Physics midterm question: wavelengths at total internal reflection interface

Physics 205B Midterm 1, spring semester 2018
Cuesta College, San Luis Obispo, CA

Light of wavelength 533 nm in a unknown material (index of refraction n₁) undergoes total internal reflection from an interface with a different unknown material (index of refraction n₂). (Drawing is not to scale.) Show that the wavelength in the n₂ material would be longer than 533 nm (assuming that light could eventually be transmitted out in the n₂ material). Explain your reasoning using the properties of light and refraction.

Solution and grading rubric:

• p:
  Discusses/demonstrates that the wavelength λ₂ in the second n₂ material would be longer than 533 nm by:
  1. qualitatively or quantitatively showing that n₂ < n₁ by appealing to the critical angle θ₂ = sin⁻¹(n₂/n₁) (where n₂ < n₁ in order to avoid a domain error in the inverse sine function); or Snell's law n₁⋅sinθ₁ = n₂⋅sinθ₂ where θ₂ = 90°, and thus n₂ < n₁; and
  2. that since index of refraction n = c/v, a smaller index of refraction n₂ results in a faster speed of light v₂; and
  3. since the wavelength λ = v/f, since the second material has a faster speed of light v₂, and the frequency of light is independent of the medium it travels through, then it will have a longer wavelength in that material.
  May combine arguments in (1)-(2) by arguing that total internal reflection occurs when light traveling in a slower material is "frustrated" in trying to travel out into a faster material.
• r:
  As (p), but argument indirectly, weakly, or only by definition supports the statement to be proven, or has minor inconsistencies or loopholes. Typically one of arguments (1)-(3) missing, incomplete, or problematic.
• t:
  Nearly correct, but argument has conceptual errors, or is incomplete. Typically two of arguments (1)-(3) missing, incomplete, or problematic.
• v:
  Limited relevant discussion of supporting evidence of at least some merit, but in an inconsistent or unclear manner. Some garbled attempt at applying Snell's law and/or critical angles, indices of refraction, wave speed, frequency and wavelengths.
• x:
  Implementation/application of ideas, but credit given for effort rather than merit. No clear attempt at applying Snell's law and/or critical angles, indices of refraction, wave speed, frequency and wavelengths.
• y:
  Irrelevant discussion/effectively blank.
• z:
  Blank.
  

Grading distribution:
Sections 30882, 30883
Exam code: midterm01cVdP
p: 17 students
r: 6 students
t: 9 students
v: 3 students
x: 0 students
y: 0 students
z: 0 students

A sample "p" response (from student 3673):
A sample "p" response (from student 1929):