Physics quiz question: minimum possible path length

Physics 205B Quiz 3, spring semester 2014
Cuesta College, San Luis Obispo, CA

Cf. Giambattista/Richardson/Richardson, Physics, 2/e, Problem 25.1

A radio antenna receives a constructive interference signal from two transmitters that broadcast in phase at the same wavelength of 420 m. The distance from the first transmitter to the radio antenna is 800 m. The minimum possible distance from the second transmitter to the radio antenna is:
(A) 240 m.
(B) 380 m.
(C) 420 m.
(D) 800 m.

Correct answer (highlight to unhide): (B)

For in phase sources, the condition for constructive interference is given by:

∆l = m·λ,

where the difference in path lengths is ∆l = l₁ – l₂, and solving for l₂ yields:

l₂ = l₁ – m·λ,

To find the minimum (positive-definite) possible value for l₂, we put in progressively larger (positive) values for m = 0, +1, +2, +3, ..., such that:

l₂ = (800 m) – m·(420 m) = 800 m, 380 m,

of which 380 m is the minimum possible distance from the second transmitter to the receiving radio antenna. (Plugging in negative values for m would result in other possible distances from the second transmitter to the receiving radio antenna greater than 800 m.)

(Response (A) is λ/2; response (C) is λ; response (D) is the second smallest possible distance from the second transmitter to the receiving radio antenna.)

Sections 30882, 30883
Exam code: quiz03cDvD
(A) : 2 students
(B) : 19 students
(C) : 12 students
(D) : 7 students

Success level: 48%
Discrimination index (Aubrecht & Aubrecht, 1983): 0.45