Refractive Index: Homework Answers

[Pushoam]

Homework Statement

Homework Equations

The Attempt at a Solution

I think that $\frac { \sin \theta } {\sin \theta ' } = \frac { n_2} { n_1}$

I am taking the dielectric constant as 1 as the potential is 0 i.e. the medium is air..

So, $n_1 = 1$

For the 2nd medium,

The dielectric constant $\epsilon_r = \frac { E } { E- V_0}$

So, $n_2 = \sqrt{ \frac { E } { E-V_0 } } = 1+ \frac { V_0 } {2E }$

What is wrong here?

 

[Charles Link]

I haven't solved it in its entirety yet, but I think the problem makes use of the Schrödinger equation. Try writing out the Schrödinger equation in two dimensions with an incident plane wave particle with $\psi (\vec{r})=e^{i \vec{k}_o \cdot \vec{r} }$ and $\vec{k}_o=k_{xo}\hat{i}+k_{yo} \hat{j}$. $\\$ Editing: To complicate the problem, you might also get a partial reflection at the boundary, with angle of incidence =angle of reflection. (I'm not sure because I didn't solve it completely yet).