# The process of constructing new knowledge: an undergraduate laboratory
exercise facilitated by a vacuum capacitor-resistor circuit

Frank V. Kowalski  Physics Department, Colorado School of Mines

###### Abstract

The process of constructing knowledge is typically taught to students by
having them reproduce established results (e.g., homework problems). An
alternative pedagogical strategy is to illustrate this process using an open
problem, such as voltage decay in an RC circuit as described below. Analyzing
data from this circuit in an undergraduate physics laboratory course reveals a
discrepancy between the data and the exponential decay model found in
textbooks. As students attempt to reconcile this discrepancy, the instructor
can provide guidance in the process of validating data, modeling, and
experimental design. This undergraduate laboratory exercise also provides an
engaging transition from classroom learning to real world experience.

###### pacs:

Valid PACS appear here

##  I  Introduction

The process of constructing new knowledge in physics may best be demonstrated
with an open problem, rather than with a problem whose solution is found in a
textbook, on the web, or by using artificial intelligence software. This is
particularly difficult in the undergraduate laboratory setting, where student
understanding and budget constraints limit the sophistication of the problem
and the apparatus. One strategy that can be used is to revise a typical first
year laboratory experiment by improving its accuracy, since the validity of
every scientific model is in this manner challenged.

Unlike the skills students acquire in a first year course applying Newton’s
laws to solve a problem, there exists no recipe for creating models that
support the data in an open problem. It is not within the scope of this paper
to attempt to describe such a generic method to attack an open problem in
physics; rather, a process is outlined for refining understanding of one
example, an RC circuit. The purpose here is to provide a simple illustration
of scaffolding student knowledge construction in a undergraduate laboratory
setting. This is not a new pedagogical approach. It is found in senior
projects and graduate school. Although applying it earlier in the curriculum
is unusual, this may provide students with beneficial exposure into how
knowledge is constructed by physicists.

Different aspects of the results presented below were effectively used to
facilitate learning in a junior level advanced laboratory course (over 200
students during a 5 year span) and senior design projects. The emphasis in the
former course was on scientific writing, experimental techniques, and critical
thinking about data used to support models while that in the latter was on the
processes of modeling and experimental design.

##  II  Methods

The RC circuit shown in fig. 1 consists of a parallel resistor and a vacuum
capacitor with $C=2.789$ nF (Comet model CFMN-2800BAC/8-DE-G). The $1\%$ metal
film resistors are $1/4$ W. A $98$ V power supply energizing the circuit is
disconnected using a mechanical switch. The needed improvement in accuracy
(over a typical 8-bit laboratory oscilloscope) to illustrate deviations from
single exponential decay is accomplished with a Keysight 34465A digital
voltmeter (used to collect all the data shown here).

The current decay in this circuit is shown as the solid lines from top to
bottom in fig. 1 corresponding to circuit resistances
$R=47~{}\mathrm{k}\Omega$, $R=182~{}\mathrm{k}\Omega$,
$R=337~{}\mathrm{k}\Omega$, and $R=637~{}\mathrm{k}\Omega$. In clear contrast
to the predictions of Kirchhoff’s laws these data deviate from the expected
single exponential decay that is shown as the dashed line.

Figure 1: Logarithm of the current vs time in time constant units for the
circuit in the inset. A vacuum capacitor is used while the the lines from top
to bottom correspond to data for increasing resistances. The expected RC decay
for the lowest trace is shown as the dashed line. It is offset for clarity.

##  III  Constructing knowledge

Creating knowledge about this circuit can be classified into two parts. The
first involves determining that the results are repeatable and known with
certainty to reflect the behavior of the RC circuit; that is, establishing
confidence that the results are valid. The second involves creating a model
that the data support within error.

