Physics midterm question: destructively interfering out-of-phase radio transmitters

Physics 205B Midterm 1, spring semester 2019
Cuesta College, San Luis Obispo, CA

Two vertical radio transmitters broadcast at the same wavelength, and are spaced 6.0 m apart along the east-west direction. A receiver held by a Physics 205B student located to the east of both transmitters detects a destructive interference signal, and a receiver held by another Physics 205B student located to the north also detects a destructive signal. Show (a) why the transmitters must be out-of-phase sources, and (b) find a plausible numerical value for the wavelength. Explain your reasoning using the properties of source phases, path lengths, and interference.

Solution and grading rubric:

• p:
  Correct. Discusses/demonstrates:
  1. that the radio transmitters are out of phase, as there is no path length difference for the student located to the north, and destructive interference (as is the case here) can only occur if the sources are out of phase; and
  2. for the student located to the east, for these out of phase sources to interfere destructively the path length difference must be a whole number of wavelengths, and since the path length difference is 6.0 m, plausible (non-zero) wavelength values would be 6.0 m, 3.0 m, 1.5 m, etc.
• 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. At least discusses why (1) transmitters are out of phase, but (2) does not use the correct destructive interference condition (whole number of wavelengths) for out of phase sources for the student located to the east.
• v:
  Limited relevant discussion of supporting evidence of at least some merit, but in an inconsistent or unclear manner. Some garbled attempt at discussing source phases, path lengths, and interference.
• x:
  Implementation/application of ideas, but credit given for effort rather than merit. Approach other than that of discussing source phases, path lengths, and interference.
• y:
  Irrelevant discussion/effectively blank.
• z:
  Blank.
  

Grading distribution:
Sections 30882, 30883
Exam code: midterm01Ft6G
p: 13 students
r: 4 students
t: 13 students
v: 12 students
x: 0 students
y: 0 students
z: 0 students

A sample "p" response (from student 5250), identifying the correct path length difference condition for two sources that are out of phase:
Another sample "p" response (from student 1810), with a graphical representation of the interfering waves: