# Fermat's principle to derive lens formula

• Srv C
In summary, the person is trying to derive the lens maker's equation and is having difficulty understanding the equations. A helper could help, but is not necessary.
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.

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 ?

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.

## 1. What is Fermat's principle to derive lens formula?

Fermat's principle is a fundamental concept in geometric optics that states that light will always take the path that requires the least time to travel from one point to another. This principle is used to derive the lens formula, which is a mathematical equation that relates the focal length, object distance, and image distance of a lens.

## 2. How is Fermat's principle used to derive lens formula?

In order to derive the lens formula using Fermat's principle, one must consider the path of light as it travels through a lens. By considering different paths and using the principle of least time, one can derive the lens formula by equating the time taken by light to travel from an object to the lens and from the lens to the image.

## 3. What is the significance of Fermat's principle in optics?

Fermat's principle is a fundamental concept in optics as it provides a conceptual understanding of how light behaves and interacts with objects. It is used in various applications, such as in the design and analysis of lenses, telescopes, and other optical systems.

## 4. Can Fermat's principle be applied to all types of lenses?

Yes, Fermat's principle can be applied to all types of lenses, including convex, concave, and compound lenses. However, the exact mathematical derivation of the lens formula may differ depending on the type of lens being considered.

## 5. How does Fermat's principle relate to other principles in optics?

Fermat's principle is closely related to other principles in optics, such as the laws of reflection and refraction. All of these principles are based on the idea that light takes the path that requires the least time, and they are used to explain and predict the behavior of light in various optical systems.

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