\(tan(\theta_2-\alpha^{'}_{2s})=\cfrac{\mu_2}{\mu_1}tan(\theta_1-\alpha)\)
when \(\theta_1\) is large, such that
\(\theta_1-\alpha\gt90^o\)
Let \(x+90^o=\theta_1-\alpha\) then
\(tan(\theta_1-\alpha)=tan(x+90^o)=-cot(x)\)
So,
\(tan(\theta_2-\alpha^{'}_{2s})=-\cfrac{\mu_2}{\mu_1}cot(\theta_1-\alpha-90^o)\)
\(cot(\theta_2-\alpha^{'}_{2s}+90^o)=\cfrac{\mu_2}{\mu_1}cot(\theta_1-\alpha-90^o)\)
When \(\mu _{ 2 }\gt\mu _{ 1 }\), since \(cot(x)\) is a decreasing function,
\(\theta_2-\alpha^{'}_{2s}+90^o\lt\theta_1-\alpha-90^o\)
\(\alpha^{'}_{2s}\gt180^o+\alpha+\Delta \theta\)
Since the \(\alpha^{'}_{2s}\) has been reflected back into medium 1, \(\Delta \theta=0\),
\(\alpha^{'}_{2s}\gt180^o+\alpha\)
When \(\mu _{ 2 }\lt\mu _{ 1 }\),
\(\theta_2-\alpha^{'}_{2s}+90^o\gt\theta_1-\alpha-90^o\)
\(\alpha^{'}_{2s}\lt180^o+\alpha\)
We can also have,
\(tan(\theta_2+\alpha^{'}_{2p})=\cfrac{\mu_2}{\mu_1}tan(\theta_1+\alpha)\)
when \(\theta_1\) is large, such that
\(\theta_1+\alpha\gt90^o\)
Let \(x+90^o=\theta_1+\alpha\) then
\(tan(\theta_1+\alpha)=tan(x+90^o)=-cot(x)\)
So,
\(tan(\theta_2+\alpha^{'}_{2p})=-\cfrac{\mu_2}{\mu_1}cot(\theta_1+\alpha-90^o)\)
\(cot(\theta_2+\alpha^{'}_{2p}+90^o)=\cfrac{\mu_2}{\mu_1}cot(\theta_1+\alpha-90^o)\)
When \(\mu _{ 2 }\gt\mu _{ 1 }\),
\(\theta_2+\alpha^{'}_{2p}+90^o\lt\theta_1+\alpha-90^o\)
\(\alpha^{'}_{2p}\lt\alpha-\Delta\theta-180^o\)
Since, \(\Delta\theta=0\)
\(\alpha^{'}_{2p}\lt\alpha-180^o\)
When \(\mu _{ 2 }\lt\mu _{ 1 }\),
\(\theta_2+\alpha^{'}_{2p}+90^o\gt\theta_1+\alpha-90^o\)
\(\alpha^{'}_{2p}\gt\alpha-\Delta\theta-180^o\)
Since, \(\Delta\theta=0\)
\(\alpha^{'}_{2p}\gt\alpha-180^o\)
This might seem to be the same results as the post "More Bending Of Light" dated 13 Aug 2015, but the swing of \(\alpha_s\) and \(\alpha_p\) are different,
When \(\mu _{ 2 }\gt\mu _{ 1 }\), \(\alpha^{'}_{2s}\gt180^o+\alpha\)
When \(\mu _{ 2 }\lt\mu _{ 1 }\), \(\alpha^{'}_{2p}\gt\alpha-180^o\)
Previously,
When \(\mu_2\gt\mu_1\), \(\alpha_{2s}\lt-180^o+\alpha\)
When \(\mu_2\lt\mu_1\), \(\alpha_{2s}\gt-180^o+\alpha\)
This cases show the swing of \(\alpha_{2s}\) and \(\alpha_{2p}\) as \(\theta_1\) changed. There is no Brewster angle here.
