Chapter 18, Problem 99A

To determine

To show:The ray configuration cannot be achievedfor the given refractive index.

Answer to Problem 99A

The ray configuration is not achievable when $n_{\text{glass}}=1.5$ .

Explanation of Solution

Given:

The index of refraction is $n_{\text{g}}=1.5$ .

The given figure is shown below.

  

Formula used:

The expression for Snell’s law of refraction is,

  $n_1\sin {\theta }_1=n_2\sin {\theta }_2$

Here, $n_1$ is the index of refraction of the first medium, ${\theta }_1$ is the angle of incidence of the first medium, $n_2$ is the index of refraction of the second medium, and ${\theta }_2$ is the angle of incidence of the second medium

Consider the given Figure.The ray from the light enters the glass at an angle of ${\theta }_1$ and refracted the glass at an angle of ${\theta }_2$ .The angle of incidence must be in air.

Hence, the index of refraction of the first medium $n_{\text{A}}=1.000$ .

The angle of refraction by using Snell’s law of refraction is,

  $\begin{array}{l}{n}_{\text{A}}\sin {\theta }_1=n_{\text{g}}\sin {\theta }_2\\ \sin {\theta }_2=\frac{n_{\text{A}}\sin {\theta }_1}{n_{\text{g}}}\\ {\theta }_2={\sin }^{-1}\left(\frac{\left(1.00\right)\sin 90°}{1.5}\right)\\ {\theta }_2=42°\end{array}$

The angle ${\theta }_{2^{\prime }}$ is,

  $\begin{array}{l}{\theta }_{2^{\prime }}=90°-{\theta }_2\\ {\theta }_{2^{\prime }}=90°-42°\\ {\theta }_{2^{\prime }}=48°\end{array}$

The critical angle of glass is,

  $\begin{array}{l}\sin {\theta }_{\text{c}}=\frac{n_{\text{A}}}{n_{\text{g}}}\\ {\theta }_{\text{c}}={\sin }^{-1}\left(\frac{1.00}{1.5}\right)\\ {\theta }_{\text{c}}=42°\end{array}$

The angle ${\theta }_{2^{\prime }}>{\theta }_{\text{c}}$ . So the light reflects back into the glass and one cannot see out of an adjacent side.Hence, the ray configuration is not achievable when $n_{\text{glass}}=1.5$ .

Conclusion:

Thus, it is proved that the ray configuration is not achievable when $n_{\text{glass}}=1.5$ .