# Why does light take a longer path in a rarer medium?

## Summary:

Is Fermat's principle of least time applied when light travels from an optically denser to a rarer medium?
When travelling into an optically denser medium, the speed of light reduces and as per the principle of least time, light bends towards the normal and takes the shortest path.

But why isn't this followed when light passes into a rarer medium? With its speed increased in the rarer medium, if it takes a shorter path, time will be lesser than the path taken (by bending away from the normal).

## Answers and Replies

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Dale
Mentor
Surely you are familiar with this effect. If you wish to go to a nearby city sometimes the shortest route is all on surface streets but the fastest overall route involves driving a little out of your way to use the freeway.

etotheipi
Gold Member
2019 Award
When travelling into an optically denser medium, the speed of light reduces and as per the principle of least time, light bends towards the normal and takes the shortest path.
Note that you're after the shortest total time across the entire path.

vanhees71
sophiecentaur
Gold Member
Note that you're after the shortest total time across the entire path.
Yebbut how does it know which way to go when it's setting off?

etotheipi
etotheipi
Gold Member
2019 Award
Yebbut how does it know which way to go when it's setting off?
Well if it goes the wrong way then it's not going to get to B anyway . If light is emitted radially from A in all possible directions, the only light that reaches B will be that which is emitted along our calculated path.

But I suspect you already know this, so now I worry I have missed something. So I will preface this with "at least that's what happens classically, I have no idea if this holds up in more advanced Physics ".

Merlin3189
Homework Helper
Gold Member
I thought it went all ways, but only the paths very close to Fermats route interfered constructively: the sum of all the other paths interfere destructively.
I know that's very vague and arm waving, but I did pinch it from a real physicist.

vanhees71 and etotheipi
sophiecentaur
Gold Member
so now I worry I have missed something.
Nah - I was just being cheeky!!
@Merlin3189 's post make sense. And he's not muckin' you about.

But that question of mine was typical of the sort of question that used to be asked in the past.

Merlin3189 and etotheipi
etotheipi
Gold Member
2019 Award
No that's very cool, I didn't know that. I had a look at the very last section here:

https://www.feynmanlectures.caltech.edu/I_26.html

where he describes the process of summing the phasors for each path together. Close paths only differ beyond first order (I suppose because we're at a minimum of the curve) and will add together nicely, and phasors rotated either side of the average will generally cancel.

Please do correct me if I've made a mistake

sophiecentaur
A.T.
Summary:: Is Fermat's principle of least time applied when light travels from an optically denser to a rarer medium?
Yes, least time refers to the total time in both media.

When travelling into an optically denser medium, the speed of light reduces and as per the principle of least time, light bends towards the normal and takes the shortest path.

But why isn't this followed when light passes into a rarer medium? With its speed increased in the rarer medium, if it takes a shorter path, time will be lesser than the path taken (by bending away from the normal).
Swapping the media is the same as reversing the direction of the light. And if it's the quickest path in one direction, then it's also the quickest path if you reverse the direction.

etotheipi and sophiecentaur
Thank you, everyone!

Dale