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Ch 34: Ray Optics
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
All textbooksKnight Calc 5th EditionCh 34: Ray OpticsProblem 81a
Chapter 34, Problem 81a

CALC FIGURE CP34.81 shows a light ray that travels from point A to point B. The ray crosses the boundary at position x, making angles θ1 and θ2 in the two media. Suppose that you did not know Snell's law. You've proven that Snell's law is equivalent to the statement that 'light traveling between two points follows the path that requires the shortest time.' This interesting way of thinking about refraction is called Fermat's principle. Write an expression for the time t it takes the light ray to travel from A to B. Your expression should be in terms of the distances a, b, and w; the variable x; and the indices of refraction n1 and n2.

Verified step by step guidance
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Step 1: Begin by understanding Fermat's principle, which states that light travels between two points along the path that requires the shortest time. To calculate the total time t, we need to sum the time taken by the light ray in each medium.
Step 2: The time taken by light to travel a distance d in a medium is given by t = d / v, where v is the speed of light in that medium. The speed of light in a medium is related to the index of refraction n by v = c / n, where c is the speed of light in a vacuum. Thus, t = n * d / c.
Step 3: In the first medium (with index of refraction n1), the distance traveled by the light ray is d1. Using trigonometry, d1 can be expressed as d1 = sqrt(a^2 + x^2). The time taken in the first medium is t1 = n1 * d1 / c = n1 * sqrt(a^2 + x^2) / c.
Step 4: In the second medium (with index of refraction n2), the distance traveled by the light ray is d2. Using trigonometry, d2 can be expressed as d2 = sqrt(b^2 + (w - x)^2). The time taken in the second medium is t2 = n2 * d2 / c = n2 * sqrt(b^2 + (w - x)^2) / c.
Step 5: Combine the times from both media to write the total time t. The total time is t = t1 + t2 = (n1 * sqrt(a^2 + x^2) / c) + (n2 * sqrt(b^2 + (w - x)^2) / c). This is the expression for the time it takes the light ray to travel from A to B.

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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 refracts when it passes between two different media. It states that the ratio of the sines of the angles of incidence and refraction is equal to the ratio of the indices of refraction of the two media. Mathematically, it is expressed as n1 * sin(θ1) = n2 * sin(θ2), where n1 and n2 are the indices of refraction, and θ1 and θ2 are the angles of incidence and refraction, respectively.
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Snell's Law

Fermat's Principle

Fermat's Principle posits that light travels between two points along the path that takes the least time. This principle can be used to derive Snell's Law and provides a conceptual framework for understanding refraction. By applying this principle, one can analyze the time taken for light to traverse different paths and determine the optimal route based on the varying speeds of light in different media.
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Optical Path Length

The optical path length is a concept that combines the physical distance light travels with the refractive index of the medium. It is defined as the product of the distance traveled in a medium and the refractive index of that medium. In the context of the question, the total time taken for light to travel from point A to B can be expressed in terms of the distances a, b, and w, and the indices of refraction n1 and n2, allowing for the calculation of the shortest time path.
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Related Practice
Textbook Question

The mirror in FIGURE CP34.79 is covered with a piece of glass whose thickness at the center equals the mirror's radius of curvature. A point source of light is outside the glass. How far from the mirror is the image of this source?

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Textbook Question

A fortune teller's 'crystal ball' (actually just glass) is 10 cm in diameter. Her secret ring is placed 6.0 cm from the edge of the ball. The crystal ball is removed and a thin lens is placed where the center of the ball had been. If the image is still in the same position, what is the focal length of the lens?

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Textbook Question

A 2.0-cm-tall object is placed in front of a mirror. A 1.0-cm-tall upright image is formed behind the mirror, 150 cm from the object. What is the focal length of the mirror?

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Textbook Question

A fortune teller's 'crystal ball' (actually just glass) is 10 cm in diameter. Her secret ring is placed 6.0 cm from the edge of the ball. An image of the ring appears on the opposite side of the crystal ball. How far is the image from the center of the ball?

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Textbook Question

Consider a lens having index of refraction n₂ and surfaces with radii R₁ and R₂. The lens is immersed in a fluid that has index of refraction n₁. A symmetric converging glass lens (i.e., two equally curved surfaces) has two surfaces with radii of 40 cm. Find the focal length of this lens in air and the focal length of this lens in water.

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