Physics midterm question: enlarged virtual image from converging lens

Physics 205B Midterm 1, spring semester 2017
Cuesta College, San Luis Obispo, CA

An object is placed in front of a converging lens, which results in a virtual image that is exactly 2× larger than the size of the object. Discuss why the object must be placed exactly halfway between the focal point and the lens for this to happen. Explain your reasoning by using ray tracings and/or thin lens equations, properties of lenses, images, and magnification.

Solution and grading rubric:

• p:
  Correct. Discusses/demonstrates that a virtual image that twice the size of an object must be placed halfway between the focal point and a converging lens:
  1. uses the linear magnification equation m = –d_i/d_o to find that d_i = –2·d_o (where m = +2, as virtual images are upright); and
  2. from the thin lens equation, substituting d_i = –2·d_o to eliminate d_i, and solving for d_o in terms of the focal length f results in d_o = f/2, thus the object must be placed halfway between the focal point and a converging lens; and/or
  3. assumes numerical values for the focal length, object distance and/or image distance to quantitatively result in a linear magnification factor of +2, with an object distance that is half of the numerical value of the focal length; and/or
  4. draws a carefully, properly scaled ray tracing diagram.
• r:
  As (p), but argument indirectly, weakly, or only by definition supports the statement to be proven, or has minor inconsistencies or loopholes. May have negative sign errors in (1)-(2).
• t:
  Nearly correct, but argument has conceptual errors, or is incomplete. May have (1), but does not explicitly prove that d_o = f/2.
• v:
  Limited relevant discussion of supporting evidence of at least some merit, but in an inconsistent or unclear manner. Some garbled attempt at applying ray tracings and/or thin lens equations, properties of lenses, images, and linear magnification.
• x:
  Implementation/application of ideas, but credit given for effort rather than merit. No clear attempt at applying ray tracings and/or thin lens equations, properties of lenses, images, and linear magnification.
• y:
  Irrelevant discussion/effectively blank.
• z:
  Blank.
  

Grading distribution:
Sections 30882, 30883
Exam code: midterm01AhC4
p: 6 students
r: 4 students
t: 10 students
v: 7 students
x: 2 students
y: 0 students
z: 1 student

A sample "p" response (from student 6819):