In this lesson we are going to look at a derivation of Snell's Law based on the Principle of Least Time.

 

 

In the diagram shown above, two mediums are juxtapositioned one below the other. A ray of light beginning in the top medium travels through the interface into the bottom medium. When the ray enters the second medium (which we are assuming in the more optically dense medium) its speed will be reduced. Therefore the angle at which it enters the second medium is smaller than the angle from which it left the first medium. Note that the angles are measured between the rays and the normal, NOT between the rays and the interface (aka, surface).

To determine the best path of these two rays whose starting and ending positions are fixed we are going to allow the value of x, or the distance the ray travels in the top medium, the ability to vary. That is, the ray may enter the bottom medium at any point along the interface.

 

 

 

To generalize our calculation, we are going to set up defined quantities based on the two shaded right triangles below.

 

Using the Pythagorean Theorem, we know that (Equations 1)

 

 

Since light travels at a constant speed in each medium, we also know that (Equations 2)

 

 

The total time that the light ray requires to travel between its predetermined starting and ending points can now be written as  (Equation 3)

 

 

In calculus to minimize or maximize a quantity, we takes its derivative and set it equal to zero. (Equations 4)

 

 

Simplifying the derivative gives us, (Equations 5)

 

 

From our original diagram of the two shaded right triangles, notice that (Equations 6)

 

 

giving us (Equation 7)

 

 

 

Now, our final step involves remembering the definition of the index of refraction, (Equation 8)

 

Substituting out terms  in our equation gives us the familiar expression for Snell's Law. (Equations 9)

 

 

Hopefully you now understand why Snell's Law is often referred to as the path of least time between our two points.