Example 23-2 Critical Angles
(a) A laser is aimed from under the water toward the surface,
as in Figure 23-11. Find the critical angle of the light incident
in the water beyond which total internal reflection occurs.
(b) Find the critical angle if the liquid in the tank were replaced
by water containing a high concentration of dissolved sugar.
The light undergoes total internal reflection
Air
Water
θ1
Figure 23-11 ​Total internal reflection in water Light from a laser aimed
from under the water toward the surface is totally internally reflected. What is
the minimum incident angle for which total internal reflection will occur?
Set Up
Before hitting the surface (the boundary between
water and air), the light is propagating in water, so
this is ­medium 1 with index of refraction n1. In
part (a), medium 1 is ordinary water with n1 = 1.33;
in part (b), medium 1 is sugar water with n1 = 1.49. In
both parts, medium 2 on the other side of the surface is
air with index of refraction n2 = 1.00. We’ll use Equation
23-10 to solve for the critical angle in each case.
Solve
(a) Find the critical angle if medium 1 is water.
Incident
light
Reflected
light
Normal
Critical angle for total internal reflection:
uc = sin-1 a
n2
b (23-10)
n1
With n1 = 1.33 and n2 = 1.00, the critical angle is
uc = sin-1 a
1.00
b = sin-1 0.752 = 48.8
1.33
If the angle of incidence u1 is 48.8° or greater, the light will
undergo total internal reflection and no light will go into the
air. (Note that u1 in Figure 23-11 is approximately 60°, which
is indeed greater than 48.8°.)
(b) Find the critical angle if medium 2 is sugar water.
With n1 = 1.49 and n2 = 1.00, the critical angle is
uc = sin-1 a
Reflect
1.00
b = sin-1 0.671 = 42.2
1.49
Adding sugar to water results in a lower speed of light, which increases the index of refraction. The minimum angle of
incidence for which total internal reflection occurs is therefore smaller for sugar water than for pure water. The smaller
the critical angle, the larger the range of angles at which light experiences total internal reflection.