###  III.1  Establishing experimental validity

One method to determine data validity involves questioning the function and
influence of each system component. For example, the switch not only
disconnects the conducting path but thereafter it has an intrinsic
capacitance. The wire loop of the circuit generates a self-inductance and
therefore an electromotive force during the decay. The capacitor response is
often modeled as having capacitance, series resistance, and inductance, the
effects of which need to be considered (self resonance typically occurs at
frequencies above $100$ MHz murata ). The resistance varies with temperature,
which is a function of the electrical power dissipated in the resistor (which
is a function of time during the decay). The capacitor constricts as a
function of voltage, thereby changing its value. The analog-to-digital
converter behaves non-linearly at some level of accuracy. Aware of the
potential influence of each of these system components, the student can at
this point design experiments or perform calculations to determine the impact
of these effects (none of which invalidate the data presented).

However, a major issue in acquiring data that is precise and valid is
associated with the small capacitance values that are found in vacuum or air
capacitors. Since the resolution of digital voltmeters and oscilloscopes
decreases with increasing bandwidth the student may at first attempt to employ
large resistance values in the RC circuit to increase the time constant and
therefore decrease the bandwidth associated with the decay. It soon becomes
apparent that this increases the background noise. One solution for fixed
background noise is increasing the energizing voltage. However, the student
must then consider safety issues and design the circuit with resistors that
have the appropriate wattage.

High bandwidth oscilloscopes with $16$ bit resolution at $10$ MS/s are
available (e.g. Picoscope model $4262$). However, even for RC circuits with
ten microsecond time constants the frequency response of the oscilloscope
probe can introduce oscillations in the decay data or, due to its input
impedance, load the circuit. Such issues of data validation can be addressed
by collecting data from two probes that are placed at the same point in the
circuit. Different data in this case is often the consequences of improperly
compensated probes while loading of the circuit by the probe impedance changes
the magnitude of the measured responses between using one and both probes.

This RC laboratory exercise exposes students to generic experimental
techniques that are often needed to collect reliable and repeatable data.
Examples of the latter are the advanced triggering of an oscilloscope that is
needed to mitigate the effect of switch bouncing. bounce  The use of a
shielded metal box to mitigate noise is another important method. Taking
multiple samples and averaging over these is yet another method. Setting the
data acquisition rate much higher than the time constant and using a moving
average on the logarithm of the data effectively introduces a low pass filter
that also increases precision. Using larger capacitance polypropylene
capacitors to collect decay data with much larger time constants illustrates
the advantages of power line period averaging available in many digital
voltmeters. These methods were applied in collecting the data presented here.

###  III.2  Constructing models

Aware of the discrepancy yet confident of the validity of their data, the
student can now be guided by the instructor to construct new models to explain
their observations.

####  III.2.1 Using analogical models

The student can be exposed to the use of analogical models in an attempt to
provide insight into this circuit behavior. In this regard and in an effort to
help the student make connections, the instructor may first challenge the
student to suggest physical processes that exhibit exponential decay and then
to consider how such decays may possibly be modified to exhibit the behavior
shown in fig. 1. peshkin

####  III.2.2 Fundamental principles

The student should now clearly recognize that the physics curriculum is
focused on understanding fundamental principles rather than on manipulating a
litany of equations. When a novel phenomenon is encountered it is considered
from the perspective of these principles; this emphasis on fundamental
principles should stimulate questions associated with validity of these laws.

The RC decay is most often modeled using Kirchhoff’s and Ohm’s laws. This
emphasis on fundamental principles should arouse curiosity about the validity
of these laws. Although Ohm’s law is not fundamental it is expected to be
appropriate in the analysis of the circuits described here, while Kirchhoff’s
laws are strictly valid for steady state behavior.

In recognizing that the steady state regime may not be appropriate, the
student might ask if the data reproduces Kirchhoff’s result for much larger
time constants. However, that does not appear to be the case for the data
shown in fig. 1.

Kirchhoff’s laws are superseded by Maxwell’s equations. Only a few circuits,
involving steady state behavior, have been treated in this manner
analytically. sommerfeld ; heald ; chabay ; klee ; muller ; moreau  Maxwell’s
equations have been solved numerically for an RC circuit. preyer  However, the
data presented here do not support the results of that calculation.

