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When light travels from a denser medium to a less dense one, like from water to air, the critical angle is the angle of incidence beyond which the light reflects entirely back into the denser medium instead of refracting out. This phenomenon occurs because at the critical angle, the refracted angle becomes exactly 90 degrees, meaning the light travels along the boundary without passing into the other medium. Understanding the critical angle helps in applications where total internal reflection is crucial, such as in fiber optics. A practical example in the given exercise involves the light within a liquid reaching the liquid-air boundary. When the critical angle is exactly 39 degrees, it means any light hitting at or above this angle will not pass into the air but will instead be completely reflected within the liquid. Some key points about the critical angle include:

• Total internal reflection only occurs when moving from a denser to a less dense medium.
• The critical angle is specific to the pair of materials involved (e.g., water-air).
• At the critical angle, the angle of refraction is 90 degrees.

The refractive index is a fundamental property of materials that quantifies how much they bend, or refract, light. Each transparent material has its own refractive index, which can be thought of as how many times slower light travels in the material compared to the vacuum. For example, a refractive index of 1.57 indicates light travels 1.57 times slower in the liquid than in a vacuum or air. The step-by-step solution calculates the refractive index of the liquid using the critical angle and Snell's Law. Knowing this refractive index is crucial, not just for solving the Brewster's angle but also in optical applications, like designing lenses and prisms.

A lower refractive index means less bending of light as it enters the material.

The refractive index of air is typically close to 1, simplifying calculations in many practical situations.

Refractive index guides understandings of optical density, with higher values indicating that light slows more when entering the medium.

Snell’s Law is the mathematical equation that describes how light refracts when transitioning between two different media. It states that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is constant, and this constant is equivalent to the ratio of the refractive indices of the two media.This law is expressed as:\[ n_1 \sin \theta_1 = n_2 \sin \theta_2\]Where:

• \(n_1\) and \(n_2\) are the refractive indices of the first and second medium respectively,
• \(\theta_1\) and \(\theta_2\) are the angles of incidence and refraction respectively.

In the given problem, Snell's Law is applied to relate the critical angle (39 degrees) to the refractive index of the liquid. By recognizing that the angle of refraction at the critical angle is 90 degrees, Snell's Law simplifies, which allows finding the refractive index directly.A few key uses of Snell's Law are:

• To predict how light will change direction when entering a new medium.
• To design optical devices like lenses and prisms.
• To find the critical angle, which can be used in calculating phenomena like total internal reflection and Brewster's angle.