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Conceptual difficulties with rates
of change, or Zeno redux
Ronald Newburgh
Extension School, Harvard University, Cambridge, MA 02138, USA
and Noble and Greenough School, Dedham, MA 02026, USA
E-mail: rgnew@bellatlantic.net
Abstract
A simple circuit problem treating an inductor, resistor and battery in series
has uncovered a basic misconception of first-year students. The
misconception is not about circuits or electrical properties but concerns
rather the meaning and interpretation of the rate of change (instantaneous
slope or derivative) of physical quantities. Resolving the misconception
required much dialogue with the students until I could understand their
thinking. How it was resolved can serve as a model to help students confront
their misconceptions. An amusing footnote is that they were thinking exactly
as did Zeno over 2000 years ago when he formulated his motion paradoxes.
Keywords: H, A; TLA, Elec
Introduction
Conceptual problems are perhaps harder than any
other for a teacher to recognize. It is far easier to
treat problems in applying techniques such as the
solution of a given type of equation. To correct
a physical misconception is much more difficult.
The student does not see the misconception as
existing. He feels that he understands and thus
cannot explain why he is floundering with the
material. The teacher must first perceive that
the student has trouble and then guess where the
misunderstanding lies. Often he is not certain
that he has succeeded. As a result he gives a
series of prescriptions for treating the problem,
thereby giving the student a set of rules but no
real clarification.
Recently an assignment contained a problem
that many students solved incorrectly. I had not
thought that it was a difficult problem, but I was
clearly mistaken. The problem treated an inductor,
resistor and battery in a series circuit. The
misconception was not about circuits or electrical
0031-9120/02/020147+05$30.00
quantities but rather about the meaning and use
of the time rate of change of physical quantities.
I was surprised because these students had gone
through about a year of physics and nearly all had
taken calculus. Forcing them to confront their
fundamental misconceptions led to real learning
and understanding.
Statement of the problem
Consider a battery with an emf E , an inductor L
and a resistor R connected in series with an open
switch. The switch is then closed. The students
were asked when is the voltage across the inductor
a maximum and when is it a maximum across
the resistor. They were also asked to justify their
answers. Note that these are qualitative questions.
No calculations are necessary. As an aside I have
found that such questions often reveal more about
the students’ comprehension than do quantitative
ones. The quantitative question, unless carefully
crafted, all too often puts a premium on the
© 2002 IOP Publishing Ltd
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manipulation of equations and does not probe
physical understanding.
Misconceptions
The answers to the problem are that the voltage
across the inductor is a maximum when the switch
is closed because the time rate of change of current,
dI /dt, is a maximum, while the initial value of
the current is zero. The current itself climbs
exponentially with time, reaching a maximum
(and constant value) at steady state. At this time
the voltage across the resistor is a maximum and
that across the inductor is zero since dI /dt is zero.
More than half the student responses were
wrong. Only half of those that were right had
correct justifications. The most common answer
was that the current could not be zero when the
switch was closed; otherwise there would never
be current in the circuit. Others argued that the
voltage across the inductor was a maximum only
when steady state was reached. They also said that
the maximum value of voltage across the resistor
occurred when the switch was thrown.
How does one explain these wrong answers?
The explanation required long discussions with the
students so that I could follow their reasoning. In
a sense I asked them to justify their justifications.
In doing this together we were able to identify
their misconceptions. As the title indicates, the
majority involved the meaning of rate of change
as slope or derivative. To say that the current
cannot be zero when the switch is closed because
that would imply zero current at all times shows a
genuine confusion between the current, I , and its
rate of change, dI /dt.
I then recalled a similar confusion from
mechanics. Consider a ball held in a hand. At
zero time one opens one’s fingers and lets the
ball drop. There are always those who argue that
the acceleration must be zero at the moment of
release, since the velocity is zero at that moment.
Others argue that the velocity cannot be zero then;
otherwise the ball would never fall.
To convince them of their errors one must
make them confront the implications of their
arguments.
When asked to consider the
implications of zero acceleration, they agree (after
some discussion) that zero acceleration implies
constant velocity.
