A hiker stands on an isolated mountain peak near sunset and...

A hiker stands on an isolated mountain peak near sunset and observes a rainbow caused by water droplets in the air 8.00 $\mathrm{km}$ away. The valley is 2.00 $\mathrm{km}$ below the mountain peak and entirely flat. What fraction of the complete circular arc of the rainbow is visible to the hiker? (See Fig. $35.24 . )$

Key Concepts

Rainbow Formation and Geometry

A rainbow is formed when sunlight is refracted, internally reflected, and then refracted again inside water droplets, leading to dispersion into constituent colors. The resulting phenomenon follows a precise geometric configuration that produces a circular arc with a fixed angular radius relative to the incoming light direction, which is key to understanding the overall appearance and structure of the rainbow.

Geometric Optics

Geometric optics is the branch of optics dealing with the ray description of light propagation. It explains how the bending (refraction) and bouncing (reflection) of light inside water droplets lead to the formation of a rainbow, providing the theoretical framework to calculate angles and trajectories that determine the observed arc.

Observer’s Position and Horizon Effects

The location of the observer significantly affects the visible fraction of a rainbow. An observer's elevation and the surrounding terrain determine the unobstructed portion of the circular arc, as the Earth's curvature or local topography, like valleys or mountains, may limit the viewable section of the rainbow.

Angular Extent and Visibility Fraction

Determining the fraction of the complete circular arc of a rainbow involves calculating the angular extent of the arc visible above the horizon. This geometric concept uses the relationships between angles, distances, and the observer’s elevation to ascertain what portion of the full 360° circle, or the 180° primary arc typically seen, is actually visible.

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