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Mathematical Methods in the Physical Sciences

Mary L. Boas

Chapter 9

Calculus of Variations - all with Video Answers

Educators

Section 1

Introduction

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Problem 1

The speed of light in a medium of index of refraction $n$ is $v=d s / d t=c / n .$ Then the time of transit from $A$ to $B$ is $t=\int_{A}^{B} d t=c^{-1} \int_{A}^{B} n d s .$ By Fermat's principle above, $t$ is stationary. If the path consists of two straight line segments with $n$ constant over each segment, then $\int_{A}^{B} n d s=n_{1} d_{1}+n_{2} d_{2}$ and the problem can be done by ordinary calculus. Thus solve the following problems:
Derive the optical law of reflection. Hint: Let light go from the point $A=\left(x_{1}, y_{1}\right)$ to $B=\left(x_{2}, y_{2}\right)$ via an arbitrary point $P=$ $(x, 0)$ on a mirror along the $x$ axis. Set $d t / d x=(n / c) d D / d x=$
$0,$ where $D=$ distance $A P B,$ and show that then $\theta=\phi$.

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 2

The speed of light in a medium of index of refraction $n$ is $v=d s / d t=c / n .$ Then the time of transit from $A$ to $B$ is $t=\int_{A}^{B} d t=c^{-1} \int_{A}^{B} n d s .$ By Fermat's principle above, $t$ is stationary. If the path consists of two straight line segments with $n$ constant over each segment, then $\int_{A}^{B} n d s=n_{1} d_{1}+n_{2} d_{2}$ and the problem can be done by ordinary calculus. Thus solve the following problems:
Derive Snell's law of refraction: $n_{1} \sin \theta_{1}=n_{2} \sin \theta_{2}$ (see figure).

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 3

The speed of light in a medium of index of refraction $n$ is $v=d s / d t=c / n .$ Then the time of transit from $A$ to $B$ is $t=\int_{A}^{B} d t=c^{-1} \int_{A}^{B} n d s .$ By Fermat's principle above, $t$ is stationary. If the path consists of two straight line segments with $n$ constant over each segment, then $\int_{A}^{B} n d s=n_{1} d_{1}+n_{2} d_{2}$ and the problem can be done by ordinary calculus. Thus solve the following problems:
Show that the actual path is not necessarily one of minimum time. Hint: In the diagram, $A$ is a source of light; $C D$ is a cross section of a reflecting surface, and $B$ is a point to which a light ray is to be reflected. $A P B$ is to be the actual path and $A P^{\prime} B, A P^{\prime \prime} B$ represent varied paths. Then show that the varied paths:
(a) Are the same length as the actual path if $C D$ is an ellipse with $A$ and $B$ as foci.
(b) Are longer than the actual path if $C D$ is a line tangent at $P$ to the ellipse in
(a).
(c) Are shorter than the actual path if $C D$ is an arc of a curve tangent to the ellipse at $P$ and lying inside it. Note that in this case the time is a maximum!
(d) Are longer on one side and shorter on the other if $C D$ crosses the ellipse at $P$ but is tangent to it (that is, $C D$ has a point of inflection at $P$ ).

Victor Salazar
Victor Salazar
Numerade Educator