Deriving the Lens Maker formula of a plano-convex lens using Fermat's Principle

[Clara Chung]

Homework Statement

Homework Equations

The Attempt at a Solution

I only need help on part c. I tried to add up t1 and t2 and differentiate it. However what variables should I differentiate with respect to? If I differentiate with respect to f I got f=root(2) * h, if i differentiate with respect to R i get R tends to infinity. What should I do?

 

[TSny]

You want different rays all to take the same time. What parameter selects different rays?

 

[Clara Chung]

You want different rays all to take the same time. What parameter selects different rays?

I should differentiate with respect to h. However, why do different rays all take the same time? Shouldn't there be different quickest light path for different h?

 

[TSny]

I should differentiate with respect to h. However, why do different rays all take the same time? Shouldn't there be different quickest light path for different h?

Feynman has a nice discussion here
http://www.feynmanlectures.caltech.edu/I_26.html

You can skip to the paragraph just before Fig 26-9 and read this as well as the next paragraph. (But if you do skip, you will miss Feynman's derivation of Snell's law (eq. 26.4) from Fermat's principle that doesn't use calculus!)

To relate this to your problem, you will want to consider the point $P$ as placed "infinitely far" to the left of the lens. Then $P'$ will be at a focal point of the lens. All the rays from $P$ that pass through the lens will be essentially horizontal. So, changing $h$ will correspond to different rays from $P$. There will also be different rays that hit the lens in a circle at the same $h$ corresponding to rotating the figure about the horizontal axis of the lens. But, by symmetry, these will clearly take the same time from $P$ to $P'$.