A 4.00-m-long pole stands vertically in a freshwater lake...

A 4.00-m-long pole stands vertically in a freshwater lake having a depth of $2.00 \mathrm{m}$. The Sun is $40.0^{\circ}$ above the horizontal. Determine the length of the pole's shadow on the bottom of the lake.

Key Concepts

Trigonometric Analysis in Ray Geometry

Geometric relationships, particularly those involving sine, cosine, and tangent functions, are used to relate vertical distances and horizontal displacements in the path of the light ray. This analysis is key for determining the overall lateral shift of the light–and hence the length of the shadow–as it travels through media of different depths.

Continuity of the Horizontal Component of a Light Ray

At a flat interface between two media, the horizontal component of the light ray’s propagation remains unchanged despite refraction. This continuity is important because it allows the horizontal displacements in the air and water to be summed directly, thereby making it possible to compute the total horizontal distance from the object’s base to the point where the ray meets the bottom.

Snell's Law

This fundamental principle in optics states that when light passes from one medium to another, the product of the medium’s refractive index and the sine of the angle made with the normal remains constant. It is essential for determining how the light ray changes direction at the interface between air and water, influencing the path taken by the light that creates the shadow.

Angle of Incidence and Refraction

Understanding the relationship between the angle at which light hits an interface (the angle of incidence) and the angle it travels in the new medium (the angle of refraction) is crucial. This concept enables one to calculate the altered direction of the ray after entering the second medium, which affects the position of the shadow.

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