Chapter 9.7, Problem 38E

a.

To determine

To show: The derivation of Snell’s law.

Explanation of Solution

Given information:

When light changes the medium then angle then angle of travel also changes. This is the phenomenon of refraction shown as below.

  

Calculation:

The phenomenon of refraction is shown as below.

  

The relationship between angles ${\theta }_1\text{ and }{\theta }_2$ can be derived from below equation.

  $\frac{n_1}{\sqrt{{\cot }^2{\theta }_1+1}}=\frac{n_2}{\sqrt{{\cot }^2{\theta }_2+1}}$

Simplifying above equation,

  $\begin{array}{l}\Rightarrow \frac{n_1}{\sqrt{{\csc }^2{\theta }_1}}=\frac{n_2}{\sqrt{{\csc }^2{\theta }_2}}\\ \Rightarrow \frac{n_1}{\csc {\theta }_1}=\frac{n_2}{\csc {\theta }_2}\\ \Rightarrow n_1\sin {\theta }_1=n_2\sin {\theta }_2\left\{\text{Snell's law}\right\}\end{array}$

b.

To determine

To show: The value of $n_1$ .

Explanation of Solution

Given information:

Same as part $a$ of this exercise.

  ${\theta }_1={55}^0,{\theta }_2={35}^0,\text{ and }{n}_2=2$ .

Calculation:

The relationship between angles ${\theta }_1\text{ and }{\theta }_2$ can be derived from below equation.

  $n_1\sin {\theta }_1=n_2\sin {\theta }_2\left\{\text{Snell's law}\right\}$

Substitute ${\theta }_1={55}^0,{\theta }_2={35}^0,\text{ and }{n}_2=2$ ,

  $\begin{array}{l}\Rightarrow n_1\sin {55}^0=2\sin {35}^0\\ \Rightarrow n_1=\frac{2\sin {35}^0}{\sin {55}^0}\\ \Rightarrow n_1=\frac{2\times 0.57357}{0.81915}\\ \Rightarrow n_1=1.4\end{array}$

Hence, value of $n_1$ is $1.4$ .

c.

To determine

To show: The value of $n_1\text{ and }{n}_2$ .

Explanation of Solution

Given information:

Same as part $a$ of this exercise and ${\theta }_1={\theta }_2$ .

Calculation:

The relationship between angles ${\theta }_1\text{ and }{\theta }_2$ can be derived from below equation.

  $n_1\sin {\theta }_1=n_2\sin {\theta }_2\left\{\text{Snell's law}\right\}$

Substitute ${\theta }_1={\theta }_2$ ,

  $\begin{array}{l}\Rightarrow n_1\sin {\theta }_1=n_2\sin {\theta }_1\\ \Rightarrow n_1=n_2\end{array}$

In real word, this means light is travelling in same medium and angle of travel do not change. In other words light travels in a straight line in a medium.