Physics midterm question: equally bright, different resistance light bulbs

Physics 205B Midterm 2, spring 2016
Cuesta College, San Luis Obispo, CA

An 6.0 V emf source is connected to a resistor r, and two light bulbs with different resistances R₁ and R₂. Both light bulbs have the same brightness. Discuss why R₁ > R₂. Explain your reasoning using Kirchhoff's rules, Ohm's law, and electric power.

Solution and grading rubric:

• p:
  Correct. Discusses/demonstrates how R₁ > R₂ using at least one of two similar arguments:
  1. the current that flows through the R₂ light bulb and emf splits in parallel through both the resistor and the R₁ light bulb, such that I₂ is greater than I₁, and given P₁ = P₂, then (I₁)²⋅R₁ = (I₂)²⋅R₂, and since I₂ is greater than I₁, then R₂ must be less than R₁; or
  2. the current that flows through the R₂ light bulb and emf splits in parallel through both the resistor and the R₁ light bulb, such that I₂ is greater than I₁, and given P₁ = P₂, then I₁⋅ΔV₁ = I₂⋅ΔV₂, and since I₂ is greater than I₁, then ΔV₂ is less than ΔV₁, such that I₂⋅R₂ is less than I₁⋅R₁, and again from I₂ is greater than I₁, thus R₂ must be less than R₁.
• 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. May use the statement to be proved (R₁ > R₂) and given information (P₁ = P₂) to instead demonstrate how I₂ is greater than I₁ and/or ΔV₂ is less than ΔV₁; or does not sufficiently show how either I₂ is greater than I₁ and/or ΔV₂ is less than ΔV₁ in order to prove that R₁ > R₂.
  
• v:
  Limited relevant discussion of supporting evidence of at least some merit, but in an inconsistent or unclear manner. Some attempt at applying properties of Kirchhoff's rules, Ohm's law, and electric power.
  
• x:
  Implementation/application of ideas, but credit given for effort rather than merit. Approach other than that of applying Kirchhoff's rules, Ohm's law, and electric power.
  
• y:
  Irrelevant discussion/effectively blank.
  
• z:
  Blank.
  

Grading distribution:
Sections 30882, 30883
Exam code: midterm02Mc4s
p: 8 students
r: 4 students
t: 11 students
v: 15 students
x: 4 students
y: 0 students
z: 0 students

A sample "p" response (from student 1157):