Why is the following situation impossible? A laser beam...

Why is the following situation impossible? A laser beam strikes one end of a slab of material of length $L=42.0 \mathrm{~cm}$ and thickness $t=3.10 \mathrm{~mm}$ as shown in Figure $\mathrm{P} 34.36$ (not to scale). It enters the material at the center of the left end, striking it at an angle of incidence of $\theta=50.0^{\circ} .$ The index of refraction of the slab is $n=1.48 .$ The light makes 85 internal reflections from the top and bottom of the slab before exiting at the other end.

Key Concepts

Snell's Law

This principle determines how light bends as it passes between media with different refractive indices. In the context of the problem, it is used to find the angle of propagation inside the slab when the beam enters at a known external angle. Snell’s law is fundamental for calculating the refracted angle, which in turn affects the geometric path the light takes inside the material.

Geometric Ray Tracing in a Slab

This concept involves using the geometry of the slab—its thickness and length—to analyze the light ray’s path after entering the material. By applying trigonometric relationships, one can determine the horizontal distance the beam travels between successive reflections on the top and bottom surfaces. This analysis is crucial to assess whether the number of internal reflections is consistent with the finite dimensions of the slab.

Path Length and Reflection Constraints

For a light beam undergoing internal reflections, there is a fixed relationship between the vertical spacing (set by the slab’s thickness) and the horizontal displacement after each reflection. The number of reflections is limited by the available horizontal path length within the slab. If the prescribed number of reflections would require a horizontal distance exceeding the actual length of the slab, the scenario becomes geometrically impossible. This concept ensures that the configuration adheres to the finite dimensions of the medium.

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