April: The Chain of Command

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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
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