Fermat's principle to derive lens formula

[Srv C]

1. For the past 1 hour, I'm trying to derive lens maker's equation using fermat's principle, which of course is our homework. Any help would be great regarding the topic.



2. According to Fermat's Principle, optical path length OPL = PA + AQ, here PA and AQ are two rays. Now using this I have to derive Lens makers Formula.



3. Now the diagrams and eqn's i don't know how to put in here. Upto the point where I found PA + AQ, I have completed.
PA = (x^2 + h^2)^1/2 = x + 1/2 X h^2/x
AQ = (y^2+ h^2)^1/2 = y + 1/2 X h^2/y
Now, PA + AQ = x + y + h^2/2 X (1/x + 1/y)
Now kindly help me if you happen to know the next steps.

 

[BvU]

Hello C and welcome to PF. Some good spirit will appear and change your boldface into normal font.
The idea of the template is not that you replace the headings (if only to remind you what was there in the first place) but that you fill in your stuff on the blank lines following them.

Under 1 you can then formulate Fermat's priciple, which is somewhat different from what you write down.
It looks as if you are also supposed to have the lens maker formula as a given, so you know where to end up.

Equations can be put in by typing them. If you need $\theta$ and are not satisfied with writing e.g. (theta), invest some time to experiment with 'Go Advanced'.

A helper can help you effectively if you write down what you think are the relevant equations. Personally, I find the expressions for PA and AQ difficult to interpret, but perhaps you already acquired the proper jargon ?

And under 2. you can also sum up the approximations you intend to make good use of. There are quite a few to make here.

So if we get 1. and 2. sorted out a little better, we can turn to 3.: what you 've done to get going.

As a direct answer to your direct question, all I can propose at this moment is:
A logical next step would involve inserting the lens, because so far you don't have anything that can lengthen the optical path.
Bring in the refractive index between A on the one side of the lens and A on the other side of the lens.

Did you notice I also invested quite some time in your problem ?

 

[rude man]

What you want to do is derive Snell's law from Fermat's principle.

Then going from Snell to the lensmaker's fomula is in all the textbooks.