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Physics Challenge for Teachers and Students Boris Korsunsky, Column Editor Weston High School, Weston, MA 02493 [email protected] Solution to April 2013 Challenge w The Chain of Command In the infinite circuit shown in the diagram, each battery has emf e and internal resistance r. Each resistor has resistance 2r. Find the emf and the internal resistance of the equivalent battery. Solution: We can model the circuit as a voltage source with emf E in series with resistance R, shown below on the left (let us call it “the simple model”). Because the basic combination of “the real battery + 2r resistance” is repeated indefinitely, the simple model must also be equivalent to the original battery (emf ε in series with r) followed with the 2r resistor in parallel with the simple model. (See the figures below.) We can find the equivalent resistance of the circuit on the right by “shorting” the voltage sources. This yields r in series with the parallel combination of 2r and R. This must be equal to the resistance of the left circuit, R: R =r+ 2rR . ( R + 2r ) Solving this for R, we get a quadratic with a positive solution R = 2r. The open-circuit voltage of the simple model is E. The open-circuit voltage of the circuit on the right can be found by simple analysis. There is no current in resistor r, and the current in the 2r resistor is E/(2r+R) = E/(4r). The resulting voltage across the 2r resistance is then E/(4r)*2r = E/2. The total voltage across the open nodes is therefore ε+ E/2. Now we have E = e+ E , or E = 2e. 2 (Contributed by Bill Nettles, Union University, Jackson, TN) The Physics Teacher ◆ Vol. 51, 2013 We would also like to recognize the following successful contributors: Phil Cahill (The SI Organization, Inc., Rosemont PA) Don Easton (Lacombe, Alberta, Canada) Fernando Ferreira (Universidade da Beira Interior, Covilhã, Portugal) Gerald E. Hite (TAMUG, Galveston, TX) Art Hovey (Galvanized Jazz Band, Milford, CT) José Ignacio Íñiguez de la Torre (Universidad de Salamanca, Salamanca, Spain) Daniel Mixson (Naval Academy Preparatory School, Newport, RI) Carl E. Mungan (U. S. Naval Academy, Annapolis, MD) Jorge Salazar (Pontificia Universidad Católica del Perú, Perú) Asif Shakur (Salisbury University, Salisbury, MD) Jason L. Smith (Richland Community College, Decatur, IL) Clint Sprott (University of Wisconsin – Madison, WI) Many thanks to all contributors and we hope to hear from many more of you in the future! Guidelines for contributors: – We ask that all solutions, preferably in Word format, be submitted to the dedicated email address [email protected] Each message will receive an automatic acknowledgment. – The subject line of each message should be the same as the name of the solution file. – The deadline for submitting the solutions is the last day of the corresponding month. – We can no longer guarantee that we’ll publish every successful solver’s name; each month, a representative selection of names will be published, both in print and on the web. – If your name is—for instance—Joseph Ratzinger, please name the file “May13Ratzinger” (do not include your first initial) when submitting the May solution. – If you have a message for the Column Editor, you may contact him at [email protected]; however, please do not send your solutions to this address. As always, we look forward to your contributions and hope that they will include not only solutions but also your own Challenges that you wish to submit for the column. Boris Korsunsky, Column Editor