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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

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.

Chapter 4.7, Problem 71E, Let v1 be the velocity of light in air and v2 the velocity of light in water. According to Fermats

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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