Therefore if the original
velocity is zero, so it will remain and the ball will
levitate. Faced with an experimental impossibility
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Figure 1. Normalized current as a function of time for
the LR circuit. Time is plotted in units of time
constant.
they recognize that the acceleration must be nonzero. Hence the velocity cannot remain zero
and must change instantaneously. Note that
this argument avoids any discussion of forces or
Newton’s second law.
Resolution
To resolve the misconceptions let us look at the
behaviour of the current in the LR circuit as a
function of time. As is well known from the theory
of LR circuits, the current is
I = Imax (1 − e−t/τ ).
(1)
The maximum current Imax is
Imax = E /R
(1a)
and the time constant τ is
τ = L/R.
(1b)
Figure 1 is a graph of the current as a function
of time. Clearly, as the graph shows, the current
is zero at the instant when the switch is closed
and then increases exponentially with time. The
behaviour of dI /dt is just the opposite,
dI /dt = (Imax /τ )e−t/τ
(2)
as shown in figure 2.
To convince the students of their error it
was necessary to go over equations (1) and
(2) thoroughly as well as figures 1 and 2.
March 2002
Conceptual difficulties with rates of change, or Zeno redux
Figure 2. Normalized rate of change of current as a
function of time for the LR circuit. Time is plotted in
units of time constant.
Beginning students have difficulty relating graphs
to equations. One must stress the fact that a graph
and its equation are simply two representations
of the same physical quantity, e.g. the behaviour
of current as a function of time. They also are
unaccustomed to seeing a slope plotted explicitly
as a function of time. In kinematics we plot
displacement and velocity as functions of time
without stressing that velocity is the slope of the
displacement curve.
We know that the voltage across the inductor
VL is proportional to dI /dt. Since the latter
decreases with time, so does the voltage. The
voltage across the resistor VR is proportional to
current. Since that increases with time, so does
the voltage. We plot the absolute values of VL and
VR as functions of time in figure 3.
Dialogue with the students shows that they are
most surprised that the rate of change of current is
a maximum when the current itself is null. This
comes from their imperfect understanding of slope
or derivative, a problem that arises from the fact
that derivatives are defined in terms of infinitesimal
rather than finite quantities. Only by forcing them
to look at figures 1, 2 and 3 do they begin to
abandon their misconception. It is, of course,
related directly to the example of a dropping ball
described above.
A second source of error came from our
having treated the RC circuit just before the LR
circuit. This led to their making a false analogy
between the two. Consider a resistor R, capacitor
March 2002
Figure 3. Normalized voltage across the inductor, VL ,
and that across the resistor, VR , as a function of time
for the LR circuit. Time is plotted in units of time
constant.
C and battery with emf E in series with an open
switch. The capacitor is originally uncharged. At
the instant the switch is closed the charge on the
capacitor is zero. Therefore there is no blocking
voltage from the capacitor, and the current (the
rate of change of charge) is a maximum. As the
current flows, the capacitor becomes charged and
its voltage ultimately equals the battery emf E .
The theory of the RC circuit gives the charge
as a function of time as
Q = Qmax (1 − e−t/τ )
(3)
Qmax = C E
(3a)
τ = RC.
(3b)
with
and
The current I or dQ/dt may be written as
I = Imax e−t/τ
(4)
Imax = Qmax /τ = C E /τ.
(5)
with
Figure 4 is a graph of the charge as a function of
time and figure 5 of the current as a function of
time. Finally, in figure 6 we plot VC and VR as
functions of time.
Now we ask the students to examine closely
equations (3) and (4) and figures 4, 5 and 6.
The charge is zero at zero time, but its slope,
the current, has its maximum value then. This
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R Newburgh
Figure 4. Normalized charge on the capacitor as a
function of time for the RC circuit. Time is plotted in
units of time constant.
Figure 5. Normalized current in the circuit as a
function of time for the RC circuit. Time is plotted in
units of time constant.
behaviour is just the opposite of that of the current
in the LR circuit. The analogy between the two
circuits is false.
To summarize, the steady state in the LR
circuit is achieved when the current is a constant,
and the circuit doesn’t ‘see’ the inductor (zero
dI /dt). In the RC circuit steady state means
constant charge on the capacitor or zero current
(zero dQ/dt), and the circuit doesn’t ‘see’ the
resistor. Physically we say that the blocking
voltage from the inductor has its maximum value
at zero time, whereas that of the capacitor has a
null value at that time.
