Refer to Figure 26.4b and assume the observer is nearly...

Refer to Figure 26.4$b$ and assume the observer is nearly above the submerged object. For this situation, derive the expression for the apparent depth: $d^{\prime}=d\left(n_{2} / n_{1}\right),$ Equation 26.3 . Hint: Use Snell's law of refraction and the fact that the angles of incidence and refraction are small, so tan $\theta \approx \sin \theta$ .)

Key Concepts

Index of Refraction

The index of refraction is a measure of how much light slows down when passing through a medium relative to its speed in a vacuum. It plays a critical role in determining the degree of bending of the light ray as it crosses the interface between media with different refractive indices, directly influencing the perceived position, or apparent depth, of underwater objects.

Apparent Depth

Apparent depth is the perceived depth of an object observed through a medium where refraction occurs. Due to the bending of light rays at the interface between two media with different refractive indices, the object appears to be closer or further than its actual distance. The relationship d' = d(n?/n?) shows how the actual depth (d) is modified by the ratio of the refractive indices to yield the apparent depth (d').

Snell's Law

Snell's Law relates the angles of incidence and refraction when light passes from one medium to another, stating that n? sin?? = n? sin??, where n? and n? are the refractive indices of the first and second media respectively. This principle is fundamental in understanding how the path of light bends at the interface between two substances, which is essential for deriving equations involving apparent depth.

Small Angle Approximation

The small angle approximation assumes that for small angles (measured in radians) the sine of the angle is approximately equal to the tangent of the angle (sin? ? tan?). This simplification is crucial in optical derivations, such as calculating apparent depth, because it allows the use of simpler linear relationships instead of more complex trigonometric ones.

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