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What does equation I = V/R mean to students?
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2013 Phys. Educ. 48 279
(http://iopscience.iop.org/0031-9120/48/3/F03)
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Frontline
Current
What does equation I = V/R mean to students?
+
power
supply
–
(V)
light
bulb
(R)
voltage (V)
current (I)
slope = resistance (R)
current (I)
Figure 1. Common battery–bulb circuit illustrating
Ohm’s law.
Figure 2. Common battery–bulb graph for
discussing Ohm’s law.
Ohm’s Law is a fundamental topic at high-school
and university levels. It states that for many materials the ratio of current to voltage is a constant,
1/R, where R is the resistance. However, the electrical behaviour of some materials is not linear for
changes in current, which means that Ohm’s law
is not a fundamental law of nature, but rather an
empirical relationship valid only under certain
conditions. Without sufficient discussion of the
topic, students wrongly generalize and misinterpret the relationship.
In textbooks and on classroom blackboards it is
not surprising to read a definition of Ohm’s law
as simply ‘The linear relation between the current I and the voltage V, with an equation I = V/R.’
Alternatively, it may be presented as ‘Ohm’s law
states that in electrical circuits the current passing through two points of an element is directly
proportional to the potential difference across the
two points and inversely proportional to the resistance between these two points.’ Although there
are many college and university textbooks that
correctly identify the limitations of Ohm’s law,
even graduates tend to forget that it does not apply
universally. Furthermore, students too often find
an unknown resistance from the slope of a V versus
I graph, after plotting current and voltage values
with the independent variable on the vertical axis,
thinking of the slope of the graph as R. A familiar
but mistaken illustration for Ohm’s law is the simple battery–bulb electric-circuit diagram (figure 1)
accompanied by a linear graph of voltage versus
current (figure 2).
In a real laboratory setting, students would
encounter a contradiction between poorly taught
theory and actual practice in the form of a non-linear
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Frontline
8
7
tangent
line
6
voltage (V)
5
y 4
3
2
1
0
current (I)
0.0
0.5
1.0
1.5
x
2.0
2.5
3.0
Figure 3. A real graph obtained from battery–bulb
circuit data. Independent variable (V) is
intentionally put on the vertical axis to get R instead
of 1/R from the graph.
Figure 4. Example student work evaluating dy/dx
and y/x at a point. The slope of the tangent line at
the point is ∆y/∆x = (6.2 – 0.4) / (2.9 –1.0) = 3.05.
The ratio of y/x at the point is y/x = 2.0/1.5 = 1.33.
graph and so non-constant resistance. If the teacher
is not careful, students may try to fit their data to
a linear graph, explaining their unexpected results
as being due to experimental error.
In reality, however, the resistance of some circuit elements, such as lightbulbs, semiconductors
and diodes, changes with potential difference. For
example, the ratio V/I for a bulb increases with
the voltage across it, as its temperature rises from
room temperature to its operating temperature.
For a simple battery–bulb circuit students in the
laboratory should obtain a V versus I graph similar
to figure 3, but many students do not understand
the non-ohmic behaviour (i.e. current-dependent
resistance) of bulbs.
I wrote this paper because physics-education
journals contain little about teaching and learning errors with Ohm’s law (Johnstone and Mughol 1978, Kortemeyer 2010, Liegeois and Mullet
2002, Metioui et al 1996, O’Sullivan 1980, Walter
and Lehman 1993). It is worth remembering that
Ohm’s law applies only when temperature is kept
constant, perhaps demonstrating it with a suitable
ohmic material such as a nichrome wire. The nonohmic behaviour of a lightbulb arises when its
temperature changes significantly due to the heating effect of the current through it. Likewise, the
definition of resistance (R) is the simple ratio of V/I
and not always the slope of the V versus I graph at
a particular value of I.
Too many students calculate the resistance at
a point from the slope of the tangent line to the
curve at the point of interest. They falsely make an
analogy between finding instantaneous resistance
from a V versus I graph and finding instantaneous
velocity from a displacement versus time graph
or instantaneous acceleration from a velocity versus time graph. When discussing the definition of
resistance in relation to Ohm’s law, ask students
whether resistance can be defined as dV /dI or is
only V/I. Many students must be persuaded that
the slope of the tangent line at a point is different
from the y/x ratio at that point. In order to convince
students that they are different, I suggest drawing
the V versus I graph for a lightbulb and then asking the students to evaluate both dy/dx and y/x for
a certain point on the curve. A student’s work in
figure 4 clearly shows that the values are different.
This exercise can help students to identify and correct mistaken thinking.
280
P h ysic s E ducat ion
References
Johnstone A H and Mughol A R 1978 The
concept of electrical resistance Phys. Educ.
13 46
Kortemeyer G 2010 Experimenting with constant
current and voltage sources Phys. Teach.
48 68–9
Liegeois L and Mullet E 2002 High school
students’ understanding of resistance in
simple series electrical circuits Int. J. Sci.
Educ. 24 551
May 2013
Frontline
Metioui A, Brassard C, Levasseur J and Lavoie M
1996 Persistence of students’ unfounded
beliefs about electric circuits: the case of
Ohm’s law Int. J. Sci. Educ. 18 193
O’Sullivan C T 1980 Ohm’s law and the definition
of resistance Phys. Educ. 15 237
May 2013
Waltner S A and Lehman T A 1993 When is
Ohm’s law valid? Phys. Teach. 31 102–3
Derya Kaltakci Department of Secondary Science
and Mathematics Education, Kocaeli University,
Turkey (e-mail: kaderya@kocaeli.edu.tr)
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