Physics quiz question: reflection, refraction at cocoa butter-Pyrex® interface

Physics 205B Quiz 1, spring semester 2015
Cuesta College, San Luis Obispo, CA

Cf. Giambattista/Richardson/Richardson, Physics, 2/e, Problem 23.11

"Cocoa Butter"
John (star5112)
flic.kr/p/Agns9

Light of wavelength 585 nm in melted cocoa butter (index of refraction[*] of 1.457) is incident at an angle of 60.0° at the interface between melted cocoa butter and Pyrex® (index of refraction[*] 1.474). (Drawing is not to scale.) As measured from the normal, the reflected angle in melted cocoa butter is __________ the transmitted angle in Pyrex®.
(A) less than.
(B) equal to.
(C) greater than.
(D) (There is no actual transmitted ray in Pyrex®.)
(E) (Not enough information is given.)

[*] physics.info/refraction/.

Correct answer (highlight to unhide): (C)

Snell's law relates the indices of refraction with the incident and transmitted angles, where cocoa butter is medium 1, and Pyrex® is medium 2:

n₁·sinθ₁ = n₂·sinθ₂,

and the law of reflection relates the incident and (specularly) reflected angles in cocoa butter:

θ_incident = θ_reflected,

where the given (or assumed to be known) quantities, unknown quantities, and quantities to be explicitly solved for (or compared) are denoted.

From the law of reflection, the reflected angle in cocoa butter must also be 60.0°:

θ_incident = θ_reflected,

60.0° = θ_reflected.

From Snell's law, since the index of refraction for cocoa butter n₁ is greater than the index of refraction n₂ for Pyrex®, then the incident angle θ₁ in cocoa butter must be smaller than the transmitted angle θ₂ in Pyrex®. The transmitted angle θ₂ in Pyrex® can also be solved for directly:

n₁·sinθ₁ = n₂·sinθ₂,

(n₁/n₂)·sinθ₁ = sinθ₂,

sin^–1((n₁/n₂)·sinθ₁) = θ₂,

sin^–1((1.457/1.474)·sin(60.0°)) = θ₂,

sin^–1(0.85603732247) = θ₂,

58.8745233263° = θ₂,

and because the reflected angle in cocoa butter is 60.0°, and the transmitted angle in Pyrex® (to three significant figures) is 58.9°, then the reflected angle in cocoa butter is greater than the transmitted angle in Pyrex®.

Student responses
Sections 30882, 30883
Exam code: quiz01c0Co
(A) : 9 students
(B) : 11 students
(C) : 27 students
(D) : 1 student

Success level: 56%
Discrimination index (Aubrecht & Aubrecht, 1983): 0.32