Table of Contents
## Fermat’s Principle: A Prelude
The story begins with Fermat’s Principle, which posits that the path taken by light between two points is the one that minimizes the time taken for the journey. At its core, it’s a principle of optimization, deeply rooted in the nature of light itself.
## The Stage: The Geometry of Light Path
Imagine a beam of light traveling through air, encountering a glass slab, and emerging back into the air. The path is not straight but slightly deviated due to the change in speed of light in different media. Let’s consider this scenario mathematically:
1. 
is the distance light travels in air before entering the glass, and its length is 
.
2. 
is the initial entry into the glass slab, and its length is 
.
3. 
is the variable distance 
that light further penetrates into the glass.
4. 
is the width of the glass slab, and its length is 
.
The speed of light in air is 
and in the glass it is 
.
## The Equation of Time
For light to travel from 
to 
and then to some point 
along 
, the total time 
is given by:
![\[T = \frac{\sqrt{a^2 + (b+x)^2}}{c_a} + \frac{\sqrt{w^2 + \left(\frac{wb}{a} - x\right)^2}}{c_g}\]](https://lazying.art/wp-content/ql-cache/quicklatex.com-e4191fcba685a9f044ec4d9be3751c5f_l3.png "Rendered by QuickLaTeX.com")
## The Calculus of Light
The crux of Fermat’s Principle lies in finding the value of that minimizes . Mathematically, this is equivalent to finding 
and setting it to zero:
![\[\frac{dT}{dx} = \frac{x - \frac{bw}{a}}{c_g \sqrt{w^2 + \left(x - \frac{bw}{a}\right)^2}} + \frac{b+x}{c_a \sqrt{a^2 + (b+x)^2}} = 0\]](https://lazying.art/wp-content/ql-cache/quicklatex.com-cf87ec9274e8cfe351c439296efca10f_l3.png "Rendered by QuickLaTeX.com")
Rearranging terms gives:
![\[c_a (x - \frac{bw}{a}) \sqrt{a^2 + (b+x)^2} = - c_g (b+x) \sqrt{w^2 + \left(x - \frac{bw}{a}\right)^2}\]](https://lazying.art/wp-content/ql-cache/quicklatex.com-c186f2d1c71cf23efd9de577e975913f_l3.png "Rendered by QuickLaTeX.com")
## The Grand Finale: Snell’s Law
Finally, let’s introduce the speed of light in vacuum, 
, and relate it to the speeds in the two media: 
and 
. The equation becomes:
![\[n_a (x - \frac{bw}{a}) \sqrt{a^2 + (b+x)^2} = - n_g (b+x) \sqrt{w^2 + \left(x - \frac{bw}{a}\right)^2}\]](https://lazying.art/wp-content/ql-cache/quicklatex.com-32416ea023aea0689b9e8232fb407e19_l3.png "Rendered by QuickLaTeX.com")
And voila, this is Snell’s Law in disguise. The terms involving and are essentially the sines of the angles of incidence and refraction, and 
and 
are the indices of refraction for air and glass, respectively.
Thus, from a simple premise of light’s inherent tendency to minimize its travel time, emerges a law that holds the secrets to a multitude of phenomena—from the vivid colors of a rainbow to the intricate workings of optical fibers. Indeed, in the grand theatre of physics, Snell’s Law and Fermat’s Principle are not mere equations, but the poetic verses that describe the ballet of light.
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