Chapter MPS, Problem 14.2MPS

To determine

Rewrite Snell’s Law in terms of $\csc \theta$ .

Answer to Problem 14.2MPS

  $\frac{\csc {\theta }_2}{\csc {\theta }_1}=\frac{v_1}{v_2}$

Explanation of Solution

Given information:

Snell’s Law describes the relationship between the angles of incidence and refraction passing through a boundary between two different media, such as air and glass.

  $\frac{\sin {\theta }_1}{\sin {\theta }_2}=\frac{{\upsilon }_1}{{\upsilon }_2}$

Calculation:

Consdier the formula

  $\frac{\sin {\theta }_1}{\sin {\theta }_2}=\frac{{\upsilon }_1}{{\upsilon }_2}$

Recollect the reciprocal identity:

  $\sin \theta =\frac{1}{\csc \theta },\csc \theta \ne 0$

Use the above reciprocal identity; rewrite the given formula as,

  $\begin{array}{l}\frac{\sin {\theta }_1}{\sin {\theta }_2}=\frac{v_1}{v_2}\\ \frac{\frac{1}{\csc {\theta }_1}}{\frac{1}{\csc {\theta }_2}}=\frac{v_1}{v_2}\\ \frac{1}{\csc {\theta }_1}\times \frac{\csc {\theta }_2}{1}=\frac{v_1}{v_2}\\ \frac{\csc {\theta }_1}{\csc {\theta }_2}=\frac{v_1}{v_2}\end{array}$

Hence, the above formula in terms of $\csc \theta$ is $\frac{\csc {\theta }_2}{\csc {\theta }_1}=\frac{v_1}{v_2}$ .