Physics quiz question: wavelength in unknown material

Physics 205B Quiz 1, spring semester 2017
Cuesta College, San Luis Obispo, CA

Light of wavelength 550 nm in calcite (index of refraction[*] of 1.486) with an incident angle of 37.0° is transmitted into an unknown material with an angle of 30.5°. (Drawing is not to scale.) The wavelength of this light in the unknown material is __________ 550 nm.
(A) shorter than.
(B) equal to.
(C) longer than.
(D) (Not enough information is given.)

[*] physics.info/refraction/.

Correct answer (highlight to unhide): (A)

Snell's law relates the indices of refraction with the incident and transmitted angles:

n_calcite·sin(θ_calcite) = n_unknown·sin(θ_unknown).

Since the incident angle in calcite (θ_calcite = 37.0°) is greater than the transmitted angle in the unknown material (θ_unknown = 30.5°), then the calcite index of refraction must be less than that of the unknown material:

θ_calcite > θ_unknown,
∴n_calcite < n_unknown.

The relationships between the indices of refraction and speed of light for calcite and the unknown material are given by:

n_calcite = c/v_calcite,
n_unknown = c/v_unknown,

where c is the speed of light in vacuum. Since calcite has a lower index of refraction, then the speed of light in calcite must be faster:

n_calcite < n_unknown,
∴v_calcite > v_calcite.

The relationships between the speed of light, the frequency and wavelength in a material are given by:

λ_calcite = v_calcite/f_calcite,
λ_unknown = v_unknown/f_unknown.

Since the frequency of a wave depends on its source and not on the material it travels through, the frequency remains constant as the light starts in calcite and is then transmitted into the unknown material:

f_calcite = f_unknown.

And because the speed of light in calcite is faster than the unknown material, then the wavelength of light in calcite (λ_calcite = 550 nm) must be greater than the wavelength of light in the unknown material:

v_calcite > v_calcite,
∴λ_calcite > λ_unknown.

Student responses
Sections 30882, 30883
Exam code: quiz01Om6A
(A) : 19 students
(B) : 2 students
(C) : 7 students
(D) : 0 students

Success level: 68%
Discrimination index (Aubrecht & Aubrecht, 1983): 0.50