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Ch 33: The Nature and Propagation of Light
Young & Freedman Calc - University Physics 15th Edition
Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook
All textbooksYoung & Freedman Calc 15th EditionCh 33: The Nature and Propagation of LightProblem 20
Chapter 32, Problem 20

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
Understand the concept of total internal reflection: Light can escape from water into air only if it strikes the water surface at an angle less than the critical angle. The critical angle is determined by the refractive indices of water and air.
Calculate the critical angle using Snell's Law: The formula for the critical angle \( \theta_c \) is given by \( \sin(\theta_c) = \frac{n_{air}}{n_{water}} \), where \( n_{air} \approx 1.00 \) and \( n_{water} \approx 1.33 \).
Determine the maximum angle of incidence: Light from the ring can escape if it strikes the surface at an angle less than the critical angle. This angle will define the cone of light that can escape.
Calculate the radius of the circle at the surface: Using trigonometry, the radius \( r \) of the circle at the surface can be found using \( r = d \cdot \tan(\theta_c) \), where \( d = 10.0 \) m is the depth of the river.
Find the area of the circle: The area \( A \) of the circle is given by \( A = \pi r^2 \). Substitute the value of \( r \) from the previous step to find the area.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Snell's Law

Snell's Law describes how light bends when it passes from one medium to another, such as from water to air. It is defined by the equation n1*sin(θ1) = n2*sin(θ2), where n1 and n2 are the refractive indices of the two media, and θ1 and θ2 are the angles of incidence and refraction, respectively. Understanding this law is crucial for determining the path of light as it exits the water.
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Snell's Law

Critical Angle

The critical angle is the angle of incidence above which total internal reflection occurs when light travels from a denser medium to a less dense medium. It is calculated using the formula θc = arcsin(n2/n1), where n1 is the refractive index of the denser medium and n2 is that of the less dense medium. This concept helps determine the maximum angle at which light can escape from water into air.
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Geometry of Light Escape

The geometry of light escape involves understanding how light rays travel and form a cone as they exit the water. The base of this cone at the water's surface is a circle, and its size depends on the depth of the source and the critical angle. Calculating the radius of this circle requires applying trigonometric principles to the depth and critical angle, allowing us to find the area over which light can escape.
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Escape Velocity
Related Practice
Textbook Question

Unpolarized light with intensity I0 is incident on two polarizing filters. The axis of the first filter makes an angle of 60.0° with the vertical, and the axis of the second filter is horizontal. What is the intensity of the light after it has passed through the second filter?

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Textbook Question

A beam of light strikes a sheet of glass at an angle of 57.0° with the normal in air. You observe that red light makes an angle of 38.1° with the normal in the glass, while violet light makes a 36.7° angle. What are the speeds of red and violet light in the glass?

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Textbook Question

Light enters a solid pipe made of plastic having an index of refraction of 1.60. The light travels parallel to the upper part of the pipe (Fig. E33.15). You want to cut the face AB so that all the light will reflect back into the pipe after it first strikes that face. If the pipe is immersed in water of refractive index 1.33, what is the largest that u can be?

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Textbook Question

The indexes of refraction for violet light (λ = 400 nm) and red light (λ= 700 nm) in diamond are 2.46 and 2.41, respectively. A ray of light traveling through air strikes the diamond surface at an angle of 53.5° to the normal. Calculate the angular separation between these two colors of light in the refracted ray.

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Textbook Question

Light enters a solid pipe made of plastic having an index of refraction of 1.60. The light travels parallel to the upper part of the pipe (Fig. E33.15). You want to cut the face AB so that all the light will reflect back into the pipe after it first strikes that face. What is the largest that u can be if the pipe is in air?

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Textbook Question

The critical angle for total internal reflection at a liquid–air interface is 42.5°. If a ray of light traveling in the liquid has an angle of incidence at the interface of 35.0°, what angle does the refracted ray in the air make with the normal?

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