Unit 4 - Students Absolutely Must Learn…
Weekly Activity 7: RC Circuits - DC Source




The properties of exponential functions.
How to describe the time dependence of charging and discharging capacitors with
exponential functions.
What differential equations are and how to check their function-solutions.
Advanced features of oscilloscopes.
Weekly Activity 8: RC Circuits - AC Source



The behavior of a sinusoidally driven system including phase shifts.
How to use the solutions to the RC circuit driven by a sinusoidal voltage source and what
they mean.
Advanced features of oscilloscopes.
1
Unit 4 - Grading Guidelines
Staple the lab report, then graphs, and finally worksheets together. Please put
your worksheets in order. Turn in your work to your TA at the beginning of the
next lab meeting following the completion of the unit.
Unit Lab Report [50%, graded out of 25 points]
Write a separate section using the section titles below (be sure to label these sections in your
report). In order to save time, you may add diagrams and equations by hand to your final
printout. However, images, text or equations plagiarized from the internet are not allowed!
Remember to write your report alone as collaborating with a lab partner may make you both
guilty of plagiarism. Pay close attention to your teacher for any changes to these guidelines.

Title [0 points] – A catchy title worth zero points so make it fun.

Concepts & Equations [9 points] – {One small paragraph for each important concept, as
many paragraphs as it takes, 2+ pages.} Go over the lab activities and make a list of all
the different concepts and equations that were covered. Then simply one at a time
write a short paragraph explaining them. You must write using sentences & paragraphs;
bulleted lists are unacceptable.
Some example concepts for this unit report include (but are not limited to):
 Describe the operation and features of the digital oscilloscope.
 Discuss what differential equations are and how to check their solutions. You
should give an example (keep it simple).
 Discuss the construction process of the two differential equations that model RC
circuits with DC source. Discuss the solutions to these differential equations.
 Discuss all you know about capacitors (should be a lot from lecture).
 Discuss how to find the capacitance of a capacitor using an RC circuit with a DC
source.
 Discuss how to determine if data has an exponential relationship.
 Discuss what the half-life is of an exponential relationship is and how it works.
 Compare and contrast how to find the capacitance of a capacitor using a DC
source (square wave) versus a sinusoidal source.
 Discuss at length the three time dependent voltage equations that describe the
AC-driven RC circuit. Be sure to explain:
 impedance
 reactive capacitance
 phase shifts
 Discuss how to find the amplitude of the current through the resistor and what
combination of parameters gives this value.
 Any other equations that were used in the activities will need explained.
 Any other specific TA requests:
2

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____________________________________________________________
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____________________________________________________________
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____________________________________________________________
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____________________________________________________________
Selected In-Class Section [6 points]: {3-5 paragraphs, ~1 page}
This week's selection is: Weekly Activity 7, In-Lab Section 3
Write a "mini-report" for this section of the lab manual. Describe what you did
succinctly, and then what you found accurately. Then explain what the result means
and how it relates to some of the concepts in the previous section. You must write using
sentences & paragraphs; bulleted lists are unacceptable.
o Procedure: Do not provide a lot of specific details, but rather you should
summarize the procedure so that a student who took the course a few years ago
would understand what you did.
o Results: Do not bother to rewrite tables of data, but rather refer to the page
number on which it is found. State any measured values, slopes of ilnes-of-bestfit, etc. Do not interpret your results, save any interpretation for the discussion.
o Discussion: Analyze and interpret the results you observed/measured in terms of
some of the concepts and equations of this unit. It is all right to sound repetitive
with other parts of the report.

Open-Ended / Creative Design [6 points] – {3-5 paragraph, ~1 page} Choose one of the
open-ended experiments from the two weekly activities to write about. Describe your
experimental goal and the question you were trying to answer. Explain the ideas you
came up with and what you tried. If your attempts were successful, explain your results.
If your attempts resulted in failure, explain what went wrong and what you would do
differently in the future. You must write using sentences & paragraphs; bulleted lists are
unacceptable.

