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 \(\left(x_{1}, y_{1}\right)\) to point \(B\) at \(\left(x_{2}, y_{2}\right)\) 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.

Step by step solution

01

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 \( t = \int_{A}^{B}kdx \sqrt{1+(y'(x))^2} \).

02

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: \( \frac{\partial F}{\partial y} - \frac{d}{dx}\frac{\partial F}{\partial y'} = 0 \) where \( F = k \sqrt{1+(y')^2} \). Solving the equation leads to the equation \( y'/\sqrt{1-(y')^2} = C \) where \( C \) is a constant

03

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 \( y' = \sin \theta \). By Snell's law \( n \sin \theta \) is a constant.

04

Deriving Expression for Total Travelling Time

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

05

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: \( n_1 \sin \theta_1 = n_2 \sin \theta_2 \).

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