Question

You are looking straight down into a swimming pool, that is, at an angle of \(0^{\circ}\) relative to the vertical, and you see a coin at the bottom of the pool. This coin appears to you to be at a) a lesser depth than it really is. b) the same depth as it really is. c) a greater depth than it really is.

Short answer

Question: When looking straight down into a swimming pool, a coin at the bottom appears to be at: a) a lesser depth than it really is b) the same depth as it really is c) a greater depth than it really is Answer: b) the same depth as it really is.

Step-by-step solution

Diagram the Situation

First, draw a diagram to represent the situation. Draw the vertical side of the pool, representing the actual depth of the pool, and the observer's line of sight (also the line of the light beam) as it moves from the coin to the observer. Label the angle of incidence (in the water) as \(i\) and the angle of refraction (in the air) as \(r\).

Applying Snell's Law

Snell's Law states that 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 refractive indices of the two mediums: \(n_1 \cdot \sin{i} = n_2 \cdot \sin{r}\) Where \(n_1\) is the refractive index of the first medium (water) and \(n_2\) is the refractive index of the second medium (air). In this case, the refractive index for air is approximately \(1.00\) and the refractive index for water is approximately \(1.33\).

Calculate the Angle of Refraction

Since the observer is looking straight down into the pool, the angle of incidence \(i = 0^{\circ}\). Now, we will use Snell's Law to find the angle of refraction \(r\): \(1.33 \cdot \sin{0^{\circ}} = 1.00 \cdot \sin{r}\) Since \(\sin{0^{\circ}} = 0\), the equation simplifies to: \(0 = \sin{r}\) Therefore, \(r = 0^{\circ}\).

Conclude the Perceived Depth

Since both the angle of incidence and angle of refraction are \(0^{\circ}\), the coin appears to be at the same depth as it actually is. Therefore, the answer is: b) the same depth as it really is.

Key concepts

Snell's Law

Snell's Law is a fundamental principle in optics that describes how light changes direction, or refracts, as it passes from one medium into another with a different refractive index. The key formula for Snell's Law is:
\[\begin{equation} n_1 \cdot \sin{i} = n_2 \cdot \sin{r}\end{equation}\]
Here, n_1 and n_2 are the refractive indices of the first and second media, respectively. The variables i and r represent the angle of incidence and the angle of refraction, respectively. When light enters a new medium at an angle, it bends towards the normal if entering a denser medium or away from the normal if entering a less dense medium. This bending effect can influence how we perceive the position of an object under water, such as a coin at the bottom of a pool. Applying Snell's Law properly in exercises allows us to predict the behavior of light as it crosses the boundary between two different media.

Angle of Incidence

The angle of incidence is the angle between the incident ray of light approaching a surface and the perpendicular, or normal, to that surface at the point of contact. A light ray approaching straight on has an angle of incidence of 0 degrees, meaning it will not refract and continue on its original path. However, with any other angle, refraction occurs according to Snell's Law. Understanding this concept helps to predict how much light will bend when entering a new medium. For example, when looking straight down into water, the angle of incidence is 0 degrees, and thus, no refraction occurs, and objects appear at their true depth—a key factor in optical activities like pool depth estimation.

Refractive Index

The refractive index, often denoted as n, is a dimensionless number that describes how light propagates through a particular medium. It is defined as the ratio of the speed of light in a vacuum to the speed of light in the medium. Materials with a higher refractive index will bend light more compared to those with a lower refractive index. Common refractive indices include about 1.00 for air and 1.33 for water. These values are critical when applying Snell's Law, since they determine the extent to which light bends when transitioning between air and water, ultimately affecting how we perceive the position and depth of underwater objects.

Perceived Depth in Water

Perceived depth in water refers to the optical illusion that occurs when we observe objects submerged in water; they often appear shallower or closer to the surface than they really are. This phenomenon can be attributed to the refractive index of water and Snell's Law. When light rays from a submerged object travel to the observer's eye, they bend away from the normal as they move from the denser water to the less dense air. This phenomenon can lead to inaccurate estimations of depth if not accounted for. For instance, a diver might perceive the depth of the pool's bottom differently from its true depth. However, in the case of looking straight down, where the angle of incidence is 0 degrees, the perceived depth is accurate, as the light does not bend—it travels straight upwards into the observer's eye.