The intent of this exercise is not for the student to develop a novel solution
to RC decay using Maxwell’s equations, but rather to use this open ended
problem as an example of how a physicist begins to make sense of such data.
What often distinguishes an expert from a novice is knowing what to question
in light of confounding data. The experimental results described above allow
the instructor to awaken and further encourage such awareness.

####  III.2.3 Constructing models that eliminate all extraneous variables

The essence of a problem can often be revealed by a model that eliminates all
extraneous variables (as illustrated by the spherical cow metaphor wiki ). The
decay of a vacuum or air capacitor provides a system where this approach can
be fruitful.

Students can be asked to construct the simplest possible system in RC decay.
As an example, they may consider replacing the vacuum capacitor with a single
spherical conductor (having only self capacitance) attached to ground (or to
infinity) via a resistor through which the charge flows in the decay. This
model eliminates the effect of the second plate on the decay and appears to
contain the bare minimum number of components needed to generate RC decay.

Having established this thought experiment the student may focus on how to
implement it: using the metal sphere available from a Van de Graaf
demonstration apparatus, or soldering the terminals of the vaccum capacitor
together. A careful design is first required to determine feasibility,
particularly since the capacitance of the measuring apparatus typically
overwhelms the small self capacitance of the sphere or the capacitor with its
terminals soldered together.

Figure 2: The decay currents through $R_{1}=337~{}\mathrm{k}\Omega$ (upper
trace) and $R_{2}=337~{}\mathrm{k}\Omega$ (lower trace) as a function of time.
The vacuum capacitor has $C=2.789$ nF.

To more accurately model the vacuum capacitor the student may choose to add
another spherical conductor, connected to ground through a different resistor,
and brought near to the first spherical conductor. The system then has both
self and mutual capacitance.

Such a revised thought experiment generates questions about the effect of the
charges on one sphere attracting those on the other and thereby modifying the
decay process, potentially causing the non-exponential decay in the vacuum
capacitor. The charges on one plate repel each other, leading to the voltage
decay, while the opposite charges on the nearby plate counteract this decay.

The addition of this second spherical conductor may lead the student to the
idea of connecting each plate of the vacuum capacitor to ground via different
resistors. One advantage of including another resistor in the circuit is that
the voltage across each resistor reveals the currents flowing into or out of
each capacitor plate. The student might then revise the experiment as shown in
the circuit schematic of fig. 2.

####  III.2.4 Testing and refining the new model

A cursory examination of this data in the upper frame appears to support equal
currents flowing into and out of the capacitor as expected from Kirchhoff’s
law. The student may be tempted to go no further in their pursuit of
experimental confirmation of the Kirchhoff model. However, when prodded to use
a logarithmic scale, discrepancies are apparent, as shown in the lower frame
of fig. 2. This illustrates the importance of attention to detail and how an
improvement in precision challenges a scientific model.

The current from the right capacitor plate almost matches that from a single
exponential decay while that from the left plate has a distinct tail. This is
evidence for a time varying net charge on the capacitor. However, the initial
charge is not measured with this circuit.

The student may choose a different modification of the single charged sphere
model: placing a neutral isolated spherical conductor near to the charged
spherical conductor. In this case the exponential decay of the latter is
modified by the charges induced on the isolated sphere (rather than those
drawn to the second sphere through its resistor connected to negative side of
the battery). Perhaps the charge on the first sphere does not decay to zero
due to the mutual attraction between it and the induced charges on this
isolated sphere.

To test this the student might use two capacitors in series as shown in upper
frame of fig. 3. The right plates are similar to the isolated sphere. By
measuring the voltage across $R_{2}$ the current between the right plates is
determined (such current data are not shown).

Initial net negative charge on the right plate of $C_{1}$ and no net charge on
the right plate of $C_{2}$, in the upper frame of fig. 3, is generated when
the two switches are closed. The decay is measured when these two switches are
opened simultaneously (within a millisecond using a double pole single throw
mechanical switch for a $338$ millisecond time constant). After the two
switches open some of the charge on the right plate of $C_{1}$ relaxes to
right plate of $C_{2}$. The voltage between the plates of $C_{2}$ decreases
below its equilibrium value $V_{f}$, as shown in 3 (b) by the dashed line,
before returning to it. On a logarithmic scale of the absolute value of
$V_{M}-V_{f}$ vs time this minimum results in a dip as shown in fig. 4.

