Chapter 13.3, Problem 32PPS

a.

To determine

To prove: $n=\sqrt{3}\sin \frac{a}{2}+\cos \frac{a}{2}$ .

Explanation of Solution

Given Information: Index of refraction of the glass with respect to air is

  $n=\frac{\sin \left[\frac{1}{2}\left(a+b\right)\right]}{\sin \frac{b}{2}}$ , where $\text{'}a\text{'}$ is the measure of the deviation angle and $\text{'}b\text{'}$ is the measure of the prism apex angle.

Proof:

Prism apex angle, $b=60°$

  $\begin{array}{l}n=\frac{\sin \left[\frac{1}{2}\left(a+b\right)\right]}{\sin \frac{b}{2}}\\ \Rightarrow n=\frac{\sin \frac{a}{2}\cos \frac{b}{2}+\sin \frac{b}{2}\cos \frac{a}{2}}{\sin \frac{b}{2}}\\ =\sin \frac{a}{2}\cot \frac{b}{2}+\cos \frac{a}{2}\\ =\sin \frac{a}{2}\cot 30°+\cos \frac{a}{2}\\ =\sqrt{3}\sin \frac{a}{2}+\cos \frac{a}{2}\end{array}$

So, $n=\sqrt{3}\sin \frac{a}{2}+\cos \frac{a}{2}$ .

Hence, proved.

b.

To determine

To find: $\text{'}n\text{'}$ for the given prism.

Answer to Problem 32PPS

  $n=\sqrt{3}$ .

Explanation of Solution

Given Information: Index of refraction of the glass with respect to air is $n=\frac{\sin \left[\frac{1}{2}\left(a+b\right)\right]}{\sin \frac{b}{2}}$ , where $\text{'}a\text{'}$ is the measure of the deviation angle and $\text{'}b\text{'}$ is the measure of the prism apex angle.

Calculation:

Angle of deviation, $a=180°$

From part (a),

  $\begin{array}{l}n=\sqrt{3}\sin \frac{a}{2}+\cos \frac{a}{2}\\ =\sqrt{3}\sin 90°+\cos 90°\\ =\sqrt{3}(1)+0\\ =\sqrt{3}\end{array}$

Thus, $n=\sqrt{3}$ .