Ch. 32 - Light: Reflection and Refraction

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 32 - Light: Reflection and Refraction

Problem 57

Chapter 31, Problem 57

(II) A ray of light, after entering a light fiber, reflects at an angle of 14.5° with the long axis of the fiber, as in Fig. 32–57. Calculate the distance along the axis of the fiber that the light ray travels between successive reflections off the sides of the fiber. Assume that the fiber has an index of refraction of 1.55 and is 1.60 x 10^-4 m in diameter.

Verified step by step guidance

Step 1: Understand the geometry of the problem. The light ray reflects off the sides of the fiber at an angle of 14.5° with the long axis of the fiber. This means the light ray forms a right triangle with the fiber's diameter as one leg and the distance traveled along the axis between reflections as the other leg.

Step 2: Identify the relationship between the angle of reflection and the geometry of the fiber. The tangent of the angle (14.5°) is equal to the ratio of the fiber's diameter (1.60 × 10⁻⁴ m) to the distance traveled along the axis between reflections. Use the formula:

\tan (θ) = \frac{d}{L}

, where θ is the angle, d is the diameter, and L is the distance along the axis.

Step 3: Rearrange the formula to solve for L, the distance along the axis:

L = \frac{d}{\tan (θ)}

. Substitute the given values: d = 1.60 × 10⁻⁴ m and θ = 14.5°.

Step 4: Calculate the tangent of 14.5° using a scientific calculator or trigonometric table. This will give you the value needed to compute L.

Step 5: Divide the fiber's diameter (1.60 × 10⁻⁴ m) by the tangent of 14.5° to find the distance L along the axis of the fiber between successive reflections. Ensure units are consistent and verify the calculation.

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Key Concepts

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

Total Internal Reflection

Total internal reflection occurs when a light ray traveling in a denser medium hits a boundary with a less dense medium at an angle greater than the critical angle. This phenomenon is crucial in optical fibers, as it allows light to be trapped and guided along the fiber's length. The angle of incidence must exceed the critical angle, which depends on the refractive indices of the two media.

Refractive Index

The refractive index is a dimensionless number that describes how light propagates through a medium. It is defined as the ratio of the speed of light in a vacuum to the speed of light in the medium. In this problem, the fiber has a refractive index of 1.55, indicating that light travels slower in the fiber than in a vacuum, which affects the angles of reflection and refraction.

Geometry of Light Path in Fibers

The geometry of the light path in optical fibers involves understanding how light travels and reflects within the cylindrical structure of the fiber. The distance traveled along the fiber's axis between reflections can be calculated using trigonometric relationships, considering the angle of reflection and the diameter of the fiber. This geometric approach is essential for determining the effective length of the light path.

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Related Practice

Textbook Question

A parallel beam of light containing two wavelengths, λ₁ = 461 nm and λ₂ = 656 nm, enters the silicate flint glass of an equilateral prism as shown in Fig. 32–56. At what angle does each beam leave the prism (give angle with normal to the face)? See Fig. 32–28.

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Textbook Question

A fish is swimming in water inside a thin spherical glass bowl of uniform thickness. Assuming the radius of curvature of the bowl is 32.0 cm, locate the image of the fish if the fish is located: (a) at the center of the bowl; (b) 20.0 cm from the side of the bowl between the observer and the center of the bowl. Assume the fish is small.

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Textbook Question

The critical angle for a certain liquid–air surface is 52.6°. What is the index of refraction of the liquid?

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Textbook Question

(III) A beam of light enters the end of an optic fiber as shown in Fig. 32–59. (a) Show that we can guarantee total internal reflection at the side surface of the material (at point A), if the index of refraction is greater than about 1.42. In other words, regardless of the angle α , the light beam reflects back into the material at point A, assuming air outside. (b) What if the fiber is immersed in water?

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Textbook Question

A light beam strikes a piece of glass at a 55.00° incident angle. The beam contains two wavelengths, 450.0 nm and 700.0 nm, for which the index of refraction of the glass is 1.4831 and 1.4754, respectively. What is the angle between the two refracted beams?

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Textbook Question

A beam of light is emitted 7.7 cm beneath the surface of a liquid and strikes the surface 7.2 cm from the point directly above the source. If total internal reflection occurs, what can you say about the index of refraction of the liquid?

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