# How is Snell's law formula mathematically deduced from the change in speed?

Refractive index is a function of speed in the medium. How is it related to phase velocity inside the medium?

• I would have said it's the other way around, but no matter. Either way, Fermat's Principle as in Wolphram's answer applies. Commented Dec 6, 2014 at 19:44
• Commented Jan 3, 2022 at 10:26

In optics, Fermat's principle or the principle of least time is the principle that the path taken between two points by a ray of light is the path that can be traversed in the least time. A more modern statement of the principle is that rays of light traverse the path of stationary optical length with respect to variations of the path. In other words, a ray of light prefers the path such that there are other paths, arbitrarily nearby on either side, along which the ray would take almost exactly the same time to traverse. Fermat's principle can be used to describe the properties of light rays reflected off mirrors, refracted through different media, or undergoing total internal reflection. Fermat's principle allows one to derive Snell's law of refraction and the law of reflection. Fore more details see here

• A light ray is the fastest route between two points P and Q,but what if Q is at the bottom of the sea so that light cannot reach it?Fermat's theorem is an attempt to explain the path of a light ray that progresses from P to Q, but if the light isn't going from P to Q then it doesn't apply. Commented Jan 10, 2015 at 23:31
• Wave is spread due to Huygens law;ray is a photon's linear orbit,what else? Commented Jan 24, 2020 at 12:24
• We didn't know beforehand the two points.We only know the incident point A.It is exactly the final point B we are looking for. Commented Apr 16, 2020 at 0:01

Provided that the electron & the atomic beams also exhibit refraction,it seems that this is a particle's property.Deflection angle is proportional to particle's mass/size for specific medium.Photon behaves as particle in this effect.Mass is given by de Broglie equation:m=hv/c^2 , v=frequency

• Dear Nikos Binis. It is usually frown upon to directly copy-paste identical answers. (The problem is if everybody start to copy-paste identical answers en mass.) In general in such situations, please consider one of the following options: (i) Delete one of your answers. (ii) Flag for duplicate posts and delete one of your answers. (iii) If you think the three posts are not duplicates, then personalize each answer to address the three different specific questions. Commented Dec 7, 2014 at 16:21