Question

A light ray travels from point A in a medium with index of refraction $n_1$ toward point $\mathrm{B}$ in a medium with index of refraction $n_2$. Assume that the a. Write the time functional in terms of the travel path $y(x)$. b. Apply Fermat's Principle of least time to write the Euler Equation for this functional. c. Solve the equation in part b. and show that $n\sin \theta$ is a constant. d. Let the point at which the light is incident to the interface be at $(x,0)$. Write an expression for the total time to travel from point $A$ at $(x_1,y_1)$ to point $B$ at $(x_2,y_2)$ in terms of the indices of refraction. and the coordinates $x,x_1,x_2$, and $y_2$. e. Treating the time as a function of $x$, minimize this function as a function of one variable and derive Snell's law of refraction.

Short answer

The Euler's equation for the time functional in terms of travel path $y(x)$ results in $y^{\prime }/\sqrt{1-(y^{\prime }{)}^2}=C$. It was shown that $n\sin \theta$ is a constant which led to Snell's law, $n_1\sin {\theta }_1=n_2\sin {\theta }_2$

Step-by-step solution

Writing the Time Functional

The time taken for light to travel a path is the path length divided by the speed of light in the medium. Let $k$ be a constant and $y=y(x)$ be a function defining the path of ray. Then the time, $t$, for light to travel from point $A$ to point $B$ is $t={\int }_A^{B}kdx\sqrt{1+(y^{\prime }(x){)}^2}$.

Applying Fermat's Principle and Obtaining Euler's Equation

Fermat's principle states that the light travels a path which minimizes the time taken. Applying this to the time functional gives the Euler's equation: $\frac{\partial F}{\partial y}-\frac{d}{dx}\frac{\partial F}{\partial y^{\prime }}=0$ where $F=k\sqrt{1+(y^{\prime }{)}^2}$. Solving the equation leads to the equation $y^{\prime }/\sqrt{1-(y^{\prime }{)}^2}=C$ where $C$ is a constant

Show that $n\sin \theta$ is Constant

Label the angle between the ray and the normal in the first medium as ${\theta }_1$ and in the second medium as ${\theta }_2$. Applying trigonometry on the Euler’s equation gives $y^{\prime }=\sin \theta$. By Snell's law $n\sin \theta$ is a constant.

Deriving Expression for Total Travelling Time

Assuming light changes medium at $x$, the total time is expressed as $t=k(n_1\sqrt{(x-x_1{)}^2+y_1^2}+n_2\sqrt{(x_2-x{)}^2+y_2^2})$.

Deriving Snell's law of Refraction

Setting derivative $\frac{dt}{dx}$ equal to zero, the minimum time is given where $n_1\frac{(x_1-x)}{\sqrt{(x_1-x{)}^2+y_1^2}}=n_2\frac{(x_2-x)}{\sqrt{(x_2-x{)}^2+y_2^2}}$. Using the fact that $\sin \theta =\frac{y}{\sqrt{x^2+y^2}}$, the result is the Snell's law: $n_1\sin {\theta }_1=n_2\sin {\theta }_2$.

Key concepts

Snell's Law

Snell's Law describes how light behaves when it crosses the boundary between two different media. For example, moving from air to water. It's named after Willebrord Snell, who discovered the relationship in the 17th century. To visualize, imagine you are looking at a stick partially in water. The stick appears bent at the water's surface. This bending of light as it changes speed from one medium to another is refraction, and Snell’s Law predicts the angle of this bending. Snell's Law states: $$n_1\sin {\theta }_1=n_2\sin {\theta }_2$$ where:

• $n_1$ and $n_2$ are the indices of refraction for the initial and second medium respectively.
• ${\theta }_1$ is the angle of the incoming light relative to the normal at the boundary.
• ${\theta }_2$ is the angle relative to the normal in the second medium.

These angles are measured from the normal, an imaginary perpendicular line at the boundary. Understanding this law helps predict how lenses work or how to design glasses and other optical devices.

Euler's Equation

Euler's Equation is a fundamental tool in calculus of variations. It helps find the path or shape of objects that minimize or maximize a certain quantity. In optics, Euler's Equation is used to apply Fermat's Principle of least time. Fermat’s Principle states that light travels between two points following the path that takes the least time. This concept helps us derive essential laws like Snell's Law. Mathematically, Euler's equation looks like this: $$\frac{\partial F}{\partial y}-\frac{d}{dx}\frac{\partial F}{\partial y^{\prime }}=0$$ Here, $F$ represents a functional, often related to the time light takes to travel a particular path. By solving this equation, one can determine the path taken by light. This equation is a central pillar in varying contexts, not limited to optics alone but applicable in mechanics and other variations-focused areas. Knowing it provides a stepping stone to solving complex problems about natural phenomena.

Index of Refraction

The index of refraction is a number that describes how fast light travels through a material. It's denoted by the symbol $n$. When light enters a medium from the vacuum, the speed of light inside the medium is slower than in a vacuum. The index of refraction relates to this change in speed by the formula: $$n=\frac{c}{v}$$ where:

• $c$ is the speed of light in vacuum, approximately $299,792,458$ meters per second.
• $v$ is the speed of light in the medium in question.

For instance, the index of refraction for water is about $1.33$. This means light travels more slowly in water than it does in a vacuum. Similarly, for glasses like crown glass, the index might be around $1.52$. Understanding the index of refraction helps in calculating how light bends upon entering a new medium, essential for designing lenses and understanding natural phenomena like rainbows and mirages.