“Physics without maths is like trying to drive a car without fuel”
-reddit comment (xsnac)
Mathematics is believed to be the language of physics. If you are into physics, mathematics is inevitable. Mathematics concerns itself with abstract ideas. And physicists use those abstract relationships to replace it with the physical things and get the quantitative relationship. And sometime, physicists demand a new tool to solve its problems and mathematics creates it for them.
In order to taste the flavor of mathematics and understand the affair between its concept and that of physics, one must try to think of some physical example that could fit in perfectly with that mathematical concept. For example, I used to have a problem with discontinuous functions. My maths teacher used to write three different relations for the same variable, and I used to think, where I would find something like this. Something behaves in certain way up to some point and then decides to change its behavior? So I tried to think of some example in the physical world which behaves in this weird fashion, and lo and behold, we have refraction.
Consider the figure…
A light ray originates from the optically denser medium, crosses the interface and enters the rarer medium. In this example I have considered the denser medium to be glass, with refractive index 1.5, and rarer medium to be air, with refractive index of about 1. As the light ray crosses the interface, the ray gets bent away from the normal. I have considered two angles θ1 and θ2 and I have to find the relationship between them.
So we have three different relationships to form one function of θ2. Up to the critical angle (approximately 41.80 for Glass-Air interface), the relationship between them can be found by manipulating Snell’s Law of refraction. But after critical angle, the relationship suddenly changes as Total Internal Reflection (TIR) begins. The relationship becomes linear as per the law of reflection. This relationship holds until θ1 becomes a right angle after which it once again follows Snell’s law but with incident ray in the rarer medium.
The graph of this can be plotted as below. I could have used excel to plot but I wanted to kill a little more time so I plotted manually.
And finally we have some physical example for the abstract mathematical concept. I don’t know whether, this concept was derived because of some physical observation or the mathematical concept came first, but I was really happy that I could relate it. If you find any other physical example please share with me. Till then, thanks for reading.
- For Total Internal Reflection: https://en.wikipedia.org/wiki/Total_internal_reflection (Or any fundamental physics textbook)
- For the opening quote: https://www.reddit.com/r/AskPhysics/comments/44tol0/good_lectures_or_books_on_hydrostatics_without/