write my assignment 13060

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