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QUESTION
The Principle of Least Time (Fermat 1662) He argued that God made the world in a. most efficient way, in some sense. If light starts out
The variable of interest is y , and T(AB) depends on y.
Solve dT(AB)/dy =0 to find the least time.
The Principle of Least Time (Fermat 1662)He argued that God made the world in a. most "efficient" way, in some sense. If light starts out atA ( fixed) and ends up at B (fixed) and passes through the interface at O (this is theunknown), what determines the precise path A - O - B and the relation known as Snell'sLaw? Fermat guessed that light would travel from A to B in such a way as to go in the leastpossible time.For example, we may compare the paths A - O - B and A - O' - B:(since v . t = distance traveled)total time to go A - O- B =TAB (O) :TAB (O) - AO| . JOBV2|AO| = length of AO |OB| = length of OBBy=n1V= .n2Similarly, for the A - O' - B path.In general, for different possible paths, the total time for the light to travel from A to B is different. One ofthe possible paths takes the least time. Fermat showed that the path of least time corresponds toSnell's Law!Challenge problem: Derive Snell's Law from the Principle of Least Time.Hint: Set up an x-y coordinate system such that A = (XA, YA) B = (XB, yB) (fixed) O = (o, y)The variable of interest is y, and TAB depends on y.Solved'I AB = 0 to find the least time.dy