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. Find the minimum value of x for which the laser beam passes through side B and emerges into the air.

Verified step by step guidance

1. Step 1: Understand the problem setup. The laser beam enters the water at position x along side A and refracts due to the change in medium. The goal is to find the minimum value of x such that the beam passes through side B and emerges into the air. This involves applying Snell's Law and geometry.
2. Step 2: Apply Snell's Law at the point where the laser beam enters the water. Snell's Law is given by: $n_1\sin \theta =n_2\sin \theta \text{'}$, where n₁ is the refractive index of air (approximately 1), n₂ is the refractive index of water (approximately 1.33), θ is the angle of incidence, and θ' is the angle of refraction.
3. Step 3: Use geometry to relate the position x to the angle of refraction θ'. The laser beam travels diagonally through the water, forming a right triangle. The horizontal distance traveled by the beam is the width of the container (10 cm), and the vertical distance is related to x. Use trigonometric relationships: $\tan \theta \text{'}=\frac{x}{10}$.
4. Step 4: Combine Snell's Law and the trigonometric relationship. Solve for θ' using Snell's Law, then substitute θ' into the tangent equation to find x. Rearrange the equation to isolate x: $x=10*\tan \theta \text{'}$.
5. Step 5: Determine the minimum x by considering the critical angle for total internal reflection at side B. The critical angle occurs when the refracted beam just grazes side B. Use the relationship between the critical angle and the refractive indices: $\sin \theta \text{'}=\frac{n_1}{n_2}$. Calculate θ' and substitute it back into the tangent equation to find the minimum x.

Key Concepts

Refraction of Light

Refraction is the bending of light as it passes from one medium to another, caused by a change in its speed. In this scenario, the laser beam transitions from air into water, which has a different refractive index. This bending is described by 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 mathematically describes how light refracts at the interface between two media. It is expressed as n1 * sin(θ1) = n2 * sin(θ2), where n1 and n2 are the refractive indices of the first and second media, respectively, and θ1 and θ2 are the angles of incidence and refraction. This law is crucial for determining the path of the laser beam as it enters the water and exits through side B.

Critical Angle and Total Internal Reflection

The critical angle is the angle of incidence above which total internal reflection occurs when light attempts to move from a denser medium to a less dense medium. If the angle of incidence exceeds this critical angle, the light will not pass through but instead reflect back into the denser medium. Understanding this concept is essential for determining the minimum value of x, ensuring the laser beam exits the water without being totally internally reflected.