At the very end of Wagner's series of operas Ring of the Nibelung, Brünnhilde takes the golden ring from the finger of the dead Siegfried and throws it into the Rhine, where it sinks to the bottom of the river. Assuming that the ring is small enough compared to the depth of the river to be treated as a point and that the Rhine is 10.0 m deep where the ring goes in, what is the area of the largest circle at the surface of the water over which light from the ring could escape from the water?

Verified step by step guidance

1. Identify the key variables in the problem: depth of the river (h = 10.0 m) and the refractive index of water (n = 1.33 for visible light).
2. Understand the concept of critical angle, which is the angle of incidence in the denser medium (water) beyond which all light is reflected back into the water. Use Snell's Law to find the critical angle, \( \theta_c \), using the formula \( \sin(\theta_c) = \frac{n_2}{n_1} \) where \( n_1 \) is the refractive index of water and \( n_2 \) is the refractive index of air (approximately 1).
3. Calculate the critical angle, \( \theta_c \), using the values identified: \( \sin(\theta_c) = \frac{1}{1.33} \).
4. Determine the radius of the circle at the surface of the water from which light can escape. This can be done using the relationship between the depth of the river and the tangent of the critical angle: \( r = h \cdot \tan(\theta_c) \).
5. Calculate the area of the circle using the formula for the area of a circle, \( A = \pi r^2 \), where \( r \) is the radius calculated in the previous step.

Key Concepts

Refraction of Light

Refraction is the bending of light as it passes from one medium to another, such as from water to air. This phenomenon occurs because light travels at different speeds in different materials. The degree of bending is described by Snell's Law, which relates the angles of incidence and refraction to the indices of refraction of the two media. Understanding refraction is crucial for determining how light from the submerged ring can escape the water's surface.

Critical Angle

The critical angle is the angle of incidence above which light cannot pass through the boundary between two media and is instead reflected back. It occurs when light travels from a denser medium (like water) to a less dense medium (like air). The critical angle can be calculated using the formula: sin(θc) = n2/n1, where n1 and n2 are the refractive indices of the two media. This concept is essential for understanding the limits of light escape from the water's surface.

Circle of Light (Crown of Light)

The circle of light, or crown of light, refers to the area on the surface of a body of water from which light can escape when it is emitted from a point source submerged below the surface. The radius of this circle is determined by the critical angle and the depth of the water. As light escapes, it creates a circular area on the surface, which can be calculated using geometric principles. This concept is key to solving the problem of determining the area over which light from the ring can escape.