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27 Nov 2019

Fermat’s principle states that “light travels between two pointsalong the path that requires the
least time, as compared to other nearby paths.” From Fermat’sprinciple derive (a) the law of reflection
and (b) the law of refraction (Snell’s law). [Hint: Choose twoappropriate points so that a ray between
them can undergo reflection or refraction. Draw a rough path for aray between these points, and write
down an expression of the time required for light to travel thearbitrary path chosen. Then take the
derivative to find the minimum.]

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Related textbook solutions

Physics: Principles with Applications
7th Edition, 2013
Giancoli
ISBN: 9780321625922

Related questions

Fermat's principle states that "light travels between two points along the path that requires the least time, as compared to other nearby paths." From this principle, derive (a) the law of reflection and (b) the law of refraction (Snell's law). [Hint: choose two appropriate points so that a ray between them can undergo reflection or refraction. Draw a rough path for a ray between these points, and write down an expression of the time required for light to travel the arbitrary path chosen. Then take the derivative to find the minimum.]

Pierre de Fermat (1601–1665) showed that whenever light travels from one point to another, its actual path is the path that requires the smallest time interval. This statement is known as Fermat’s principle. The simplest example is for light propagating in a homogeneous medium. It moves in a straight line because a straight line is the shortest distance between two points. Derive Snell’s law of refraction from Fermat’s principle. Proceed as follows. In Figure P35.74, a light ray travels from point P in medium 1 to point Q in medium 2. The two points are, respectively, at perpendicular distances a and b from the interface. The displacement from P to Q has the component d parallel to the interface, and we let x represent the coordinate of the point where the ray enters the second medium. Let t 5 0 be the instant the light starts from P.

(a) Show that the time at which the light arrives at Q is 

 

(b) To obtain the value of x for which t has its minimum value, differentiate x with respect to t and set the derivative equal to zero. Show that the result implies

 

(c) Show that this expression in turn gives Snell's law, 

 

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