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

In summary, the conversation is discussing how to find the quickest light path for different values of h in a lens setup. The parameter that selects different rays is h, and differentiating with respect to h will give the desired result. The speaker also mentions Feynman's explanation of Snell's law from Fermat's principle, which can be applied to this problem by considering a point infinitely far to the left of the lens.
  • #1
Clara Chung
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Homework Statement


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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?
 

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  • #2
You want different rays all to take the same time. What parameter selects different rays?
 
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  • #3
TSny said:
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?
 
  • #4
Clara Chung said:
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'##.
 
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1. What is Fermat's Principle?

Fermat's Principle is a fundamental principle in optics that states that light will travel along the path that takes the least amount of time to reach its destination.

2. How does Fermat's Principle relate to the Lens Maker formula?

Fermat's Principle can be used to derive the Lens Maker formula for a plano-convex lens by considering the path of a ray of light passing through the lens and minimizing the time it takes for the ray to travel from one point to another.

3. What is the Lens Maker formula?

The Lens Maker formula relates the focal length of a lens to its refractive index and the radii of curvature of its two surfaces.

4. What are the key steps in deriving the Lens Maker formula using Fermat's Principle?

The key steps in deriving the Lens Maker formula are: 1) considering the path of a ray of light passing through the lens, 2) applying Fermat's Principle to find the minimum time path, 3) using the Snell's law to relate the angles of incidence and refraction, and 4) solving for the focal length using the resulting equations.

5. Why is the Lens Maker formula important in optics?

The Lens Maker formula is important in optics because it provides a mathematical relationship between the physical characteristics of a lens and its optical properties, such as focal length. This formula is widely used in the design and analysis of lenses for various applications, such as in cameras, telescopes, and eyeglasses.

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