Resource Lesson
A Derivation of Snell's Law
Printer Friendly Version
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.
Related Documents
Lab:
Labs -
A Simple Microscope
Labs -
Blank Ray Diagrams for Converging Lenses
Labs -
Blank Ray Diagrams for Converging, Concave, Mirrors
Labs -
Blank Ray Diagrams for Diverging Lenses
Labs -
Blank Ray Diagrams for Diverging, Convex, Mirrors
Labs -
Determining the Focal Length of a Converging Lens
Labs -
Index of Refraction: Glass
Labs -
Index of Refraction: Water
Labs -
Least Time Activity
Labs -
Man and the Mirror
Labs -
Man and the Mirror: Sample Ray Diagram
Labs -
Ray Diagrams for Converging Lenses
Labs -
Ray Diagrams for Converging Mirrors
Labs -
Ray Diagrams for Diverging Lenses
Labs -
Ray Diagrams for Diverging Mirrors
Labs -
Reflections of a Triangle
Labs -
Spherical Mirror Lab
Labs -
Student Lens Lab
Labs -
Target Practice - Revised
Resource Lesson:
RL -
Converging Lens Examples
RL -
Converging Lenses
RL -
Demonstration: Infinite Images
RL -
Demonstration: Real Images
RL -
Demonstration: Virtual Images
RL -
Dispersion
RL -
Diverging Lenses
RL -
Double Lens Systems
RL -
Lensmaker Equation
RL -
Mirror Equation
RL -
Properties of Plane Mirrors
RL -
Refraction of Light
RL -
Refraction Phenomena
RL -
Snell's Law
RL -
Snell's Law: Derivation
RL -
Spherical Mirrors
RL -
Thin Lens Equation
Review:
REV -
Drill: Reflection and Mirrors
REV -
Mirror Properties
REV -
Physics I Honors: 2nd 9-week notebook
REV -
Physics I: 2nd 9-week notebook
REV -
Spherical Lens Properties
Worksheet:
APP -
Enlightened
APP -
Reflections
APP -
The Librarian
APP -
The Starlet
CP -
Lenses
CP -
Plane Mirror Reflections
CP -
Refraction of Light
CP -
Snell's Law
CP -
Snell's Law
NT -
Image Distances
NT -
Laser Fishing
NT -
Mirror Height
NT -
Mirror Length
NT -
Reflection
NT -
Underwater Vision
WS -
An Extension of Snell's Law
WS -
Basic Principles of Refraction
WS -
Converging Lens Vocabulary
WS -
Diverging Lens Vocabulary
WS -
Lensmaker Equation
WS -
Plane Mirror Reflections
WS -
Refraction and Critical Angles
WS -
Refraction Phenomena
WS -
Refraction Through a Circular Disk
WS -
Refraction Through a Glass Plate
WS -
Refraction Through a Triangle
WS -
Snell's Law Calculations
WS -
Spherical Mirror Equation #1
WS -
Spherical Mirror Equation #2
WS -
Spherical Mirrors: Image Patterns
WS -
Thin Lens Equation #1: Converging Lenses
WS -
Thin Lens Equation #2: Converging Lenses
WS -
Thin Lens Equation #3: Both Types
WS -
Thin Lens Equation #4: Both Types
WS -
Two-Lens Worksheet
WS -
Two-Mirror Worksheet
TB -
27B: Properties of Light and Refraction
TB -
Refraction Phenomena Reading Questions
Copyright © 2007-2023
William A. Hilburn
All rights reserved.