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Ch 33: The Nature and Propagation of Light
Young & Freedman Calc - University Physics 14th Edition
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoYoung & Freedman Calc 14th EditionCh 33: The Nature and Propagation of LightProblema 20
Capitolo 33, Problema 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?

Guida verificata passo dopo passo
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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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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07:59
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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Critical Angle

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
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