Fermat's principle of least time

[Physicsfan]

According to Fermats principle of least time, light travelling from one transperent medium to another transperent medium takes a path that requires the least time.

this explains Snells Law that refractive index of second medium with respect to the first is sin i by sin r.

Now if light is travelling from a rearer medium such as air to a denser medium

why is it that it does not travel along the normal in the denser medium?

That path would require the least time right?

[Sisyphus]

No. Think of it this way. If the refractive indices of the media are exactly the same, the quickest path is just a straight line, right? Now imagine one is slightly different. The quickest path is no longer a straight line, because you want to travel less distance in the slower medium. However, you also don't want to go way out of your way, because the difference in speed is not worth the extra distance.

 

The closer the speeds are, the closer the quickest path will be to a straight line. The bigger the difference in speeds, the closer it will be to normal through the slower medium. However, it will never be exactly that unless the difference in speed is infinite (which it obviously can't be).

[swansont]

Can you set up an equation for the time it would take?

[Physicsfan]

Can you set up an equation for the time it would take?

With reference to the diagram, if velocity of light in medium=c/n

then time required to travel distance IB=IB/(c/n) =nIB/c

time required by light to travel distance MB=IBcos r/(c/n)=nIBcos r/c

time required by light to travel distance AI=AI/c

time taken by light to travel AM=(AM/AI)*(AI/c)

=(2sin i/sin i')(AI/c)

Edited May 16, 2010 by Physicsfan

[swansont]

http://cnx.org/content/m12895/latest/