The more familiar initial condition for the series capacitor configuration
results from removing the switch across $C_{2}$. For comparison the decay for
this initial condition is shown in fig. 4.

Figure 3: Voltage across $C_{2}$ for resistor values of
$R_{1}=1~{}\mathrm{k}\Omega$ and $R_{2}=337~{}\mathrm{k}\Omega$ with capacitor
values $C_{1}=C_{2}=C=2\mu$F (film metalized polypropylene axial capacitors)
for the circuit shown in the inset of the upper frame. The two switches are
initially closed to charge the circuit then open simultaneously. The dashed
line indicates the final voltage $V_{f}$. The circuit and data are from
reference $14$. Figure 4: Plots of $\log_{e}|V_{M}-V_{f}|$ for the circuit in
the upper frame of fig. 3. Four decays are are shown when the switch across
$C_{2}$ is operational (the four upper traces that appear as one solid line)
and when it is always open (the lower four traces).

The above data provide direction for constructing models. For fig. 1 the most
obvious is $I_{M}=I_{10}\exp[-\alpha t]+I_{20}\exp[-\beta t]$ where
$\alpha=1/RC$ and $\beta$ is the second decay constant. These parameters for
the data of fig. 1 are shown in fig. 5. For fig. 3 this becomes
$V_{M}=V^{in}_{f}(1-\exp[-\alpha t])+V^{out}_{f}(1-\exp[-\beta t])$ where
$V_{f}=V^{in}_{f}+V^{out}_{f}$. This is referred to as the sum of exponentials
model (fits typically overlap the data presented here within the width of the
lines in the figures).kowalski

Plots of current vs time in time constant units allow for a range of data to
be presented on one graph. However, the model parameters must be determined
from voltage vs time data.

####  III.2.5 Validity of model parameters: dynamic capacitance

One aspect of modeling is determining the validity of the parameters used.
Consider capacitance, for example. The assertion that the ratio of the charge
to the potential difference is a constant can be tested with the circuit of
fig. 1 by integrating the current leaving one capacitor plate to determine its
remaining charge at a given time. Dividing this by the voltage across the
capacitor at that time does not yield a constant value throughout the decay.
Therefore, capacitance may not be a useful circuit parameter.

However, in modeling non-linear systems a connection is often made between a
static parameter and its dynamic version. Examples are found in both static
and dynamic resistance (applied to a p-n junction, that was perhaps discussed
in an earlier course on electronics) and dynamic inductance (ferromagnetism is
often discussed in a first year course). Even the relationship between phase
velocity $\omega/k$ and group velocity $d\omega/dk$ (discussed in an earlier
course on modern physics) is a similar quantity and its derivative.

Suggesting to the student that the dynamic capacitance $C_{dyn}=dQ/dV$ may be
a useful parameter leads to the ordinary differential equation
$C_{dyn}dV/dt=-V/R$. where $V$ is the voltage across the capacitor and $dQ$
the change in charge on the left plate of the capacitor in fig. 1. Kirchhoff’s
law for this RC circuit is obtained from this relationship when
$C_{dyn}\rightarrow C$ (for exponential decay $C_{dyn}=C$). Using this method
a first order ODE has a solution that is a sum of exponentials. For example,
$C_{dyn}$ can be determined using the expression for the voltage from the sum
of exponentials model described above. The value of $C_{dyn}$ is then the
capacitance measured by a capacitance meter for $t\ll RC$ that transforms into
the much larger value associated with the $\beta$ relaxation time for $t\gg
RC$.

