In Fig. 33-47 a, a light ray in an underlying material is...

In Fig. $33-47 a,$ a light ray in an underlying material is incident at angle $\theta_{1}$ on a boundary with water, and some of the light refracts into the water. There are two choices of underlying material. For each, the angle of refraction $\theta_{2}$ versus the incident angle $\theta_{1}$ is given in Fig. $33-47 b$. The horizontal axis scale is set by $\theta_{1 y}=90^{\circ} .$ Without calculation, determine whether the index of refraction of (a) material 1 and (b) material 2 is greater or less than the index of water $(n=1.33) .$ What is the index of refraction of (c) material 1 and (d) material $2 ?$

Key Concepts

Graphical Analysis of Angle Relationships

Graphical representations of ?? versus ?? are used to qualitatively assess how different materials behave at a boundary. By analyzing the slope and intercept behavior of such graphs, one can infer whether the first medium is optically denser or rarer than the second medium, and estimate relative index values without resorting to explicit calculations.

Critical Angle and Total Internal Reflection

When light attempts to travel from a medium with a higher index of refraction to one with a lower index (such as from water to air), there exists a critical angle beyond which light is completely reflected back into the denser medium. This concept helps in understanding the behavior of refracted rays at interfaces and is useful for inferring relative indices based on the measured angles of refraction.

Snell's Law

This fundamental law of optics relates the angles of incidence and refraction when light passes between two media with different optical densities. It states that n? sin?? = n? sin??, where n? and n? are the indices of refraction of the first and second media, and ?? and ?? are the angles the ray makes with the normal in each medium. This relationship is essential for comparing the bending of light as it moves from one material to another.

Index of Refraction

The index of refraction is a measure of how much light slows down in a medium relative to its speed in a vacuum. It determines the degree to which light is refracted when it enters a new medium. Higher values indicate a greater slowing down (or higher optical density) and consequently more pronounced bending, whereas lower values indicate less bending.

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