Question

You are submerged in a swimming pool. What is the maximum angle at which you can see light coming from above the pool surface? That is, what is the angle for total internal reflection from water into air?

Short answer

Answer: The maximum angle to see light coming from above the pool surface is approximately 48.8°.

Step-by-step solution

Write Snell's Law for Water and Air

Snell's Law for refraction between water and air can be written as: n1 * sin(θ1) = n2 * sin(θ2) where n1 = refractive index of water (approx. 1.33), n2 = refractive index of air (approx. 1), θ1 = angle of incidence in water, θ2 = angle of refraction in air.

Determine the Critical Angle for Total Internal Reflection

Let's find the critical angle (θ_C) for which θ2 = 90°. When this happens, total internal reflection occurs, and the light can't escape the water. We can write Snell's Law as: 1.33 * sin(θ_C) = 1 * sin(90°) Solve for θ_C: sin(θ_C) = 1/1.33 θ_C = sin^(-1)(1/1.33)

Calculate the Critical Angle

Now, let's calculate the critical angle: θ_C = sin^(-1)(1/1.33) ≈ 48.8° The critical angle for total internal reflection between water and air is approximately 48.8°.

Determine the Maximum Angle to See Light from Above the Pool Surface

Since the critical angle is the maximum angle at which light can be seen coming from above the pool surface without having total internal reflection, the answer to the problem is: The maximum angle to see light coming from above the pool surface is approximately 48.8°.

Key concepts

Understanding Snell's Law

Snell's Law is a fundamental principle in the study of optics that describes how light bends when it passes from one medium into another. This bending, known as refraction, occurs because light travels at different speeds in different substances.

According to Snell's Law, the relationship between the angles of incidence (\theta_1) and refraction (\theta_2), and the refractive indices of the two media (n1 and n2) can be expressed with the formula: 1\[\begin{equation} n1 * \text{{sin}}(\theta_1) = n2 * \text{{sin}}(\theta_2)\end{equation}\]where \(\theta_1\) is the angle at which light hits the boundary between the two media, and \(\theta_2\) is the angle at which light exits.

The law helps us understand not just how light bends but also under what conditions phenomena like total internal reflection occur. In the exercise example where you're in a pool, understanding Snell's Law allows you to calculate the maximum angle at which you can observe light before it is totally internally reflected.

What is the Critical Angle?

The critical angle is a special angle of incidence above which total internal reflection occurs. When light travels from a medium with a higher refractive index to one with a lower refractive index, like from water to air, there comes a point where the light doesn't exit the medium but instead reflects entirely back inside.

To find the critical angle (\theta_C), you use Snell's Law by setting the angle of refraction (\theta_2) to 90 degrees, which gives you the equation:1\[\begin{equation} n1 * \text{{sin}}(\theta_C) = n2 * \text{{sin}}(90^\circ)\end{equation}\]Solving this equation for \theta_C provides the critical angle. In our swimming pool example, past this angle, all light reflecting off the surface inside the water would be trapped, causing the phenomenon of total internal reflection that allows you to see the underwater environment illuminated.

Refractive Index Explained

The refractive index (n) of a material is a number that describes how light travels through that material. It technically represents the ratio of the velocity of light in a vacuum to the velocity of light in the material. A higher refractive index means that light travels more slowly in the material and bends more when entering or leaving it.

In the given exercise, the refractive index of water is about 1.33, and the refractive index of air is approximately 1. This comparison shows why light bends when entering or exiting the water—an effect noticeable when you look at objects under water that seem distorted. The refractive index is critical in calculating the angle of incidence that will produce total internal reflection, thereby explaining why we cannot see light from above the surface of the water at certain angles.