Graphs [4 points] - {attach to typed report} Graphs must be neatly hand-drawn during
lab and placed directly after your typed discussion (before your quizzes and selected
worksheets). Your graphs must fill the entire page (requires planning ahead) and must
include: a descriptive title, labeled axes, numeric tic marks on the axes, unit labels on the
axes, and if the graph is linear, the line of best fit written directly onto the graph.
3
Thoroughly Completed Activity Worksheets [30%, graded out of 15 points]

Week 7 In-Class [7 points]: Pages assigned to turn in:
_TA signature page, Post-lab pages, ____________________________________
___________________________________________________________________

Week 8 In-Class [8 points]: Pages assigned to turn in:
_TA signature page, Post-lab pages, ____________________________________
___________________________________________________________________
The above lab report and worksheets account for 80% of your unit grade. The
other 20% comes from your weekly quizzes, each worth 10%. These will be
entered into D2L separately.
4
Weekly Activity 7: RC Circuits - DC Source
Pre-Lab
!
You must complete this pre-lab section before you attend your lab to prepare
for a short quiz. Be sure to complete all pages of the pre-lab.
Continue until you see the stop pre-lab picture:
Subsection 0-A
(If you have not yet covered capacitors in circuits in the lecture, you will need to read the first
two sections on capacitance in circuits in your text at this time.)
You will investigate two similar RC circuits: charging the capacitor and then discharging the
capacitor. First examine charging up the capacitor in an RC circuit. In this circuit (shown below)
the capacitor begins without any charge on it and is wired in series with a resistor and a
constant voltage source. The voltage source begins charging the capacitor until the capacitor is
fully charged. The charging up equation that describes the time dependence of the charge on
  t 
the capacitor is Q Cap (t)  Qmax 1 e RC . The final charge on the capacitor, Qmax is determined


by the internal structure of the capacitor (i.e. its capacitance): Qmax  C  Vsource.


¿
0-A-1
Since the voltage source is constant in time, an example may be written as
Vsource (t )  9 [V] (so it is 9 V for all times). For a capacitor with capacitance of
C  1.0 x10 3 [F] , what will the initial charge Q Cap (0) and final charge Q Cap () on
the capacitor be (in SI units)? {Hint: use Q  C V .}
5
¿

0-A-2
Use a graphing calculator (or mad graphing skills) and make a quick sketch of
Q Cap (t) vs. t on the axes. Assume that the source voltage is 9 [V], the resistance is
R  1.0 x10 3 [] and the capacitance is C  1.0 x10 3 [F] . The amount of time that
equals the resistance times the capacitance is called the time constant:   R C
( is the Greek letter 'tau'). Create your sketch so that Q(t=) is sketched above
the delineated tic mark. Be sure to include charge values along the y-axis.

6
The second circuit is the discharging of the capacitor in an RC circuit without an applied voltage.
In this circuit (shown below) the capacitor begins with some initial charge and is wired in series
with a resistor. The capacitor begins discharging through the resistor until no charge remains
on the capacitor plates. The discharging equation that describes the time dependence of the
charge on the capacitor is Q Cap (t)  Qoe

t
RC
.

It helps to think of an RC circuit with a charged capacitor that has a switch that is about to be closed
so that the capacitor can discharge:
¿
0-A-3
For a capacitor with capacitance of C  1.0 x10 3 [F] and an initial voltage across it
of Vcapacitor  9 [V] what will the initial charge Q Cap (0) on the capacitor be (in SI
units)? {Hint: use Q  C V .}
7
¿

0-A-4
Use a graphing calculator (or mad graphing skills) and sketch a graph of
Q Cap (t) vs. t on the axes below. Assume that the resistance is R  1.0 x10 3 [] and
the capacitance is C  1.0 x10 3 [F] . Find the initial charge on the capacitor by
assuming the capacitor had been charged to 9 volts Vcapacitor  9 [V] by a battery
before being discharged through the resistor. Create your sketch so that Q() is
sketched above the delineated tic mark. Be sure to include charge values along
the y-axis.
¿
0-A-5
What is the decimal value of e-1 to 3 decimal places? _________ Engineers
usually approximate this number as 1/3 (.333) in order to think quickly about
exponential decay. For example, if you plug in t= (one decay time constant),
the amount of charge left on the capacitor has decayed to approximately 1/3 of
its initial value. Approximately how much of the initial charge is left on the
capacitor in the circuit after it has discharged for three time constants t=3
seconds?
8

