Q58P - University Physics with Modern Physics

Question

Given small samples of three liquids, you are asked to determine their refractive indexes. However, you do not have enough of each liquid to measure the angle of refraction for light refracting from air into the liquid. Instead, for each liquid, you take a rectangular block of glass 1n = 1.522 and place a drop of the liquid on the top surface of the block. You shine a laser beam with wavelength 638 nm in vacuum at one side of the block and measure the largest angle of incidence ua for which there is total internal reflection at the interface between the glass and the liquid (Fig. P33.58). Your results are given in the table: Liquid A B C

Ua 1 _2 52.0 44.3 36.3 What is the refractive index of each liquid at this wavelength?

Step-by-Step Solution

Answer

$n_A=1.30,n_B=1.35,n_c=1.40$

Step 1: Calculate the refractive index of liquid A .

First, we apply Snell's law for each liquid in the form 

                                              $n_a\sin {\theta }_a=n_b\sin {\theta }_b$, (1) 

Where $n_a$ is the refractive index of the medium with the incident light, ${\theta }_a$ is the incident angle, $n_b$ , is the refractive index of the medium with the refractive beam, and ${\theta }_b$ , is the refractive angle. At the interface between liquid and the glass, the angle of the refraction is $\theta$ . So, it is related to ${\theta }_b$ , by

                                                    $\theta =90°-{\theta }_b$ (2)

For liquid A with a refractive index $n_a$ apply Snell's law for the interface between the air and the glass to get ${\theta }_b$ , as next

                                            $$\begin{gathered} n_a\sin {\theta }_a=n_b\sin {\theta }_b \\ \left(1\right)\sin \left(52.0°\right)=\left(1.52\right)\sin {\theta }_b \\ \left(1\right)\left(0.788\right)=\left(1.52\right)\sin {\theta }_b \\ \sin {\theta }_b=\left(0.518\right) \\ {\theta }_b={\sin }^{-1}\left(0.518\right) \\ {\theta }_b=31.23° \end{gathered}$$

Use equation (2) to get $\theta$ as next

                                                  $\theta =90°-31.23°=58.77°$                 

Now, apply Snell's law for the interface between the liquid and the glass to get $n_a$ as next

                                                     $$\begin{gathered} n\sin \theta =n_A\sin 90° \\ \left(1.52\right)\sin 58.77°=n_A \\ n_A=1.30 \end{gathered}$$

Step 2: Calculate the refractive index of liquid B .

For liquid B with refractive index $n_a$ apply Snell's law for the interface between the air and the glass to get ${\theta }_b$ , as next

                                               $$\begin{gathered} n_a\sin {\theta }_a=n_b\sin {\theta }_b \\ \left(1\right)\sin \left(43.3°\right)=\left(1.52\right)\sin {\theta }_b \\ \sin {\theta }_b=\left(0.459\right) \\ {\theta }_b={\sin }^{-1}\left(0.459\right) \\ {\theta }_b=27.35° \end{gathered}$$

Use equation (2) to get $\theta$ as next

                                              $\theta =90°-27.35°=62.65°$                     

Now, apply Snell's law for the interface between the liquid and the glass to get $n_b$ as next

                                                   $$\begin{gathered} n\sin \theta =n_B\sin 90° \\ \left(1.52\right)\sin 62.65°=n_B \\ n_B=1.35 \end{gathered}$$

Step 3: Calculate the refractive index of liquid C .

For liquid C with a refractive index $n_b$ apply Snell's law for the interface between the air and the glass to get ${\theta }_b$ , as next

                                                           $$\begin{gathered} n_a\sin {\theta }_a=n_b\sin {\theta }_b \\ \left(1\right)\sin \left(36.3°\right)=\left(1.52\right)\sin {\theta }_b \\ \sin {\theta }_b=\left(0.389\right) \\ {\theta }_b={\sin }^{-1}\left(0.389\right) \\ {\theta }_b=22.92° \end{gathered}$$

Use equation (2) to get $\theta$ as next

                                                            $\theta =90°-22.92°=67.08°$       

Now, apply Snell's law for the interface between the liquid and the glass to get  as next

                                                                $$\begin{gathered} n\sin \theta =n_c\sin 90° \\ \left(1.52\right)\sin 67.08°=n_B \\ n_c=1.40 \end{gathered}$$