Suppose light travels from water to another substance, with an angle of incidence of \(10.0^{\circ}\) and an angle of refraction of \(14.9^{\circ} .\) (a) What is the index of refraction of the other substance? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

The index of refraction, often denoted as n, is a dimensionless number that describes how light, or any form of electromagnetic radiation, propagates through a medium. Specifically, it quantifies the extent to which the light is slowed down when compared to its speed in a vacuum. The higher the index of refraction, the more the light is slowed. For example, the index of refraction of vacuum is exactly 1, meaning light travels through a vacuum without being slowed down at all. In contrast, water has an index of refraction of about 1.33, indicating that light travels slower in water than in a vacuum.

In the context of the exercise, determining the index of refraction for an unknown substance can be enlightening as it provides insights into the optical properties of that material. However, as seen in the solution, an erroneous calculation led to a value of 0.970, which is less than 1. This suggests that light would travel faster in the substance than in a vacuum, which is physically impossible. It's crucial to remember that the index of refraction will always be equal to or greater than 1.

The angle of incidence is defined as the angle between a ray incident on a surface and the line perpendicular to the surface at the point of incidence, often called the normal. This angle is measured in the incident medium, which in our exercise is water. The angle of incidence is crucial in determining the angle of refraction through Snell's Law, and it is an essential factor when analyzing light's behavior at the interface of two different media.

When light passes from one medium to another, an incident ray that represents the incoming light is refracted due to changes in speed caused by the differing indices of refraction in the two media. For instance, if light travels from air into water, the angle of incidence affecting how much the light ray will bend towards the normal upon entering the water. The concept of the angle of incidence is fundamental to understanding how light behaves when it reaches the boundary between two different materials and applies to a wide range of phenomena, from the basic refraction occurring in a glass of water to complex optics in lenses and instruments.

Conversely, the angle of refraction is the angle between the refracted ray and the normal to the interface between two different media. It is measured in the refractive medium and is indicative of the direction the light ray will take after passing through the interface. Snell's Law, which relates the angle of incidence to the angle of refraction, is a critical tool for predicting the path of a light ray after refraction occurs.

Returning to the original problem, the angle of refraction is given as 14.9 degrees when the light exits the water into an unknown substance. The ratio of the sine of the angle of incidence to the sine of the angle of refraction is equal to the ratio of the indices of refraction of the two media according to Snell's Law. This relationship allows us to solve for unknown optical parameters when provided with sufficient information. Misconceptions about the angles or incorrect measurements can result in calculations that defy physical laws, such as an index of refraction less than one. Correctly understanding the angle of refraction is thus not only crucial for grasping basic optical principles but also for ensuring that physical realities are not mistakenly contradicted.