Chapter 7.4, Problem 57E

Refraction of Light It has been observed since ancient times that light refracts, or “bends,” as it travels from one medium to another (from air to water, for example). If v₁, is the speed of light in one medium and v₂ is its speed in another medium, then according to Snell’s Law,

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

where θ₁ is the angle of incidence and θ₂ is the angle of refraction (see the figure). The number v₁/v₂ is called the index of refraction. The index of refraction for several substances is given in the table.

If a ray of light passes through the surface of a lake at an angle of incidence of 70°, what is the angle of refraction?

Substance	Refraction from air to substance
Water	1.33
Alcohol	1.36
Glass	1.52
Diamond	2.41

To determine

To find: The angle of refraction ${\theta }_2$ if a ray of light passes through the surface of a lake at an angle of incidence of $70°$.

Answer to Problem 57E

The angle of refraction is $\underset{¯}{{\theta }_2=45.23}$.

Explanation of Solution

Given:

Angle of incidence is ${\theta }_1=70°$.

Index of refraction from air to water is $\frac{v_1}{v_2}=1.33$.

Result used:

Snell’s Law:

If $v_1$ is the speed of light in one medium and $v_2$ is its speed in another medium, then $\frac{\sin {\theta }_1}{\sin {\theta }_2}=\frac{v_1}{v_2}$, where ${\theta }_1$ is the angle of incidence and ${\theta }_2$ is the angle of refraction. The number $\frac{v_1}{v_2}$ is called the index of refraction.

Calculation:

Use Snell’s Law and obtain the value of ${\theta }_2$.

Substitute the given value of ${\theta }_1$ and $\frac{v_1}{v_2}$ in the equation $\frac{\sin {\theta }_1}{\sin {\theta }_2}=\frac{v_1}{v_2}$.

$\begin{array}{l}\frac{\sin 70°}{\sin {\theta }_2}=1.33\\ \sin {\theta }_2=\frac{0.94}{1.33}\\ \sin {\theta }_2=0.71\\ {\theta }_2={\sin }^{-1}\left(0.71\right)\end{array}$

That is, ${\theta }_2=45.23$.

Thus, the angle of refraction is $\underset{¯}{{\theta }_2=45.23}$.