As shown in Fig. 37-6, a small luminous body, at the bottom...

As shown in Fig. $37-6$, a small luminous body, at the bottom of a pool of water $(n=4 / 3) 2.00 \mathrm{~m}$ deep, emits rays upward in all directions. A circular area of light is formed at the surface of the water. Determine the radius $R$ of the circle of light. The circular area is formed by rays refracted into the air. The angle $\theta_{c}$ must be the critical angle, because total internal reflection, and hence no refraction, occurs when the angle of incidence in the water is greater than the critical angle. We have, then, $$ \sin \theta_{c}=\frac{n_{a}}{n_{w}}=\frac{1}{4 / 3} \quad \text { or } \quad \theta_{c}=48.6^{\circ} $$ From the figure, $$ R=(2.00 \mathrm{~m}) \tan \theta_{c}=(2.00 \mathrm{~m})(1.13)=2.26 \mathrm{~m} $$

As shown in Fig. $37-6$, a small luminous body, at the bottom of a pool of water $(n=4 / 3) 2.00 \mathrm{~m}$ deep, emits rays upward in all directions. A circular area of light is formed at the surface of the water. Determine the radius $R$ of the circle of light.
The circular area is formed by rays refracted into the air. The angle $\theta_{c}$ must be the critical angle, because total internal reflection, and hence no refraction, occurs when the angle of incidence in the water is greater than the critical angle. We have, then,
$$
\sin \theta_{c}=\frac{n_{a}}{n_{w}}=\frac{1}{4 / 3} \quad \text { or } \quad \theta_{c}=48.6^{\circ}
$$
From the figure,
$$
R=(2.00 \mathrm{~m}) \tan \theta_{c}=(2.00 \mathrm{~m})(1.13)=2.26 \mathrm{~m}
$$

Key Concepts

Refractive Index

The refractive index of a medium measures how much the speed of light is reduced inside the medium compared to a vacuum. It is a critical property that explains how light bends, or refracts, when entering or leaving a medium, and is fundamental in describing the optical behavior at material interfaces.

Snell's Law

Snell's Law is a formula used to describe the relationship between the angles of incidence and refraction when light passes between two media with different refractive indices. It quantitatively expresses how much a light ray will bend during refraction and is essential for analyzing optical phenomena at boundaries.

Critical Angle

The critical angle is the minimum angle of incidence in a denser medium for which light, when striking the interface with a less dense medium, is refracted at 90 degrees along the boundary. It is pivotal in determining the conditions under which total internal reflection occurs.

Total Internal Reflection

Total internal reflection is the phenomenon where a light ray, traveling from a medium with a higher refractive index to one with a lower refractive index, is completely reflected back into the original medium when the angle of incidence exceeds the critical angle. This concept is not only important in understanding light behavior at interfaces but also has practical applications in devices such as optical fibers.

Trigonometric Relationships in Optics

Trigonometric relationships, such as those involving tangent and sine functions, are used to relate geometric dimensions in optical setups. These relationships allow calculation of distances, angles, and radii of light patterns (for example, the circular area of light on a surface), thereby linking geometric configurations with physical optics principles.

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