Q.What is Brewster's law? Derive the formula for Brewster angle.

[!TLDR] Brewster's law, $\tan\theta_B=n_2/n_1$, follows from combining the geometric condition $\theta_B+\theta_r=90°$ (reflected ray completely polarized, perpendicular to the refracted ray) with Snell's law. [!ANSWER] Brewster's Law: $ …

When unpolarized light strikes a boundary between two transparent media (refractive indices $n_1$, $n_2$) at a general angle of incidence, both the reflected and refracted beams end up only PARTIALLY polarized. Sir David Brewster discovered experimentally (1812) that there exists one special angle of incidence, the BREWSTER ANGLE $\theta_B$, at which the REFLECTED beam becomes COMPLETELY plane polarized, with its electric field oscillating entirely perpendicular to the plane of incidence, while the refracted beam remains only partially polarized. At this special angle, the reflected and refracted rays are found (experimentally, and derivable from a fuller treatment of the reflection/transmission coefficients for the two polarization components, beyond this chapter's scope) to be exactly PERPENDICULAR to one another: $\theta_B+\theta_r=90°$, where $\theta_r$ is the corresponding angle of refraction.

To derive the formula for $\theta_B$, combine this geometric condition with the ordinary law of refraction (Snell's law) applied at this specific angle of incidence: $n_1\sin\theta_B=n_2\sin\theta_r$. Since $\theta_r=90°-\theta_B$, we have $\sin\theta_r=\sin(90°-\theta_B)=\cos\theta_B$. Substituting: $n_1\sin\theta_B=n_2\cos\theta_B$. Dividing both sides by $n_1\cos\theta_B$: $\dfrac{\sin\theta_B}{\cos\theta_B}=\dfrac{n_2}{n_1}$, i.e. $\tan\theta_B=\dfrac{n_2}{n_1}$ -- this is BREWSTER'S LAW. …

Combine the perpendicularity condition between the reflected and refracted rays at the Brewster angle ($\theta_B+\theta_r=90°$) with Snell's law, substituting $\sin\theta_r …

Forgetting the perpendicularity condition ($\theta_B+\theta_r=90°$) that Brewster's law is actually built on, and instead trying to derive a formula for $\theta_B$ from Snell's law a …