A scuba diver training in a pool looks at his instructor as shown below. What angle does the ray from the instructor's face make with the perpendicular to the water at the point where the ray enters? The angle between the ray in the water and the perpendicular to the water is \(25.0^{\circ}\).

The incident ray in optics is the ray that strikes a surface, making contact with it at a specific point and angle. In the context of Snell's Law, the incident ray is a crucial concept because it helps determine the path of light as it moves between different media with varying optical densities.

When light travels through different substances, it bends at the interface; this bending is due to the change in speed of light in different materials. The incident ray and the refracted ray are related to each other by Snell's Law, which is written as: \[n_1 \sin(\theta_1) = n_2 \sin(\theta_2)\].
Here, \(n_1\) and \(n_2\) are the refractive indices of the two media, and \(\theta_1\) is the angle of incidence—the angle between the incident ray and the perpendicular to the surface at the point of contact. This law can be employed to solve various optical phenomena such as the bending of light when moving from air to water, which is a common scenario for swimmers and divers.

The refractive index is a dimensionless number that indicates how much the speed of light is reduced in a particular medium compared to the speed of light in a vacuum. When light moves from a medium with a low refractive index to a medium with a higher refractive index, it bends towards the normal, or perpendicular line, at the point of incidence. Conversely, it bends away from the normal when moving into a less dense medium.

In the given scuba diver problem, the refractive indices provided for water and air are essential in understanding the behavior of light and using Snell's Law effectively. Interestingly, temperature and wavelength can affect the refractive index, although these considerations are not part of this particular problem. An important point for students to remember is that the refractive index helps us predict how much an incident ray will bend upon entering a different medium.

The angle of refraction is the angle between the refracted ray and the normal to the surface at the point where the light enters the new medium. In Snell's Law, this angle demonstrates how much the path of the incident ray has been altered due to entering a new medium.

In the exercise involving the scuba diver, we calculated this angle to understand the change in direction of the ray from the instructor's face as it leaves the water. It is important to grasp that the angle of refraction is dependent on both the angle of incidence and the ratio of the refractive indices of the two media. If the incident ray enters the medium at a larger angle with respect to the normal, the refracted ray will bend more significantly, and the angle of refraction will reflect this change.
When solving for the angle of refraction, there's usually a step involving inverse trigonometric functions, as demonstrated in the 'step by step solution'. Acknowledging that the angle of refraction can also give us information about the perceived position of objects under or above the water will help students understand phenomena like the 'bent pencil' illusion or why the pool's depth appears shallower than it really is.