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Kirchhoff's Gambit
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by David Castro
Charles A. Dana Center
University of Texas
Austin, Texas
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Introduction
When solving circuit problems, many students rely exclusively on simplifying complicated circuits and then applying Ohm's law. While this technique works for many circuits, it is often more efficient and more elegant to take advantage of Kirchhoff's two circuit rules. This question asks students to solve a puzzle by creating a circuit diagram from 16 different circuit elements. There is only one solution to the puzzle (as far as I can tell), and students must apply both of Kirchhoff's circuit rules repeatedly. Three variants of the puzzle are presented so that the instructor can vary the level of challenge. All three variants have the same solution but differ in the amount of initial information provided to students.
Students will manipulate the puzzle pieces extensively, so I recommend that the instructor print copies of the puzzle on card stock.
Problem Statement
In 1845, while he was still a student, Gustav Kirchhoff formulated two rules to help solve circuit problems. Kirchhoff's point rule states that the total current entering a junction must equal the total current leaving that junction. This rule is depicted below.
Kirchhoff's loop rule states that the sum of the voltage gains and drops around any closed loop in a circuit must total zero. If the loop is drawn so that the path flows "downstream" (in the direction of conventional current), a battery is considered to have a positive voltage (energy is added to the circuit), and a resistor is considered to have a negative voltage (energy is subtracted from the circuit).
If a loop is traced "upstream," the potential difference of a battery is considered to be negative, and the potential difference of a resistor is considered to be positive.
Note that in this example, parallel resistors are shown to have the same voltage.
The 16 puzzle pieces associated with this problem represent different circuit elements. When arranged correctly, they form a four-by-four-piece square, with the "sun" symbol appearing somewhere within the puzzle. If all of the puzzle pieces are placed appropriately, the sun will be in a specific position -- ask your instructor to confirm your solution!
 Circuits Puzzle Pieces: Easy Version (.pdf/24KB)
Circuits Puzzle Pieces: Medium Version (.pdf/24KB)
Circuits Puzzle Pieces: Hard Version (.pdf/24KB)
Note that you must apply Ohm's law and Kirchhoff's point rule to several of the individual pieces before they can be correctly placed. The arrows indicate the direction of conventional current flow.
Click here to view the answers and commentary!
David Castro taught AP Physics (B and C), AP Calculus (AB and BC), and AP U.S. and European History in a teaching career spanning 14 years, including 5 years as a master AP Physics teacher. In 1997, he received a Special Recognition Teaching Award, and in 2002 his combined AP Physics and AP Calculus syllabus was published in the AP Physics Teacher's Guide. Active as an AP Physics consultant in the Southwest Region since 1995, his areas of expertise include Pre-AP middle school science, AP Vertical Teams, as well as interdisciplinary physics/calculus. He also serves as a Reader for AP Physics. Mr. Castro recently joined the Charles A. Dana Center at the University of Texas, where he continues to focus on providing support for science educators.
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