# Simple proof of Snell's law without calculus

shihab-kol
Well, I have checked out the ones with calculus but I was just wondering if there was one without calculus
I tried it but could not do it
I think Fermat's principle can be used to do it but I am not being successful
So, anyone please help

## Answers and Replies

Mentor
2022 Award
I think Fermat's principle can be used to do it but I am not being successful
Yes. I've found it on Wikipedia, but I do not know what you mean by "without calculus". I even don't know, how to formulate it without calculus.

weirdoguy
I think Fermat's principle can be used to do it but I am not being successful

No surprise since Fermat's principle is a type of variational principle, so it is somewhat connected to calculus.

Staff Emeritus
Science Advisor
Education Advisor
Well, I have checked out the ones with calculus but I was just wondering if there was one without calculus
I tried it but could not do it
I think Fermat's principle can be used to do it but I am not being successful
So, anyone please help

This question doesn't make any sense. It is like asking if one can do kinematics with something that has variable velocity and acceleration but without calculus. In many of these cases, not using calculus is NOT an option!

And as stated already, Fermat Principle itself is based on calculus, i.e. finding the path of least time, which is a variational calculus application (see Pg. 8 of this document). So I have no idea why you'd think that using Fermat Principle is not using calculus.

Zz.

shihab-kol
This question doesn't make any sense.

Zz.

Oh yes it does.
And as stated already, Fermat Principle itself is based on calculus, i.e. finding the path of least time, which is a variational calculus application (see Pg. 8 of this document). So I have no idea why you'd think that using Fermat Principle is not using calculus.

But it can be done WITHOUT calculus too.

You can prove snell's law too WITHOUT calculus.

I have seen the proof and it is by R.P.Feynman.
So, thanks but no thanks

Staff Emeritus
Science Advisor
2022 Award
I have seen the proof and it is by R.P.Feynman.

Feynman's "proof" isn't really a proof. At least the example I've seen from his lectures isn't. It explains how you can use geometry to find an answer in a specific case where there's a single boundary and two non-varying refractive indices. A real proof inherently requires calculus because it has to deal with continuously varying variables. Page 8 in Zz's link contains this exact situation and you cannot use geometry by itself to solve it.

weirdoguy
Higgsono
Can't you use the property of isotropy of space and the time invariance of physical law? But that may only prove that incident angle is equal to the reflected angle for total reflection.

Science Advisor
Gold Member
Oh yes it does.

But it can be done WITHOUT calculus too.

You can prove snell's law too WITHOUT calculus.

I have seen the proof and it is by R.P.Feynman.
So, thanks but no thanks

since you are going against good advice and say it can be done
then the onus is on you to show us how it is done and we can see if it makes sense

Science Advisor
You only have to match the wavevectors parallel to the surface of the light inside and outside the medium. This doesn't involve calculus, merely vector arithmetics.

But when fermat devised his principle at that time there was no calculus. As calculus came much after by Newton and leibnitz.Fermat must have done in his own way.Or he just gave us an intuitional principle with no proof or he had find maxima or minima in his own way.

I gues this question/topic needs importance.

http://aapt.scitation.org/doi/10.1119/1.1514235

Moderator's edit: File substituted by link due to potential copyright violation.

Last edited by a moderator:
shihab-kol
Thanks.
Being an Indian,perhaps you have gone through H.C.Verma .There also you will find a simple but elegant proof of the above.
I did not post anything since by the time I started the thread the book has been returned and I could not include the proof on the site .
Thanks once again.