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2002
A Nomograph for Resistors in Parallel
Tom Greenslade
Kenyon College, greenslade@kenyon.edu
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Recommended Citation
“A Nomograph for Resistors in Parallel”, The Physics Teacher, 40, 458 (2002)
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A Nomograph for Resistors in Parallel
Thomas B. Greenslade Jr.
Citation: The Physics Teacher 40, 458 (2002); doi: 10.1119/1.1526612
View online: http://dx.doi.org/10.1119/1.1526612
View Table of Contents: http://scitation.aip.org/content/aapt/journal/tpt/40/8?ver=pdfcov
Published by the American Association of Physics Teachers
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A Nomograph for
Resistors in Parallel
Thomas B. Greenslade Jr., Kenyon College, Gambier, OH 43022; greenslade@kenyon.edu
I
n addition to a large collection of 19th-century textbooks, I also have a large collection
of textbooks from the earlier part of the 20th
century. While browsing through one of them recently, I ran across the following quotation:1
“Let R1 and R2 be two resistances in parallel.
Using rectangular coordinates, lay off R1 on
the x-axis and R2 on the y-axis [Fig. 1]. Join
the points determined in this way by a
straight line. Draw a 45-degree line from the
origin, thus determining the point P. Then
either coordinate of P will give the value of
R, the resistance of the branched circuit.
This method may be used to find either of
the quantities R1, R2 or R, when the other
two are known.”
This seemed to be a far remove from 1/R1 +
1/R2 = 1/R, so I worked out the geometry behind
the construction. Triangles ABC and PBD are
similar, which gives
R1/R2 = (R1 – R)/R.
The similarity of triangles CAB and EAP gives
R2/R1 = (R2 – R)/R.
Inverting the second equation and setting it
equal to the first, followed by a little algebra,
gives the standard equation for the equivalent
resistance of two resistors in parallel. My colleague Tim Sullivan points out that the equation
of the diagonal line is
y = R1 – (R1/R2) x.
A
E
R
P
R2
R
45⬚
D
C
B
R1
Fig. 1. A construction for finding the equivalent resistance of
two resistors in parallel. The letters A, B, C, D, and E were not in
the original diagram.
458
At point P both x and y have the value R.
Substituting this value into the equation above
and doing some algebra gives the equation for
resistors in parallel.
Today calculators take reciprocals and store
them, making it easy to make the calculation. As
an undergraduate in the 1950s, I would have welcomed this construction, which I have not seen
before and whose motivation is obscure.
Reference
1. Louis Bevier Spinney, A Textbook of Physics, 3rd ed.
(MacMillan, New York, 1925), pp. 337–338.
THE PHYSICS TEACHER ◆ Vol. 40, November 2002
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