Physics Challenge for
Teachers and Students
Solution to March 2008 Challenge
◗ The Chain Rules!
4r
The circuit shown in the diagram extends to the right
into infinity. Each battery has the emf ε (unknown) and
the internal resistance r. Each resistor has the resistance
4r. The reading of the ideal ammeter shown in the diagram is I. Find the value of ε in terms of I and r.
4r
4r
4r
ε,r
ε,r
A
r
Because the elements are purely resistive, the chain
can be replaced by an equivalent resistance, R.
Then using the time-honored tradition of adding one more element to the chain and insisting
that the resistance be unchanged since the chain is
already infinitely long, we get this:
ε,r
4r
Solution: Each of the real batteries can be replaced
with an ideal cell and the battery’s internal resistance. The circuit would then look like this:
4r
4r
r
R
ε
4r
r
ε
r
ε
A
A
r
x
ε
The resistance of the circuit is given by
R=(
1 1 −1
+ ) + 4r .
R r
The line labeled x marks points that are at the
same potential; they can be joined by an ideal wire This can be rearranged to
without affecting the circuit. An equivalent circuit
R 2 – 4rR – 4r 2 = 0.
can be drawn with a single ideal cell and purely
resistive array (enclosed by the broken line) like
Keeping only the positive-valued solution gives
this:
4r
A
4r
r
4r
r
R = (2 + 2 2 )r .
r
The value of ε is related to I and R by ε = IR. So,
ε = (2 + 2 2 )Ir
ε
or
The elements of the one-dimensional array look
like this:
THE PHYSICS TEACHER ◆ Vol. 46, 2008
ε ≈ 4.82 Ir .
Interestingly, the equivalent resistance of the
chain, 4.82r, is not very different from the
1
resistance, 5r, when the chain is reduced to a
single element. This suggests that the equivalent
resistance rapidly approaches its asymptotic value.
This is borne out by analyzing a two-element
chain using Kirchhoff ’s circuit laws for a twoelement chain.
For a two-element chain, the effective resistance is
R=
29
r
6
or R ≈ 4.83r .
(Contributed by Don Easton, Lacombe, Alberta,
Canada)
We would also like to recognize the following
contributors:
André Bellemans (Université Libre de Bruxelles,
Belgium)
Marianne Breinig (The University of Tennessee,
Knoxville, TN)
Phil Cahill (Lockheed Martin Corporation, North
Yorkshire, United Kingdom)
Dario Costello (Vasquez High School, Acton, CA)
Alan J. DeWeerd (University of Redlands,
Redlands, CA)
F. Javier Doblas (Escuela Técnica Superior de
Ingenieros, Sevilla, Spain)
John Dowd (Naples, FL)
Fernando Ferreira (Universidade da Beira Interior,
Covilhã, Portugal)
John F. Goehl, Jr. (Barry University, Miami Shores,
FL)
Fredrick P. Gram (Cuyahoga Community College,
Cleveland, OH)
Art Hovey (Milford, CT)
THE PHYSICS TEACHER ◆ Vol. 46, 2008
Yung Huh (South Dakota State University,
Brookings, SD)
J. Iñiguez (Universidad de Salamanca, Salamanca,
Spain)
Jim Mallmann (Milwaukee School of Engineering,
New Berlin, WI)
Stephen McAndrew (Trinity Grammar School,
Summer Hill, NSW, Australia)
Jeff Melmed (Eastern Maine Community College,
Bangor, ME)
Carl E. Mungan (U. S. Naval Academy, Annapolis,
MD)
Gustavo A. Sánchez (TenarisSiderca R&D,
Campana, Buenos Aires, Argentina)
Scott Schneider (Lawrence Technological
University, Southfield, MI)
Jim Schreve (Hamilton College, Clinton, NY)
Daniel Schumayer (University of Otago, Dunedin,
New Zealand)
Clint Sprott (University of Wisconsin – Madison,
WI)
Michael Threapleton (Centralia College, Centralia,
WA)
Marie van Nieuwenhuizen (Great Neck North
High School, Great Neck, NY)
Chee Leong Wong (student, NUS High School,
Singapore)
Many thanks to all contributors and we hope to
hear from you in the future!
Please send correspondence to:
Boris Korsunsky
korsunbo@post.harvard.edu
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