Shown from above in FIGURE P34.54 is one corner of a rectangular box filled with water. A laser beam starts 10 cm from side A of the container and enters the water at position x. You can ignore the thin walls of the container. If x = 15 cm, does the laser beam refract back into the air through side B or reflect from side B back into the water? Determine the angle of refraction or reflection.

Verified step by step guidance

1. Identify the key concepts involved in the problem: refraction and total internal reflection. Refraction occurs when light passes from one medium to another, changing its speed and direction. Total internal reflection occurs when light traveling in a denser medium (water) hits the boundary with a less dense medium (air) at an angle greater than the critical angle, causing it to reflect back into the denser medium.
2. Determine the geometry of the problem. The laser beam enters the water at position x = 15 cm. The distance from side A to the entry point is 10 cm. Use this information to calculate the angle of incidence at side B. The angle of incidence can be found using trigonometry: \( \tan(\theta_i) = \frac{\text{opposite side}}{\text{adjacent side}} \), where the opposite side is the distance from the entry point to side B, and the adjacent side is the depth of the water.
3. Apply Snell's Law to determine whether the light refracts or reflects. Snell's Law is given by \( n_1 \sin(\theta_i) = n_2 \sin(\theta_r) \), where \( n_1 \) and \( n_2 \) are the refractive indices of water and air, respectively, \( \theta_i \) is the angle of incidence, and \( \theta_r \) is the angle of refraction. Use the refractive index of water (approximately 1.33) and air (approximately 1.00).
4. Calculate the critical angle for total internal reflection. The critical angle \( \theta_c \) is given by \( \sin(\theta_c) = \frac{n_2}{n_1} \). If the angle of incidence \( \theta_i \) is greater than \( \theta_c \), total internal reflection occurs, and the light reflects back into the water. Otherwise, the light refracts into the air.
5. Compare the calculated angle of incidence \( \theta_i \) with the critical angle \( \theta_c \). If \( \theta_i > \theta_c \), the laser beam reflects back into the water. If \( \theta_i \leq \theta_c \), the laser beam refracts into the air. Use the calculated values to determine the angle of refraction or reflection as required.

Key Concepts

Refraction

Refraction is the bending of light as it passes from one medium to another, caused by a change in its speed. When light travels from air (less dense) into water (more dense), it slows down and bends towards the normal line. The degree of bending can be calculated using Snell's Law, which relates the angles of incidence and refraction to the indices of refraction of the two media.

Snell's Law

Snell's Law describes the relationship between the angles of incidence and refraction when light passes between two different media. It is mathematically expressed as n1 * sin(θ1) = n2 * sin(θ2), where n1 and n2 are the indices of refraction for the two media, and θ1 and θ2 are the angles of incidence and refraction, respectively. This law is essential for determining how much the light will bend when entering the water from air.

Total Internal Reflection

Total internal reflection occurs when a light ray attempts to move from a denser medium to a less dense medium at an angle greater than the critical angle, resulting in the light being completely reflected back into the denser medium. The critical angle can be calculated using the indices of refraction of the two media. Understanding this concept is crucial for determining whether the laser beam will reflect off side B or refract into the air.