A layer of oil (n=1.45) floats on water (n=1.33). A ray of...

A layer of oil $(n=1.45)$ floats on water $(n=1.33)$. A ray of light shines onto the oil with an incidence angle of $40.0^{\circ}$. Find the angle the ray makes in the water. (See Fig. $37-5 .$ ) At the air-oil interface, Snell's Law gives $$ n_{\text {air }} \sin 40^{\circ}=n_{\text {oil }} \sin \theta_{\text {oil }} $$ At the oil-water interface, we have (using the equality of alternate angles) $$ n_{\text {oil }} \sin \theta_{\text {oil }}=n_{\text {water }} \sin \theta_{\text {water }} $$ Thus, $n_{\text {air }} \sin 40.0^{\circ}=n_{\text {water }} \sin \theta_{\text {water }}$; the overall refraction occurs just as though the oil layer were absent. Solving gives $$ \sin \theta_{\text {water }}=\frac{n_{\text {air }} \sin 40.0^{\circ}}{n_{\text {water }}}=\frac{(1)(0.643)}{1.33} \quad \text { or } \quad \theta_{\text {water }}=28.9^{\circ} $$

A layer of oil $(n=1.45)$ floats on water $(n=1.33)$. A ray of light shines onto the oil with an incidence angle of $40.0^{\circ}$. Find the angle the ray makes in the water. (See Fig. $37-5 .$ )
At the air-oil interface, Snell's Law gives
$$
n_{\text {air }} \sin 40^{\circ}=n_{\text {oil }} \sin \theta_{\text {oil }}
$$
At the oil-water interface, we have (using the equality of alternate angles)
$$
n_{\text {oil }} \sin \theta_{\text {oil }}=n_{\text {water }} \sin \theta_{\text {water }}
$$
Thus, $n_{\text {air }} \sin 40.0^{\circ}=n_{\text {water }} \sin \theta_{\text {water }}$; the overall refraction occurs just as though the oil layer were absent. Solving gives
$$
\sin \theta_{\text {water }}=\frac{n_{\text {air }} \sin 40.0^{\circ}}{n_{\text {water }}}=\frac{(1)(0.643)}{1.33} \quad \text { or } \quad \theta_{\text {water }}=28.9^{\circ}
$$

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Key Concepts

Snell's Law

Snell's Law is a fundamental principle in optics that describes how light bends when it passes between media with different refractive indices. It states that the product of the refractive index and the sine of the angle of incidence in one medium is equal to the product for the angle of refraction in the other medium. This law is essential for understanding and predicting the path of light as it transitions from one material to another.

Refractive Index

The refractive index is a measure of how much the speed of light is reduced inside a medium compared to its speed in a vacuum. It is a key property in determining the degree of bending of a light ray when it enters a new medium and plays a critical role in applications ranging from simple lenses to complex optical systems.

Composite Refraction Through Multiple Interfaces

When light passes through multiple layers or interfaces, such as transitioning from air to oil to water, the overall effect on the light ray can be determined by applying Snell's Law at each interface sequentially. In cases where the intermediate layer has parallel boundaries, the net refraction can be equivalent to light passing directly from the initial to the final medium, simplifying the analysis of the light’s path.

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