Calculate the (a) upper and (b) lower limit of the Brewster angle for white light incident on fused quartz. Assume that the wavelength limits of the light are 400 and 700 nm.

• The white light is incident on fused quartz.
• The wavelength limits of the light are.$400\text{\hspace{0.17em}nmand700\hspace{0.17em}nm}$

The special angle of incidence that produces a 90 degrees angle between the reflected and refracted ray is called the Brewster angle. If an unpolarized beam of light is incident on a boundary between two media at Brewster angle, the reflected wave will be fully polarized. For this situation, the tan of the Brewster angle is equal to the ratio of refractive indices of the two media.

Formula:

The Brewster angle is related to the refractive indices of the two mediaas,

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

Here, the two media are fused quartz and air. So$n_1=1$. The value of n₂ will be different for the two given wavelengths.

The standard values are$n_2=1.47$for wavelength 400 nm and$n_2=1.455$for wavelength = 700 nm considering refracting index is inversely proportional to wavelength.

Thus, we calculate the upper limiting value of the Brewster angle using equation (1) as follows:

localid="1664201175448" $\begin{array}{c}{\left({\theta }_B\right)}_{up}={\tan }^{-1}\left(\frac{1.47}{1}\right)\\ =55.8°\end{array}$

Hence, the value of the upper limit is.$55.8°$

Now, we calculate the lower limiting values of the Brewster angle using equation (i) as follows:

$\begin{array}{c}{\left({\theta }_B\right)}_{low}={\tan }^{-1}\left(\frac{1.455}{1}\right)\\ =55.5°\end{array}$

Hence, the value of the lower limit is.$55.5°$