   Chapter 4.7, Problem 71E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Let v1 be the velocity of light in air and v2 the velocity of light in water. According to Fermat’s Principle, a ray of light will travel from a point A in the air to a point B in the water by a path ACB that minimizes the time taken. Show that sin θ 1 sin θ 2 = v 1 v 2 where θ1 (the angle of incidence) and θ2 (the angle of refraction) are as shown. This equation is known as Snell’s Law. To determine

To show: The Snell’s Law, that is sinθ1sinθ2=v1v2

Explanation

Given:

The angle of incidence=θ1.

The angle of refraction=θ2.

The velocity of light in air =v1

The velocity of light in water =v2

Formula used:

Pythagorean Theorem: If c denotes the length of the hypotenuse and a and b denote the length of the other two sides of a right angle triangle, then the Pythagorean theorem can be expressed as the Pythagorean equation: a2+b2=c2

Calculation:

A ray of light travels from a point A to a point C in the air.

Then the ray travels from the point C to the point B in the water.

In Figure 1,

Let ED=d,EC=x.

Hence, CD=dx

By Pythagorean Theorem for the triangle ΔAEC,

AC=a2+x2

The velocity of light in air =v1

The time consumed for travelling from A to C =a2+x2v1

For the triangle ΔAEC,

sinθ1=xa2+x2

By Pythagorean Theorem for the triangle ΔBDC,

BC=b2+(dx)2

The velocity of light in air =v2

The time consumed for travelling from A to C=b2+(dx)2v2

For the triangle ΔBDC,

sinθ2=dxb2+(dx)2

The total time consumed,

T=a2+x2

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Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 