####  III.2.6 Return to fundamental principles

It is surprising that the capacitor, whose dimensions do not change
appreciably, can have a capacitance that varies as suggested by the dynamic
capacitance model. This conundrum may stimulate the student to focus on
acquiring a fundamental understanding of circuit behavior using Maxwell’s
equations before addressing the data. Such solutions typically involve steady
state behavior, utilizing a surface charge on a wire to confine its current.
muller ; sommerfeld ; heald ; chabay ; moreau  A numerical calculation
involving relaxation in an RC circuit yields an initial non-exponential decay
due to transit time effects but does not match the above data. preyer

Surface charge effects are demonstrated with circuit wire that has a kink. If
the current into the kink is greater than that out, “Then the charge piles up
at the ‘knee,’ and this produces a field aiming away at the kink. The field
opposes the current flowing in (slows it down) and promotes the current
flowing out (speeding it up) until these currents are equal, at which point
there is no further accumulation of charge and equilibrium is established.”
griffiths2

Drawing an analogy between the charge on the kink in a wire and the capacitor
in the circuit of fig. 1 the student may divide the charge into two parts: one
that generates a field only between the capacitor plates, $Q^{in}$, that
exerts no force on the current in the wires and one that confines the current
to flow in the wires, $Q^{out}$. This is referred to as the sum of charges
model.

For example, before the switches are opened in the circuit of fig. 3,
$Q^{out}$ generates zero electric field in the wire segment from the upper
node along the branch to the capacitor while the field is non-zero in the
segment from the upper node along the other branch to the resistor, thereby
directing the current from the battery into $R$ and not to the capacitor. The
response due to $Q^{in}$ and $Q^{out}$ is the sum of their individual
responses from the superposition principle and is manifest as the first (given
by the standard solution for the RC circuit) and second terms in the sum of
exponentials model.

The student may attempt to extend the kink in a wire analogy to the RC circuit
in fig. 1: a larger charge may be needed to redirect a larger current from the
capacitor plate into the resistor. The student might then conjecture that this
charge results in an inverse relationship between $I_{20}$ and the resistance
in the circuit. The student can check this in a plot of $I_{20}$ as a function
of R and compare it to $I_{10}\propto 1/R$ as illustrated in fig. 5.

Figure 5: Parameters of the sum of exponentials model as a function of R
determined from a fit of the data in fig. 1. The data with error bars in frame
(a) corresponds to $I_{20}$ while the solid circles indicate $I_{10}/4200$
(note that this data is proportional to $1/$R). The points with error bars in
frame (b) are the $\beta$ values while the solid circles are the corresponding
values of $\alpha/100$.

Another method to test the sum of charges model is to vary the amount of
$Q^{out}$ compared with $Q^{in}$ on a capacitor plate. To do this the student
might modify the circuit in fig. 2 by adding a different voltage source,
$V_{2}$, across $R_{2}$ using a switch that opens synchronously with the other
switch. As $V_{2}$ approaches $V$ the field inside the capacitor is reduced
(as is $Q^{in}$) while $Q^{out}$ on the plates has not been diminished (it is
still required to direct current through the resistors). Data for such a
modification are also found in the literature.kowalski

The sum of charges model attributes the unusual behavior of $C_{dyn}$ to the
$Q^{out}$ charges on the capacitor (and the circuit wires) that are not
included in the definition of capacitance, $C=Q^{in}/\Delta V$. Since $Q^{in}$
and $Q^{out}$ decay at different rates the capacity to hold charge at a given
voltage is not constant.

####  III.2.7 Modeling with a dimensionless variable

Decay data expressed in terms of the dimensionless variable $\tau=$ t/RC
facilitates the determination of the variables that influence the decay. For
example, if the relaxation is a function of only R and C then a point on the
decay plot at a particular value of $\tau$ will be the same for different
combinations of R and C that yield this value of $\tau$ (apart from variations
due to the initial voltage). The decay plot is therefore the same for various
R and C values. That is the case for the faster but not slower decay shown in
fig. 1. Therefore a model that this data supports must include more variables
than R and C. From the sum of charges model the student might conjecture that
$Q^{out}$ and its distribution on the outside of the circuit wires and
capacitor plates is such a variable.

