Recent content by Fraqtive42

The refractive index and \theta2 are related through Snell's Law.

If the material is layered infintesimally so that the index of refraction is proportional to the y, which I stated in the problem, then y is related to \theta2 because the index of refraction is related to \theta2

But because y is a function of time, that also makes \theta_{2} a function of time.

My work: So far I know that v=\frac{c}{n_{2}} is the speed of the light beam, which is also equal to v=\frac{dy}{dt}. So a differential equation to solve would be \frac{dy}{dt}=\frac{c}{n_{2}}

I think that I have made my proof on the assumption that for any functions satisfying these conditions, \prod_{j=1}^{N}(f(j)-1)\leq\prod_{j=1}^{N}f(j) for any N\in\mathbb{N}. I don't see why f(n)=n+1 is an exception though.

A ray of light travels through a medium with an index of refraction n_{1} and strikes an layered medium such that the index of refraction is n_{2}=ky+1 where y is the depth of the medium and k is a constant. If it hits at an angle of \theta_{1} with respect to the normal, find the angle...

Hmm... that is weird. My proof had some kind of error in it...

You are saying that f(n)=n+1 doesn't work?

Prove that \prod_{j=1}^{\infty}\left(1-\frac{1}{f(j)}\right)>0 for all f:\mathbb{N}^{+}\to\mathbb{R}^{+} which satisfy f(1)>1 and f(m+n)>f(m), where m,n\in\mathbb{N}^{+}. I found the problem "Prove that \prod_{j=1}^{\infty}\left(1-\frac{1}{2^{j}}\right)>0" and felt the need to make a...