Now examine the time dependence of the voltage across the capacitor for the same discharging
capacitor in previous questions. As the charge on the capacitor changes, the voltage difference
across the capacitor plates also changes. In fact, the definition of capacitance easily relates
Q (t)
VCap (t) and QCap (t) by a constant: VCap (t)  Cap . Therefore, the equation describing the time
C
t

Q
dependent decay of the voltage across the capacitor is simply VCap (t)  Voe RC , where Vo  o .
C
You will experimentally test
 this equation later in this lab.


0-A-6
Sketch a graph of VCap (t) vs. t on the axes below using your answer to the
previous question (graph of QCAP). Be sure to include voltage values along the yaxis.
¿

9
As the capacitor discharges, it causes a current to flow through the resistor. Because energy
must be conserved, the magnitude of the voltage across the resistor is the same as the voltage
across the capacitor (they are the only circuit components!). Because the resistor is ohmic, the
current through the resistor can be related to its voltage and resistance, I=V/R. This gives a
time dependent equation for the current through the resistor of I Res(t)  Ioe

t
RC
.
You should notice that the time dependence of the charge on the capacitor, the voltage across
the capacitor, and the current through the resistor all exhibit the same exponential decay

function, and the constant values in front of the exponentials are simply related to each other
using properties you already know, V=Q/C and I =V/R.
¿
0-A-7
Relate the time dependent equation for the current going through the resistor
I Res(t)  Ioe

t
RC
Q Cap (t)  Qoe
to the time dependent equation for the charge on the capacitor
t

RC
. In other words, relate I o to Q o .


10
Subsection 0-B
A differential equation is an equation that involves derivatives. Most equations designed to
model reality in the physical sciences make use of differential equations, so a good working
knowledge of this type of mathematics is essential to any working scientist or engineer,
especially if they need to model something that changes in time (electronics, animal
populations, chemical reactions, biological processes,…). The following table compares
algebraic equations to differential equations using two specific examples:
¿
0-B-1
Examine the differential equation
d 2 y(t)
 9y(t) .
dt 2
One solution to this
differential equation is y(t)  4 sin3t . Check the solution by plugging 4 sin3t 
into the differential equation for y (t ) , perform the mathematical operations on
each side of the equation, andsee if the left hand side equals the right hand
have  36 sin(3t )  36 sin(3t ) .}
side. {You should
11
When analyzing circuits, you often must write a differential equation describing the behavior of
the circuit. This is most easily done by using conservation of energy to write a voltage equation.
Then one would use V=Q/C and V=IR to relate voltage to charge on the capacitor or current
through the resistor to create a new differential equation for Q(t).
Examine the discharging circuit for today’s lab and the construction of the differential equation
that describes it.
The Discharging RC Circuit (no voltage source)
1) First use conservation of energy to write a voltage equation:
0  Vres (t )  Vcap (t )
2) Next V=Q/C and V=IR to relate voltage to Q and I:
0  R  I res (t ) 
3) Notice that I resistor (t ) 
dQ cap (t )
dt
Qcap (t )
C
because charge leaving the capacitor must travel through the
0R
resistor:
dQcap (t )
dQcap (t )
4) Rearrange to have "attractive looking" equation:
dt
dt