Fig. 6 illustrates the decay as a function of t/RC while C varies, where
R$=R_{1}+R_{2}$, for fixed $R_{1}=R_{2}=337.8~{}{\textrm{k}}\Omega$. The data
are obtained using four vacuum capacitors, two with C$=0.5177$ nF and two with
C$=2.789$ nF. The upper trace is the current through $R_{1}$ with C$=0.5177$
nF and the trace just below it (that almost overlaps with the upper trace) is
the current through $R_{1}$ for C$=1.0354$ nF (a parallel combination of two
C$=0.5177$ nF capacitors). The next two traces below these are for the current
through $R_{1}$ with C$=2.789$ nF and C$=5.578$ nF (a parallel combination of
two C$=2.789$ nF capacitors). These traces completely overlap. The lowest
trace is the decay data for the current through $R_{2}$ for all four values of
C.

Consider the above discussion about the determination of variables in plots of
the decay as a function of $\tau$ for the upper pair of the traces that are
assumed (for the sake of argument) to overlap. The doubling of C does not
change the plot (it of course does in a plot vs t). This implies that R and C
are the only variables required to model the data (assuming for the moment
that variation in R also yields the same plot).

Similarly, the overlap of the plots of the current decay vs $\tau$ for
C$=2.789$ nF and C$=5.578$ nF implies that R and C are the only variables
required to model the decay from these capacitors (again assuming for the
moment that variation in R also yields the same plot). However, a new variable
must exist to model both pairs of data since the decay differs for the two
pairs of lines. One then looks for a variable that is the same for each pair
of capacitors but differs between them. Their geometry is such a variable.
From the sum of charges model the student might again propose that the
distribution of $Q^{out}$ due to this geometrical difference is the needed
variable and then propose experiments to test this hypothesis.

Figure 6: Current relaxation for the circuit in the inset for fixed resistance
$R_{1}=R_{2}=337.8~{}{\textrm{k}}\Omega$ and different values of the
capacitance. The trace just above the bottom trace consists of two decays that
overlap. The bottom trace corresponds to all currents through $R_{2}$. A $98$
V power supply was used to energize the circuit. The circuit and data are from
reference $14$.

##  IV Summarizing the new knowledge constructed

The knowledge gained from the above exercise is divided into two parts: the
knowledge acquired in obtaining precise and valid data (experimental
techniques that are typically unknown to the student but not novel) and the
consequences of the models that these data support.

###  IV.1 Knowledge the student gains in constructing precise and valid data

  * •

Advanced triggering methods (often available in an oscilloscope) required to
obtain repeatable data;

  * •

The technique of averaging these repeatable data sets to improve precision;

  * •

Filtering techniques in the form of a moving average over the oversampled data
to increase precision;

  * •

Shielding methods to mitigate noise (including a properly grounded metal
enclosure of the circuit);

  * •

The technique of increasing the signal but not the noise (by increasing the
energizing voltage);

  * •

Safety and design considerations in applying a high voltage energizing source
to the circuit;

  * •

Understanding that high impedances increase noise pickup;

  * •

Methods of validating the data such as placing two probes at the same point in
the circuit to determine the effect of their impedance on the circuit or to
determine if the probes have different frequency responses;

  * •

Use of power line period averaging available in digital voltmeters to mitigate
power line noise.

###  IV.2 Knowledge gained in modeling this data

  * •

At least two relaxation mechanisms occur during the decay with disparate time
constants. The first decay constant is that expected from Kirchhoff’s laws
while the second is unrelated to the first;

  * •

Unequal currents flow into and out of the capacitor in RC decay;

  * •

Net charge is generated on the capacitor;

  * •

The sum of exponentials model is supported by the data;

  * •

The series capacitor circuit provides a method to explore how charge flows
between isolated conductors during the decay;

  * •

Valid data for RC decay, obtained for dielectric capacitors in series,
indicate a change in current direction during the decay;

  * •

The use of dynamic capacitance in modeling this data is of limited utility
since it does not present a microscopic understanding of the process and
therefore has little predictive power;

  * •

The sum of charges model supports the sum of exponentials model since the
initial decay time constant, RC, is the same for all data sets while the
parameters of the the second term in the sum of exponentials model need to be
adjusted to match the particular decay. For example, $Q^{out}$ for the left
plate in fig. 2 differs from that on the right plate in order for the sum of
exponentials model to match the data from each plate, yet the initial decay
term for both plates involves only the time constant $(R_{1}+R_{2})C$.

