate the expected Brewster's angle for the diamond-oil interface. Rearrange the equation to solve for $\theta$_B: $\theta$_B=tan^-1(n_diamond/n_oil). Substitute the values $2.42$ for $n$_diamond and $1.43$ for $n$_oil.
4. Compare the calculated Brewster's angle with the given angle of $56°$. If the calculated angle matches the given angle, the material is likely diamond. If not, it is not diamond.
5. Conclude whether the material is diamond based on the comparison. If the calculated Brewster's angle does not match $56°$, the material is not diamond. Provide reasoning based on the refractive index and Brewster's angle relationship.

Key Concepts

Snell's Law

Snell's Law describes how light refracts when it passes from one medium to another with different refractive indices. It is mathematically expressed as n1 * sin(θ1) = n2 * sin(θ2), where n is the refractive index and θ is the angle of incidence or refraction. This principle is crucial for understanding how light behaves at the interface between the diamond and the oil.

Brewster's Angle

Brewster's Angle is the angle of incidence at which light with a particular polarization is perfectly transmitted through a transparent dielectric surface, with no reflection. It can be calculated using the formula θ_B = arctan(n2/n1), where n1 and n2 are the refractive indices of the two media. In this case, the angle of 56° indicates that the reflected light is polarized, suggesting a specific relationship between the refractive indices of diamond and oil.

Refractive Index

The refractive index (n) of a material is a dimensionless number that describes how light propagates through that medium. It is defined as the ratio of the speed of light in a vacuum to the speed of light in the medium. For the diamond in question, with a refractive index of 2.42, comparing it to the oil's refractive index of 1.43 helps determine whether the material behaves like a diamond when subjected to polarized light.