Qcap (t )
C
Qcap (t )
R C
Since many students have not been trained on how to solve differential equations, the solution
is provided: Q Cap (t)  Qoe
¿

t
RC
, which should look familiar.
0-B-2
dQcap (t )
Examine
the differential equation


dt

Qcap (t )
R C
.
The solution to this
t
RC

t
RC
differential equation is Q Cap (t)  Qoe . Check the solution by plugging Q o e
into the differential equation for Q Cap (t ) , perform the mathematical operations
on each side of the equation, and see if the left hand side equals the right hand

side. {You should have 
t

Q o  RCt
Q
e
  o e RC in only two steps.}
R C
R C
12
Notice that no numerical value is needed for the initial charge Qo in this checking process. This
seems to indicate that the functional behavior of the circuit is the same regardless of the
specific numerical details, the real life quantities. The capacitor is going to discharge with
exponential decay no matter what the starting value of charge is on the capacitor.
Technically speaking then, a differential equation describes the general behavior of a system,
but it cannot provide all the information needed. One also needs to specify the actual starting
values of the system. These are called initial conditions or boundary conditions. This is only
mentioned for completeness, you won't have to worry about this for now.
¿
0-B-3
What trivially happens to the initial charge Qo in this checking process? Your
answer to this should allow you to see that the initial amount of charge on the
capacitor Qo is not determined by the differential equation. Basically you
choose the initial amount of charge to put on the capacitor plates and the
differential equations determines how quickly that charge discharges through
the resistor.
13
The other circuit for today’s lab, charging the capacitor, also has a differential equation to
describe its behavior in time:
CHARGING WITH CONSTANT SOURCE
1) First use conservation of energy to write a voltage equation:
Vsource  Vres (t )  Vcap (t )
2) Next V=Q/C and V=IR to relate voltage to Q and I:
Vsource  R  I res (t ) 
3) Notice that I resistor (t ) 
dQ cap (t )
dt
Qcap (t )
C
because charge leaving the capacitor must travel through the
resistor:
4) Rearrange to have "attractive looking" equation:
Vsource  R
dQ cap (t )
dQcap (t )
Vsource

R
R C
dt

dt

Q cap (t )
C
Qcap (t )
Since many students have not been trained on how to solve differential equations, the solution
  t 
is provided: Q Cap (t)  C  Vsource1 e RC , which should look familiar.



14
In-Lab Section 1: examining slow RC circuits with a stopwatch
A large capacitance C and large resistance R translate into a slow time constant =RC so that
you may easily measure the rate of decay with a stopwatch. You are supplied with a 1000 [F]
electrolytic capacitor. Electrolytic capacitors are “one-way” capacitors.
!
Be careful to only apply voltage correctly to the electrolytic capacitor or you
will damage it (the negative terminal is clearly marked on the capacitor).
You will discharge your capacitor in an RC circuit with approximately 10 [k Remember the
time dependent equation for the voltage across a discharging capacitor VCap (t)  Voe

t
RC
.
¿
1-1
What time constant  should you expect with R = 10 [k and C = 1000 [F]?

¿
1-2
Since approximately four time constants 4 allows the circuit to discharge to
about 2% of its initial value (because
1 1 1 1 1
   
 0.012 or more accurately
3 3 3 3 81
e 4  0.018 ), how long should you measure the decay of the capacitor’s charge in
order to make an accurate graph that doesn't take all day to collect data?
¿
1-3
Charge an electrolytic capacitor without resistance in the correct direction using
the 9-Volt battery (this happens quickly since there is very little resistance).
Then switch to discharge the capacitor through a ~10 [k resistor (if the
resistance is too small, the capacitor will discharge too rapidly to measure).
Collect (voltage, time) data by having the DMM measure voltage across the
capacitor while it discharges through the resistor using a stopwatch. You should
collect more data at the beginning when there is rapid voltage change. Record
your data here:
15
¿
1-4
Make a “raw graph” of your data by plotting Vcap(t) vs. t. {On separate graph
paper.}
Next you will linearize your data by taking the natural logarithm of your voltages. Since
VCap (t)  Voe


t
RC
, taking the natural logarithm of the function cancels the exponential:
  t 
  t 
1
RC
lnVoe  lnVo   lne RC  
t  lnVo .



