Chapter 35, Problem 31P

To determine

The expected angle of refraction in the sheet $3$.

Answer to Problem 31P

The expected angle of refraction in the sheet $3$ is $23.1°$.

Explanation of Solution

Assume $n_1$, $n_2$ and $n_3$ as the refractive indices of sheets $1$, $2$ and $3$ respectively.

The following figure shows the refraction of the laser beam when the sheet $1$ is placed on top of the sheet $2$.

Figure-(1)

Write the equation using Snell’s law in the above figure.

    $\frac{\sin i}{\sin \left(26.5°\right)}=n_1$                                                                                                         (I)

Here, $i$ is the angle of incidence on the sheet $1$.

Write the equation using Snell’s law in the figure-(1).

    $\frac{\sin \left(26.5°\right)}{\sin \left(31.7°\right)}=\frac{n_2}{n_1}$                                                                                                      (II)

The following figure shows the refraction of the laser beam when the sheet $3$ is placed on top of the sheet $2$.

Figure-(2)

Write the equation using Snell’s law in the above figure.

    $\begin{array}{c}\frac{\sin i}{\sin \left(r\right)}=n_3\\ \sin r=\frac{\sin i}{n_3}\end{array}$                                                                                                          (III)

Here, $r$ is the angle of refraction in the sheet $3$.

Write the equation using Snell’s law in the figure-(2).

    $\begin{array}{c}\frac{\sin \left(r\right)}{\sin \left(36.7°\right)}=\frac{n_2}{n_3}\\ \sin \left(r\right)=\frac{n_2}{n_3}\sin \left(36.7°\right)\end{array}$                                                                                     (IV)

Compare the equations (III) and (IV).

    $\begin{array}{c}\frac{\sin i}{n_3}=\frac{n_2}{n_3}\sin \left(36.7°\right)\\ n_2=\frac{\sin i}{\sin \left(36.7°\right)}\end{array}$                                                                                                (V)

The following figure shows the refraction of the laser beam when the sheet $1$ is placed on top of the sheet $3$.

Figure-(3)

Write the equation using Snell’s law in the above figure.

    $\begin{array}{c}\frac{\sin i}{\sin \left(r^{\prime }\right)}=n_1\\ \sin \left(r^{\prime }\right)=n_1\left(\sin i\right)\end{array}$                                                                                                   (VI)

Here, $r^{\prime }$ is the angle of refraction in the sheet $1$.

Write the equation using Snell’s law in the figure-(2).

    $\sin \left(r^{\prime }\right)=\frac{n_3}{n_1}\sin \left(\theta \right)$                                                                                                (VII)

Here, $\theta$ is the angle of refraction in the sheet $3$.

Compare the equations (VI) and (VII).

    $\begin{array}{c}{n}_1\left(\sin i\right)=\frac{n_3}{n_1}\sin \left(\theta \right)\\ \frac{\sin i}{\sin \theta }=n_3\end{array}$                                                                                            (VIII)

The angle of refraction in the sheet $1$ when the sheet $1$ is placed on top of the sheet $3$ is equal to the angle of refraction in the sheet $1$ when the sheet $1$ is placed on top of the sheet $2$.

    $r^{\prime }=26.5°$

The angle of refraction in the sheet $3$ when the sheet $3$ is placed on top of the sheet $2$ is equal to the angle of refraction in the sheet $1$ when the sheet $1$ is placed on top of the sheet $2$.

    $r=26.5°$

Rewrite the equation (IV).

    $\begin{array}{c}\frac{\sin \left(26.5°\right)}{\sin \left(36.7°\right)}=\frac{n_2}{n_3}\\ \frac{n_3}{n_2}=\frac{\sin \left(36.7°\right)}{\sin \left(26.5°\right)}\end{array}$

Rewrite the equation (II).

    $\frac{n_2}{n_1}=\frac{\sin \left(26.5°\right)}{\sin \left(31.7°\right)}$

Conclusion:

Substitute $26.5°$ for $r^{\prime }$ in the equation (VII).

    $\begin{array}{c}\sin \left(26.5°\right)=\frac{n_3}{n_1}\sin \left(\theta \right)\\ \frac{\sin \left(26.5°\right)}{\sin \left(\theta \right)}=\frac{n_3}{n_1}\\ \frac{\sin \left(26.5°\right)}{\sin \left(\theta \right)}=\frac{n_3}{n_2}\times \frac{n_2}{n_1}\end{array}$

Substitute $\frac{\sin \left(36.7°\right)}{\sin \left(26.5°\right)}$ for $\frac{n_3}{n_2}$ and $\frac{\sin \left(26.5°\right)}{\sin \left(31.7°\right)}$ for $\frac{n_2}{n_1}$ in the above equation to calculate the angle of refraction in the sheet $3$.

    $\begin{array}{c}\frac{\sin \left(26.5°\right)}{\sin \left(\theta \right)}=\frac{\sin \left(36.7°\right)}{\sin \left(26.5°\right)}\times \frac{\sin \left(26.5°\right)}{\sin \left(31.7°\right)}\\ \sin \theta =0.3923\\ \theta =23.1°\end{array}$

Therefore, the expected angle of refraction in the sheet $3$ is $23.1°$.