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Difficulty: MediumRefraction of Light, Total Internal Reflection, and Prisms

A beam of monochromatic light travels through a transparent liquid toward a boundary with air. The speed of light in the liquid is 1.80×108 m s−11.80 \times 10^8\text{ m s}^{-1} and the speed of light in air is 3.00×108 m s−13.00 \times 10^8\text{ m s}^{-1}. What is the value of the sine of the critical angle (sin⁡C\sin C) for total internal reflection at this boundary, and under what condition of propagation can total internal reflection occur?

  1. sin⁡C=0.60\sin C = 0.60, and the light ray must travel from the liquid into air.Answer
  2. B
    sin⁡C=0.60\sin C = 0.60, and the light ray must travel from air into the liquid.
  3. C
    sin⁡C=1.67\sin C = 1.67, and the light ray must travel from the liquid into air.
  4. D
    sin⁡C=1.67\sin C = 1.67, and the light ray must travel from air into the liquid.

Answer

The sine of the critical angle is sin⁡C=0.60\sin C = 0.60, and total internal reflection can only occur when light travels from the optically denser liquid into air.
Total internal reflection requires light to originate in the optically denser medium (the liquid) and travel toward the rarer medium (air), at an angle of incidence greater than the critical angle. The sine of the critical angle is calculated directly as the ratio of the speed of light in the medium to the speed of light in air: sin⁡C=vliquidvair=1.80×1083.00×108=0.60\sin C = \frac{v_{\text{liquid}}}{v_{\text{air}}} = \frac{1.80 \times 10^8}{3.00 \times 10^8} = 0.60.

Step-by-Step Solution

1
Determine the refractive index nn of the liquid relative to air.
n=cv=3.00×108 m s−11.80×108 m s−1=1.67n = \frac{c}{v} = \frac{3.00 \times 10^8\text{ m s}^{-1}}{1.80 \times 10^8\text{ m s}^{-1}} = 1.67 (or 53\frac{5}{3}).
Refractive index is the ratio of the speed of light in vacuum/air to the speed of light in the medium.
2
Calculate the sine of the critical angle sin⁡C\sin C.
sin⁡C=1n=vc=1.80×1083.00×108=0.60\sin C = \frac{1}{n} = \frac{v}{c} = \frac{1.80 \times 10^8}{3.00 \times 10^8} = 0.60.
The critical angle relationship between a medium and air is given by sin⁡C=1n\sin C = \frac{1}{n}.
3
Identify the necessary physical condition for total internal reflection to take place.
Light must travel from an optically denser medium (liquid) toward an optically less dense medium (air).
For total internal reflection to happen, the ray must bend away from the normal until the angle of refraction reaches 90∘90^\circ. This only occurs when moving from a denser to a rarer medium.

Key Concept

Critical Angle and Conditions for Total Internal Reflection
Estimated Time:1m 15s
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Topics

Refraction of Light, Total Internal Reflection, and Prisms

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