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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 4.5.56
Chapter 4, Problem 4.5.56

Snell’s Law Suppose a light source at A is in a medium in which light travels at a speed v₁ and that point B is in a medium in which light travels at a speed v₂ (see figure). Using Fermat’s Principle, which states that light travels along the path that requires the minimum travel time (Exercise 55), show that the path taken between points A and B satisfies (sinΘ₁/v₁ = (sin Θ₂) /v₂ . <IMAGE>

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Start by understanding Fermat's Principle, which states that light travels between two points along the path that requires the least time. This principle will guide us in deriving Snell's Law.
Consider the path of light traveling from point A in medium 1 to point B in medium 2. Let the point where the light crosses the boundary between the two media be point P. The total travel time T is the sum of the time taken in each medium.
Express the travel time in each medium. In medium 1, the time taken is T₁ = d₁/v₁, where d₁ is the distance traveled in medium 1. Similarly, in medium 2, the time taken is T₂ = d₂/v₂, where d₂ is the distance traveled in medium 2.
To find the path that minimizes the total travel time T = T₁ + T₂, apply calculus. Use the fact that the derivative of T with respect to the position of point P must be zero for the time to be minimized.
By applying the calculus of variations or using the geometry of the situation, derive the relationship between the angles of incidence and refraction, leading to the equation (sinΘ₁/v₁) = (sinΘ₂/v₂), which is Snell's Law.

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Key Concepts

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

Fermat's Principle

Fermat's Principle states that light travels between two points along the path that requires the least time. This principle can be applied to derive the laws of reflection and refraction, as it implies that the actual path taken by light minimizes the travel time, leading to the formulation of Snell's Law in the context of different media.
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Snell's Law

Snell's Law describes how light refracts when it passes from one medium to another, relating the angles of incidence and refraction to the speeds of light in the respective media. Mathematically, it is expressed as sin(Θ₁)/v₁ = sin(Θ₂)/v₂, where Θ₁ and Θ₂ are the angles of incidence and refraction, and v₁ and v₂ are the speeds of light in the first and second media, respectively.

Refraction

Refraction is the bending of light as it passes from one medium to another due to a change in its speed. This phenomenon is crucial in understanding how light behaves at the interface of different materials, and it is governed by Snell's Law, which quantitatively describes the relationship between the angles of incidence and refraction based on the indices of refraction of the two media.
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