Physics midterm problem: direct wave and phase-shifted in-line reflection interference

Physics 205B Midterm 1, spring semester 2012
Cuesta College, San Luis Obispo, CA

Cf. Giambattista/Richardson/Richardson, Physics, 2/e, Problems 25.1, 25.3

[20 points.] A radio transmitter sends a signal of wavelength 0.80 km to a receiver 1.40 km away. The signal also travels from the transmitter to the receiver by another path where it reflects with a 180° phase shift from a metal skyscraper. (Drawing is not to scale.) As a result, there is a constructive signal at the receiver. Find the shortest possible (non-zero) distance between the receiver and the skyscraper. Show your work and explain your reasoning.


Solution and grading rubric:

• p = 20/20:
  Correct. Applies constructive interference for out-of-phase sources Δl = (m + (1/2))·λ, where path difference Δl = 2·x. Solving for the distance between the receiver and skyscraper x = (m + (1/2))·(λ/2), and setting m = 0 to find the smallest value for x = λ/4 = 0.20 km.
  
• r = 16/20:
  Nearly correct, but includes minor math errors. Sets m = 1, and finds the next smallest value for x = 0.60 km, or has Δl = x instead of 2·x.
  
• t = 12/20:
  Nearly correct, but approach has conceptual errors, and/or major/compounded math errors. At least applies correct constructive interference for out-of-phase condition.
  
• v = 8/20:
  Implementation of right ideas, but in an inconsistent, incomplete, or unorganized manner. Some attempt at connecting interference conditions to path difference.
  
• x = 4/20:
  Implementation of ideas, but credit given for effort rather than merit.
  
• y = 2/20:
  Irrelevant discussion/effectively blank.
  
• z = 0/20:
  Blank.
  

Grading distribution:
Section 30882
Exam code: midterm01o1Ls
p: 4 students
r: 5 students
t: 8 students
v: 4 students
x: 5 students
y: 1 student
z: 0 students

A sample "p" response (from student 6969):