## richyw 2 years ago Total internal reflection question.

1. richyw

If was assume in the case of total internal reflection that there is no transmitted wave, we can reformulate$r_{\perp} = \frac{n_i\cos{\theta_i}-n_t\cos{\theta_t}}{n_i\cos{\theta_i}+n_t\cos{\theta_t}}$and$r_{\parallel}=\frac{n_t\cos{\theta_i}-n_i\cos{\theta_t}}{n_i\cos{\theta_t}+n_t\cos{\theta_i}}$ such that$r_{\perp}=\frac{\cos{\theta_i}-(n_{ti}^2-\sin^2{\theta_i})^{1/2}}{\cos{\theta_i}+(n_{ti}^2-\sin^2{\theta_i})^{1/2}}$$r_{\parallel}=\frac{n_{ti}\cos{\theta_i}-(n_{ti}^2-\sin^2{\theta_i})^2}{n_{ti}\cos{\theta_i}+(n_{ti}^{1/2}-\sin^2{\theta_i})^{1/2}}$

2. richyw

oh and i'm pretty sure that the notation they used is $n_{ti}=\frac{n_t}{n_1}$

3. richyw

my attempt: $r_{\perp} = \frac{n_i\cos{\theta_i}-n_t\cos{\theta_t}}{n_i\cos{\theta_i}+n_t\cos{\theta_t}}$$r_{\perp} = \frac{\cos{\theta_i}-n_{ti}\cos{\theta_t}}{\cos{\theta_i}+n_{ti}\cos{\theta_t}}$$r_{\perp} = \frac{\cos{\theta_i}-n_{ti}\sqrt{1-\sin^2{\theta_t}}}{\cos{\theta_i}+n_{ti}\sqrt{1-\sin^2{\theta_t}}}$then from snell's law I get$\sin{\theta_t}=n_{ti}\sin{\theta_i}$$\sin^2{\theta_t}=n_{ti}^2\sin^2{\theta_i}$

4. richyw

so then I get $r\perp=\frac{\cos{\theta_i}-n_{ti}\sqrt{1-n_{ti}^2\sin^2{\theta_i}}}{\cos{\theta_i}-n_{ti}\sqrt{1-n_{ti}^2\sin^2{\theta_i}}}$

5. richyw

which isn't what I wanted?

6. richyw

ah, I see now that I messed up snell's law. Nevermind!