Light enters a solid pipe made of plastic having an index of refraction of 1.60. The light travels parallel to the upper part of the pipe (Fig. E33.15). You want to cut the face AB so that all the light will reflect back into the pipe after it first strikes that face. (a) What is the largest that u can be if the pipe is in air?

Verified step by step guidance

1. Identify the critical angle for total internal reflection. The critical angle, \( \theta_c \), can be calculated using the formula \( \sin(\theta_c) = \frac{n_2}{n_1} \), where \( n_1 \) is the index of refraction of the plastic (1.60) and \( n_2 \) is the index of refraction of air (1.00).
2. Calculate the critical angle using the formula from step 1. This angle will determine the maximum angle at which light can strike the face AB and still be totally internally reflected.
3. Determine the angle of incidence, \( \theta_i \), of the light rays striking face AB. Since the light travels parallel to the upper part of the pipe, the angle of incidence is related to the angle at which face AB is cut.
4. Set the angle of incidence, \( \theta_i \), equal to the critical angle, \( \theta_c \), calculated in step 2. This ensures that all light hitting face AB at this angle will be totally internally reflected.
5. Solve for the angle \( u \), which is the angle at which face AB should be cut. This angle will be complementary to the critical angle, so \( u = 90^\circ - \theta_c \).

Key Concepts

Index of Refraction

The index of refraction (n) is a dimensionless number that describes how light propagates through a medium. It is defined as the ratio of the speed of light in a vacuum to the speed of light in the medium. For plastic with an index of refraction of 1.60, light travels slower than in a vacuum, which affects how it bends when entering or exiting the material.

Total Internal Reflection

Total internal reflection occurs when light attempts to move from a medium with a higher index of refraction to one with a lower index (like from plastic to air) at an angle greater than the critical angle. This phenomenon ensures that all light is reflected back into the medium, which is essential for the design of optical fibers and other applications where light confinement is necessary.

Critical Angle

The critical angle is the minimum angle of incidence at which total internal reflection occurs. It can be calculated using Snell's Law, which relates the angles of incidence and refraction to the indices of refraction of the two media. For a plastic pipe in air, knowing the critical angle helps determine the maximum angle at which light can strike the face AB to ensure it reflects back into the pipe.