Physics final exam question: shifting interfering transmitters

Physics 205B Final Exam, 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. Discuss why the radio antenna will still receive a constructive interference signal if one transmitter were moved one-half of a wavelength closer to the radio antenna, while the other transmitter was moved one-half of a wavelength farther from the radio antenna. Explain your reasoning by using the properties of waves and interference.

Solution and grading rubric:

• p:
  Correct. Shows either by a diagram or explicit demonstration of how Δl = |l₁ – l₂| is a whole number of wavelengths (where l₁' = l₁ + λ/2 and l₂' = l₂ – λ/2 results in Δl' still being equal to a whole number of wavelengths) that constructive interference still results.
• r:
  As (p), but argument indirectly, weakly, or only by definition supports the statement to be proven, or has minor inconsistencies or loopholes.
• t:
  Nearly correct, but argument has conceptual errors, or is incomplete.
• v:
  Limited relevant discussion of supporting evidence of at least some merit, but in an inconsistent or unclear manner. Some attempt at using interference, phase/path lengths differences.
• x:
  Implementation/application of ideas, but credit given for effort rather than merit.
• y:
  Irrelevant discussion/effectively blank.
• z:
  Blank.
  

Grading distribution:
Sections 30882, 30883
Exam code: finalEL7a
p: 20 students
r: 3 students
t: 7 students
v: 4 students
x: 1 student
y: 0 students
z: 0 students

A sample "p" response (from student 0001), using the path-length difference equation:
Another sample "p" response (from student 0007), using a diagram:
Another sample "p" response (from student 7810), using both the path-length difference equation and a diagram: