Q49P - University Physics with Modern Physics

Question

The prism shown in Fig. has a refractive index of 1.66, and the angles A are . Two light rays m and n are parallel as they enter the prism. What is the angle between them after they emerge?

                                                                 

Step-by-Step Solution

Answer

The angle between the two rays is 39.1°.

Step1: Calculate angle.

The refractive index of the prism is $n_a=1.66$ . We draw the two rays through the prism as shown below. The angle between both rays when they emerge is shown in the figure below. 

First, let us find the angle ${\theta }_b$ , by using Snell's law. The refractive index of an optical material is expressed by n and represents the speed of light in the vacuum divided by the speed of light in the material. Snell's law is given by an equation in the form

                                                     $n_a\sin {\theta }_a=n_b\sin {\theta }_b$

Where ${\theta }_a=25°$ and $n_b=1$ because it is the refractive index of air. Solve equation (1) for ${\theta }_b$ , and plug the values for $n_a$ ,  ${\theta }_a$ and $n_b$

                                       $\sin {\theta }_b=\frac{n_a\sin {\theta }_a}{n_b}=\frac{\left(1.66\right)\sin 25°}{1}=0.7$

Hence, the angle is

                                                         ${\theta }_b=44.55°$      

From the figure below, the angle  could be calculated by

                                              $\beta =90°-44.55°=45.55°$         

From the triangle the angle c is calculated by

                                     $c=90°-\left(A+\beta \right)=90°-\left(25°+45.55°\right)=19.55°$

Now, the angle between the two rays is calculated by

                                               $a=2c=2\left(19.55°\right)=39.1°$

Step 2: Ray diagram.