Optics - Thin Film

Consider two horizontal glass plate with a thin film of air between them. For what values of the thickness of the film of air will the film, as seen by reflected light, appear bright if it is illuminated normally from above by blue light of wavelength 488 nanometers?

A. 0, 122 nm, 244 nm
B. 0, 122 nm, 366 nm
C. 0, 244 nm, 488 nm
D. 122 nm, 244 nm, 366 nm
E. 122 nm, 366 nm, 610 nm

(GR9677 #82)

Solution:

See Thin Film.

Glass to Air: n_g  n_a → Δ_a = 0
Air to Glass: n_a  n_g →  Δ_b = 2t  +½λ

Δ = Δ_b − Δ_a = 2t  + ½λ

Blue Light → Constructive Interference: Δ = mλ
→ mλ =  2t  + ½λ
→ t = ½λ(m − ½)
with λ_a = 488 nm and m = 0,1,2,3...
m = 1 →  t = ½λ(1 − ½) = ¼ λ = 122 nm
m = 2 →  t = ½λ(2 − ½) = ¾ λ = 366 nm
m = 3 →  t = ½λ(3 − ½) = ⁵/4 λ = 610 nm

Answer: E

Classical Mechanics - Angular Speed

A thin plate of mass M, length L, and width 2d is mounted vertically on a frictionless axle along the z-axis. Initially the object is at rest. It is then tapped with a hammer to provide a torque τ, which produces an angular impulse H about the z-axis of magnitude H = ∫ τ dt. What is the angular speed ω of the plate about the z-axis after the tap? 

A. H/2Md²
B. H/Md²
C. 2H/Md²
D. 3H/Md²
E. 4H/Md²

(GR9277 #82)

Solution:

H = ∫ τ dt

Torque: τ = Iα
Angular acceleration: α = ω/t → ω = αt

H = ∫ τ dt = Iα ∫dt = Iαt = Iω

Moment inertia for the plate about the z-axis: I = ¹/₃Md²

H = ¹/₃Md²ω
ω = 3H/Md²

Answer: D

Quantum Mechanics - Spin

Let  represent the state of an electron with spin up and  the state of an electron with spin down. Valid spin eigenfunctions for a triplet state of a two-electron atom include which of the following?



A. I only
B. II only
C. III only
D. I and III
E. II and III

GR0177 #82)

Solution:

Singlet state is anti-symmetric: ψ(1,2) = −ψ(2,1)
Triplet state is symmetric: ψ(1,2) = ψ(2,1)

  is symmetric, because



  is antisymmetric, because


  is symmetric, because


Thus, II is singlet state, I and III are triplet state.

Answer: D