Fermat’s Principle of Least Time states that light traveling between two fixed points will follow the path that minimizes the total travel time relative to nearby alternative paths. When light transitions between two media with different refractive indices, its speed changes (v = c/n), which causes the path to bend at the interface rather than maintaining a single straight line. This fundamental minimization property directly yields Snell’s Law of Refraction:
Here is the complete step-by-step calculus derivation of Snell’s Law using Fermat’s Principle. 1. Set Up the Geometric Coordinate System
Consider a light ray starting at point A(0, a) in Medium 1 and traveling to point B(L, -b) in Medium 2. The flat boundary interface between the two media lies along the x-axis (y = 0).
Medium 1 has a refractive index of n₁, and the speed of light is
Medium 2 has a refractive index of n₂, and the speed of light is
Let the ray cross the boundary interface at an arbitrary point P(x, 0), where x is the horizontal distance from the origin. The horizontal distance from A to P is x. The horizontal distance from P to B is L – x. 2. Formulate the Total Time Function
By the Pythagorean theorem, we calculate the straight-line distances traveled by the ray in each medium:
Distance AP=x2+a2Distance cap A cap P equals the square root of x squared plus a squared end-root
Distance PB=(L−x)2+b2Distance cap P cap B equals the square root of open paren cap L minus x close paren squared plus b squared end-root Since time equals distance divided by velocity (
), the total travel time T(x) as a function of the boundary intersection point x is:
T(x)=x2+a2v1+(L−x)2+b2v2cap T open paren x close paren equals the fraction with numerator the square root of x squared plus a squared end-root and denominator v sub 1 end-fraction plus the fraction with numerator the square root of open paren cap L minus x close paren squared plus b squared end-root and denominator v sub 2 end-fraction 3. Differentiate the Time Equation
According to Fermat’s Principle, to find the path that minimizes time, we take the derivative of T(x) with respect to x and set it equal to zero (
dTdx=1v1⋅12×2+a2⋅(2x)+1v2⋅12(L−x)2+b2⋅2(L−x)⋅(-1)=0the fraction with numerator d cap T and denominator d x end-fraction equals the fraction with numerator 1 and denominator v sub 1 end-fraction center dot the fraction with numerator 1 and denominator 2 the square root of x squared plus a squared end-root end-fraction center dot open paren 2 x close paren plus the fraction with numerator 1 and denominator v sub 2 end-fraction center dot the fraction with numerator 1 and denominator 2 the square root of open paren cap L minus x close paren squared plus b squared end-root end-fraction center dot 2 open paren cap L minus x close paren center dot open paren negative 1 close paren equals 0 Simplifying this expression yields:
xv1x2+a2−L−xv2(L−x)2+b2=0the fraction with numerator x and denominator v sub 1 the square root of x squared plus a squared end-root end-fraction minus the fraction with numerator cap L minus x and denominator v sub 2 the square root of open paren cap L minus x close paren squared plus b squared end-root end-fraction equals 0
xv1x2+a2=L−xv2(L−x)2+b2the fraction with numerator x and denominator v sub 1 the square root of x squared plus a squared end-root end-fraction equals the fraction with numerator cap L minus x and denominator v sub 2 the square root of open paren cap L minus x close paren squared plus b squared end-root end-fraction 4. Relate Geometry to Trigonometric Angles
We define the angle of incidence θ₁ and the angle of refraction θ₂ relative to the normal line (the vertical y-axis perpendicular to the interface):
Looking at the right triangle formed in Medium 1, the sine of the angle of incidence is
Looking at the right triangle formed in Medium 2, the sine of the angle of refraction is
Substituting these trigonometric relationships back into our simplified optimization derivative gives:
sin(θ1)v1=sin(θ2)v2the fraction with numerator sine open paren theta sub 1 close paren and denominator v sub 1 end-fraction equals the fraction with numerator sine open paren theta sub 2 close paren and denominator v sub 2 end-fraction 5. Final Substitution to Yield Snell’s Law
Finally, substitute the definition of velocity based on the refractive indexes ( ) into the expression:
sin(θ1)c/n1=sin(θ2)c/n2the fraction with numerator sine open paren theta sub 1 close paren and denominator c / n sub 1 end-fraction equals the fraction with numerator sine open paren theta sub 2 close paren and denominator c / n sub 2 end-fraction
Multiplying both sides by the constant speed of light c leaves the final governing law of refraction:
n1sin(θ1)=n2sin(θ2)n sub 1 sine open paren theta sub 1 close paren equals n sub 2 sine open paren theta sub 2 close paren ✅ Restating the Result
n1sin(θ1)=n2sin(θ2)n sub 1 sine open paren theta sub 1 close paren equals n sub 2 sine open paren theta sub 2 close paren
By applying calculus optimization techniques to minimize total travel time, Fermat’s Principle mathematically derives Snell’s Law of Refraction, proving that the bending of light at an interface is a physical consequence of time minimization. If you would like to explore further,
Understand how modern quantum mechanics explains how light “chooses” this path.
Walk through an algebraic analogy involving a lifeguard saving a swimmer on a beach.
Snell’s law of refraction from Fermat’s principle of least time for light
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