  * •

Before the switch is opened $Q^{out}$ directs current into $R$ while the
electric field from $Q^{in}$ has no such influence since its field is confined
to a region between the plates (without $Q^{out}$ and before the switch is
opened the student might ask what force causes current to flow through the
resistor if the field from $Q^{in}$ does not exist in the wires to move
charges);

  * •

The description of transient behavior using a kink in a wire is related to the
behavior of the capacitor and consistent with the sum of charges model;

  * •

Exponential decay of charge is the result of $dQ/dt\propto-Q$. However, the
decay constant that makes this relationship an equality is determined by the
details of the decay mechanism which requires a solution of Maxwell’s
equations for the circuit;

  * •

The decay of the circuit in fig. 2 with a different voltage source across each
resistor (that initially energizes it) allows for a variation in $Q^{in}$
while $Q^{out}$ is essentially fixed, thereby facilitating tests of the sum of
charges model;

  * •

Plots in terms of dimensionless variables demonstrate that the geometry of the
capacitor is an important variable in the decay process;

  * •

While progress has been made a complete microscopic understanding of the decay
process, based on Maxwell’s equations, is lacking.

As with any model questions arise that probe its consequences. The student may
wonder about the effects of $Q^{out}$ in an inductor-resistor circuit, or
about the consequences of the sum of charges model when the loss mechanism
involves transfer of the electrical energy stored in the capacitor to
mechanical energy or to other circuit components. Perhaps a superconducting LC
circuit, whose loss mechanism is radiative rather than thermal, behaves in the
same manner. On a theoretical level, they may wonder why the simple sum of
exponentials model matches the data even though it is quite difficult to
obtain such a solution from Maxwell’s equation.

Through this exercise, the student not only has experienced the process of
constructing new knowledge, but also experienced the questions and
uncertainties it often exposes.

##  V  Conclusions

Students in an upper-level undergraduate physics laboratory class can be
effectively exposed to the process of constructing new knowledge in physics.
This involves the acquisition of valid data (whose importance is emphasized in
this procedure) along with the construction of models supported by the data
within error. The simple RC circuit, for which accurate measurements can be
made inexpensively, is surprisingly difficult to model on a fundamental level.
Yet the physics involves the familiar attraction and repulsion between
charges. The students should be cognizant of the learning objective: the point
is to learn how to collect valid data, practice modeling, and think critically
about the data supporting the models, not to find a unique model that the data
support. Hopefully, joy can be found in this process rather than only in the
result, for which complete success is often elusive.

This laboratory experience also provides an opportunity for students to become
aware of the differences between the simplified models ubiquitous in their
formal education and the more complex realities of the physical world. Not
only do students gain fresh insight into how new knowledge is constructed, but
they move toward a more mature and sophisticated view of physics.

###### Acknowledgements.

I wish to thank Susan Kowalski for editing comments from a non-physicist’s
perspective, as well as students for their participation and discussions in a
senior design course, particularly Justin L. Swantek, Tony D’Esposito, and
Jacob Brannum. The support from a HP Technology for Teaching Grant is
acknowledged.

##  VI  Appendix

The instruction described above is a fruitful but time intensive endeavor.
Alternately, a shorter exercise described next saves considerable course time
while still cultivating some skills essential for constructing new knowledge.
This has been successfully implemented as a pre-post assessment in a junior
level modern physics laboratory course.

Students are told that the purpose of the pre-test is to determine their
skills in performing experiments and communicating their results. Rather than
being given written instruction, they are simply told to use the equipment at
hand to collect RC decay data and write a short scientific report on their
work.

