/sci/ - Science & Math

So i can't really find a graphical way to elaborate spin with waves, so heres one that mean about the same.

Ex and Ey are out of phase, either by +- pi/2

When propagating, Ex and Ey form a circularly oscillating E field.

This constitutes as spin. Linearly polarized light are just a superposition state of the +pi/2 and -pi/2 circularly polarized lights, so light is never represented as just 1 EM wave.

tl;dr E field is never 0 when light propagates.

>>6657189
Euler's identity/formula is tricky because it seems like there is going to be some nice geometrical explanation, especially since sin and cos are being used, but there really isn't any afaik.

The red curve in the attached pic is the curve that is drawn out when you have e^(iz). It looks the same as circularly polarized light if you're familiar with that. The curve "propagates" on the z axis, but if you look dead on to the XY plane you'd see it making a circle that keeps repeating. When you have a circle you can use polar coordinates easily and write it in terms of sine and cosine. If you want to convert polar to cartesian it is just x=r*cos(theta), y=r*sin(theta). If you use the complex plane with imaginary numbers on the y axis you can write 2D coordinates as a single complex number with the real part as the x coordinate and imaginary as the y coordinate. So I think that explains how to use it but not why it exists.

Before I mentioned how there really isn't a geometrical reason behind it, and that is because the existence of the identity doesn't come from sine and cosine being ratios of sides or anything like that, but instead it comes from e^l being the solution to linear ODEs. The most straight forward way to show it is going to work is by writing sin and cosine out in exponential form in the equation, and unsurprisingly you'll get 2e^iz=e^iz+e^-iz+e^iz-e^-iz. And so now the question is, why can sine and cosine be rewritten in terms of exponentials? Well from various arguments you can set up a differential equation f(x)=-f''(x) which should give you sine or cosine, but if you solve it using the normal methods your solution is e^ix+e^-ix, so sine and cosine must be a linear combination of e^ix and e^-ix.

There are actually a ton of ways to prove Euler's formula, and I didn't give airtight reasoning above because my point is more to show that it uses that e^l is the solution of ODEs is the reason why it happens.

non-smart guy here
is this what ya'll are talking about? Like to us this pic seems amazing yet once you understand it, it seems mundane and couldn't be any other way? Anyone want to take a stab at explaining pic?

Come'on /sci/, I really need to learn this algebra. Surely someone can explain how to go between the two forms? I tried to include as general of a case as I could.