Question

When red light in vacuum is incident at the Brewster angle on a certain glass slab, the angle of refraction is$\mathbf{32}\mathbf{°}$ .What are (a) the index of refraction of the glass and (b) the Brewster angle?

Short answer

1. Theindex of refraction of glassis 1.60.
2. Brewster angle is$58°$.

Step-by-step solution

Listing the given quantities

Angle of refraction,${\theta }_r=32°$

Understanding the concepts of Brewster’s law

We use the formula for Brewster angle to find the refractive index of glass. The addition of Brewster angle and the angle of refraction is$\mathbf{90}\mathbf{°}$.Using this concept, we can find the angle of refraction.

Formulae:

${\theta }_B=ta{n}^{-1}\left(\frac{n_2}{n_1}\right)$

${\theta }_B+{\theta }_r=90°$

(a) Calculations of the index of refraction of glass

The addition of Brewster angle and the angle of refraction is $90°$, i.e.${\theta }_B+{\theta }_r=90°$,

We have${\theta }_r=32°$.

Therefore,

$$\begin{gathered} {\theta }_B+32°=90° \\ {\theta }_B=58°······\left(1\right) \end{gathered}$$

TheBrewster angle given by equation $33-49$is

${\theta }_B=ta{n}^{-1}\left(\frac{n_2}{n_1}\right)$

If the incident and reflected rays travel in air, we can approximate$n_1$as unity and let$n$be the refractive index of the second medium. Then we have

$$\begin{gathered} {\theta }_B=ta{n}^{-1}n \\ {n}_{glass}=tan{\theta }_B \\ =tan58° \\ =1.60 \end{gathered}$$

Hence, the index of refraction of glass is 1.60.

(b) Calculations of the Brewster angle

As calculated in part a), from equation (1), we can say that the Brewster angle is$58°$.

Hence, Brewster angle is$58°$.