The data that they collect using a microfarad ceramic or electrolytic
capacitor and a $1~{}\mathrm{k}\Omega<R<100~{}\mathrm{k}\Omega$ resistor with
an 8-bit oscilloscope appears to match Kirchhoff’s law during the initial
decay. However, its tail deviates significantly from exponential decay even
when analyzed with an 8-bit resolution oscilloscope. In the pre-test many of
the students conclude that this data indicates exponential decay, verifying
what was taught in a previous class (many plot the data on a linear scale). A
remarkable number of reports do not reflect an awareness that modeling is a
fundamental part of a scientific method; that is, the reports lack an argument
that the data support a model within error.

As a summative assessment later in the semester, the student repeats the
experiment (or is given a file of decay data with unique RC values for each
student to mitigate cheating). From this data the student writes a laboratory
report. Learning in the course is illustrated as the student compares their
pre-test and post-test documents. Although this alternate exercise saves
considerable course time, it has the pedagogical shortcoming of not allowing
the students practice in modeling and experimental design. Nevertheless, it
has the advantage of challenging the student to think critically about the
validity of the data, particularly in the case where the student expects the
data to support the model within error. This is an important first step in
constructing new knowledge.

## References

  * (1)
  * (2) F. V. Kowalski, “Energy relaxation in a vacuum capacitor-resistor circuit: measurement of multiple decays with divergent time constants,” AIP Advances 14, 035154 (2024) https://doi.org/10.1063/5.0190681
  * (3) ”Lumped Circuit Model”, https://www.electronicdesign.com/technologies/analog/article/21155418/logiswitch-11-myths-about-switch-bouncedebounce (Accessed: 2023-3-4).
  * (4) ”Lumped Circuit Model”, https://www.electronicdesign.com/technologies/analog/article/21155418/logiswitch-11-myths-about-switch-bouncedebounce (Accessed: 2023-3-4).
  * (5) Cadence, “Capacitor Self-resonant Frequency and Signal Integrity,” (2019), https://resources.pcb.cadence.com/blog/2019-capacitor-self-resonant-frequency-and-signal-integrity (Accessed: 2023-4-3).
  * (6) It is temping to suggest that the quantum theory of decaying states is such an example. That is disputed in the following references: M. Peshkin, A. Volya, & V. Zelevinsky, Non-exponential and oscillatory decays in quantum mechanics. EPL (Europhysics Letters). 107, 40001 (2014), https://doi.org/10.1209/0295-5075/107/40001 and P. Greenland, Seeking non-exponential decay, Nature, 335, 298 (1988).
  * (7) A. Sommerfeld, Electrodynamics. pp. 125-130 (1952).
  * (8) M. Heald, Electric fields and charges in elementary circuits. American Journal Of Physics. 52, 522-526 (1984), https://doi.org/10.1119/1.13611
  * (9) R. Chabay & B. Sherwood, Polarization in electrostatics and circuits: Computing and visualizing surface charge distributions. American Journal Of Physics. 87, 341-349 (2019), https://doi.org/10.1119/1.5095939
  * (10) M. Klee, Surface charges from a sensing pixel perspective. American Journal Of Physics. 88, 649-660 (2020), https://doi.org/10.1119/10.0001435
  * (11) R. Muller A semiquantitative treatment of surface charges in DC circuits. American Journal Of Physics. 80, 782-788 (2012), https://doi.org/10.1119/1.4731722
  * (12) W. Moreau, Charge distributions on DC circuits and Kirchhoff’s laws. European Journal Of Physics. 10, 286 (1989,10), https://dx.doi.org/10.1088/0143-0807/10/4/008
  * (13) N. Preyer, Transient behavior of simple RC circuits. American Journal Of Physics. 70, 1187-1193 (2002), https://doi.org/10.1119/1.1508444
  * (14) “Spherical Cow,” https://en.wikipedia.org/wiki/Spherical_cow (Accessed: 2023-4-2).
  * (15) From “Energy relaxation in a vacuum capacitor-resistor circuit: measurement of multiple decays with divergent time constants”, by F. V. Kowalski, AIP Advances 14, 035154 (2024) https://doi.org/10.1063/5.0190681. Licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/) Adapted with permission.
  * (16) D. J. Griffiths, _Introduction to Electrodynamics: second edition_ , Prentice-Hall, pp 277–278 (1989).