(Modeling) Fish's View of the World The figure shows a fish's view of the world above the surface of the water. (Data from Walker, J., 'The Amateur Scientist,' Scientific American.) Suppose that a light ray comes from the horizon, enters the water, and strikes the fish's eye.Assume that this ray gives a value of 90° for angle θ₁ in the formula for Snell's law. (In a practical situation, this angle would probably be a little less than 90°.) The speed of light in water is about 2.254 x 10⁸ m per sec. Find angle θ₂ to the nearest tenth.

Verified step by step guidance

1. Identify the given information: angle \( \theta_1 = 90^\circ \), speed of light in water \( v = 2.254 \times 10^8 \) m/s.
2. Recall Snell's Law: \( n_1 \sin(\theta_1) = n_2 \sin(\theta_2) \), where \( n_1 \) and \( n_2 \) are the refractive indices of air and water, respectively.
3. Use the refractive index formula: \( n = \frac{c}{v} \), where \( c \) is the speed of light in a vacuum (approximately \( 3 \times 10^8 \) m/s). Calculate \( n_2 \) for water.
4. Assume \( n_1 \approx 1 \) for air, and substitute \( \theta_1 = 90^\circ \) into Snell's Law: \( \sin(90^\circ) = 1 \).
5. Solve for \( \theta_2 \) using the equation \( \sin(\theta_2) = \frac{n_1}{n_2} \sin(\theta_1) \) and find \( \theta_2 \) to the nearest tenth.

Key Concepts

Snell's Law

Snell's Law describes how light rays change direction when they pass from one medium to another, such as from air to water. It is mathematically expressed as n₁ * sin(θ₁) = n₂ * sin(θ₂), where n represents the refractive indices of the two media, and θ represents the angles of incidence and refraction. Understanding this law is crucial for solving problems involving refraction, as it allows us to relate the angles and speeds of light in different materials.

Refractive Index

The refractive index is a dimensionless number that describes how fast light travels in a medium compared to its speed in a vacuum. It is defined as n = c/v, where c is the speed of light in a vacuum and v is the speed of light in the medium. For water, the refractive index is approximately 1.33, which indicates that light travels slower in water than in air. This concept is essential for applying Snell's Law and calculating the angles of refraction.

Angle of Incidence and Refraction

The angle of incidence (θ₁) is the angle between the incoming light ray and the normal line at the surface of the medium, while the angle of refraction (θ₂) is the angle between the refracted ray and the normal. These angles are critical in determining how light behaves at the interface of two different media. In this problem, knowing that θ₁ is 90° allows us to directly apply Snell's Law to find θ₂, which is necessary for understanding the fish's perspective of the world above the water.