Discussion
In listening to the students I finally realized that a
recurring point was their difficulty in reconciling
zero current with maximum change of current.
They were baffled. From this I saw that they
had a very imperfect understanding of the physical
meaning of instantaneous slope or derivative. This
misconception is not trivial and is well over 2000
years old. It is a new form of one of Zeno’s
paradoxes, described by Russell [1], Kline [2]
and Kaufmann [3] (see the appendix). There
are a number that concern continuous quantities,
motion and infinitesimal change. Consider an
arrow in flight. At any given instant it occupies
a definite position. At the next instant according
to Zeno it is in a new and different position.
Therefore we ask, when did it go from the first
to the second position? Our second problem is
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PHYSICS EDUCATION
Figure 6. Normalized voltage across the capacitor, VC ,
and that across the resistor, VR , as a function of time
for the RC circuit. Time is plotted in units of time
constant.
that an instant has no duration. Therefore at each
instant Zeno concludes that the arrow is at rest—
paradox! But this is precisely the argument of the
student who claims that if the current is zero at the
first instant and that instant has no duration, the
current cannot change—it is at rest.
Kline [2] resolves the paradox using Cantor’s
theory of infinite classes. Cantor’s theory shows
that the number of points on a line segment equals a
transfinite number C, a number independent of the
segment length. Similarly the number of instants
in a time interval also equals a transfinite number
C, independent of the duration of the interval.
March 2002
Conceptual difficulties with rates of change, or Zeno redux
Hence, as Kline says, “the theory allows for an
infinite number of ‘rests’ in any interval of time”.
In conclusion, the simple problem of the LR
circuit offers an opportunity to demonstrate the
connection between physical and mathematical
concepts. It has exposed a deep rooted misconception of many students that prevents real
physical understanding of the rate of change.
Finally, it offers a model to help in confronting
misconception.
One may look on it as a duration of time or as an
instant of time. It is just this confusion that causes
difficulty for the student confronted with the idea
of instantaneous rate of change.
As a footnote I might add that these
paradoxes of Zeno illustrate beautifully the
extreme discomfort of the ancient Greeks when
they were confronted with the concept and
implications of infinity.
Acknowledgments
Appendix. Zeno’s paradoxes
Zeno was born about 485 BC in Elea in Magna
Graecia. He was both a student and disciple of
Parmenides. The central idea of Parmenides and
the Eleatic School was that reality is monolithic.
There is no plurality. Zeno proposed a number
of arguments or paradoxes that have come down
to us. They include arguments against plurality,
arguments against motion and arguments against
space.
Mathematicians and physicists have been
especially intrigued by the arguments against
motion and have written much on them. Perhaps
the paradox best known to the layman is that of
Achilles and the tortoise, which argues that the
swifter cannot overtake the slower in a race. As
with the Arrow paradox, resolution comes with
Cantor’s theory of infinite classes.
The paradox closest to the problem of
instantaneous rate of change is that of the Arrow.
He argues that when the arrow is in a definite
location (described as ‘the place equal to itself’)
it must be stationary.
Therefore motion is
impossible.
The argument is based on the
confusion between the two meanings of ‘when’.
March 2002
I wish to thank Dr Kerry Parker, Editor of this
journal, for her critical reading of this paper. Her
suggestions have made the final version clearer
than was the original.
Received 21 June 2001
PII: S0031-9120(02)25988-2
References
[1] Russell B 1945 A History of Western Philosophy
(New York: Simon and Schuster) pp 804–6
[2] Kline M 1953 Mathematics in Western Culture
(New York: Oxford University Press)
pp 395–409
[3] Kaufmann W 1961 Philosophic Classics—Thales
to St Thomas (Englewood Cliffs, NJ:
Prentice-Hall) pp 27–45. Gregory Vlastos wrote
the section on Zeno.
Ronald Newburgh graduated from
Harvard College and received his
doctorate from MIT. He worked for
many years in defence-related physics
research at the Air Force laboratories at
Bedford, MA. He has taught physics
since retiring in 1987.
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