Answer to Question #126477 in Mechanics | Relativity for Stefanie Hernandez-Mendez

Question

A beam of light enters a glass tank filled with water at an incident angle of θa = 72° (see figure). The index of refraction of air is na = 1, the index of refraction of glass is ng = 1.5, and the index of refraction of water is nw = 1.3.

Calculate the value of θw, in degrees.

Answer 1

Explanations & Calculations

• Refer to the figure above.
• Since no any figure shown previously , a justified calculation which includes all the indices of refraction is provided below.
• Considering the air-glass interface & applying $n =\Large \frac{\sin\theta_i}{\sin\theta_r}$ ; n is the refractive index of the medium —in which the refractive angle is present— with respect to adjoining medium; air, for the first calculation.

$\qquad\qquad \begin{aligned} \small \frac{n_g}{n_a}&= \small \frac{1.5}{1} = \frac{\sin\theta_a}{\sin\theta_g}=\frac{\sin72}{\sin\theta_g}\\ \small \sin\theta_g &= \small 0.634\cdots\cdots\cdots(1) \end{aligned}$

• Now applying the same relationship at the glass-water interface to calculate the angle in water.

$\qquad\qquad \begin{aligned} \small \frac{n_w}{n_g}&= \small \frac{1.3}{1.5}=\frac{\sin\theta_g}{\sin\theta_w}=\frac{0.634}{\sin\theta_w}\\ \small \sin\theta_w &= \small 0.7315\\ \small \theta_w &= \small \bold{ 47.02^0} \end{aligned}$