 RC
1
t  lnVo  is the equation of a line with a slope of -1/(RC) and yThe function y(t)  
RC
intercept of ln(Vo). 
Thus, if you make the graph of lnVCap (t) vs. t on regular (Cartesian) graph
paper, you will obtain a line with a slope equal to -1/RC if your data is exponentially related.

¿

1-5
Linearize your data by taking the natural logarithm of your measured voltages
ln(V). Record your data here:
¿
1-6
Graph your linearized data by taking the natural logarithm of your measured
voltages ln(V) and plot these vs. t on regular graph paper. This should give you
a line with slope equal to -1/RC. {On separate graph paper, then calculate slope
and record here.}
¿
1-7
Find your experimentally measured value of capacitance C from the slope of
your linearized data graph and the value of the resistor's resistance R.
16
In-Lab Section 2: more oscilloscope practice
The following picture shows the digital oscilloscope and labels its most common features.
You now need to practice using the digital oscilloscope so that you are prepared to make
measurements with it. Keep in mind that the oscilloscope is simply a tool that allows you to
analyze the details of a rapidly changing voltage. With that in mind, you will now practice the
more common measurements that are made as well as their uses.
17
¿
2-1
Hook the output of the function generator directly to one of your oscilloscope
channels (and be sure the other channel is shut off). Create a sinusoidal wave
with your function generator with a very small voltage (i.e. use a special feature
of the function generator and a frequency in the 1-100 [kHz] range. Use the
autoset button to quickly get your signal on the screen so you can adjust your
function generator DC offset correctly. Be sure that your channel is on “1x
probe” and that your trigger is set to the correct source. Do this now and check
you work by interacting with students in other groups.
¿
2-2
Use the oscilloscopes measure feature to determine the average voltage of your
sine wave. Be sure to have about 7-10 full oscillations appear on the
oscilloscope screen as the oscilloscope measure feature actually uses the screen
for its data (and too few oscillations will create error in the averaging). Record
your measurement here.
¿
2-3
Get the digital oscilloscope to tell you on its screen the wave’s period and
frequency using the oscilloscope's features. Record your measurements here.
¿
2-4
Get the digital oscilloscope to tell you on its screen the wave’s amplitude.
Remember that the 'peak-to-peak' voltage value is twice the amplitude. Record
your measurement here.
18
¿
2-5
Adjust the amplitude of the wave on the function generator until you see that
the wave spends more of its time being negative than positive using the 'DCoffset' feature on the function generator. This will change your average value
for the voltage. Use a two cursor measurement of time and get the oscilloscope
to tell you on its screen how much time the sine wave spends being positive.
Then do the same thing to find out how much time the wave spends being
negative. Record your measurement here.
¿
2-6
Now use a two-cursor measurement in voltage and get the oscilloscope to tell
you on its screen the voltage drop of the wave from its maximum positive value
to zero. Record your measurement here.
¿
2-7
Change your sine wave to a triangle wave of 500,000 [Hz] and use the DC offset
so that the minimum of the triangle wave is zero volts. Examine a part of the
triangle wave that is decreasing. Use a two cursor measurement to find how
long it takes for the triangle wave to decrease from its highest value to one half
of that value. Record your measurement here.
Initially, many students become confused about the cause and effect relationship between
function generator and the oscilloscope. The function generator is creating the oscillating
voltage while the oscilloscope is merely observing it.
¿
2-8
A student is asked to change the amplitude of a voltage source and begins to
push buttons on the oscilloscope. Why is the student's TA disappointed?
¿
2-9
A student is analyzing a working circuit and is asked after to measure some
other feature of the circuit. The student starts turning knobs on the function
generator. Explain why the student's TA is yelling in a panicky voice. {Hint: the
phrase 'ruined the previous measurements' should appear in your answer.}
19
In-Lab Section 3: examining fast RC circuits with an oscilloscope
Most digital electronics make extensive use of capacitors. However, the decay rates are
typically much too rapid to measure with a DMM. In this part of the lab you will create an RC
circuit using a 0.1 F capacitor and a 1 k resistor and you will rapidly charge and discharge the
capacitor with an oscillating square wave.
¿
3-1
Calculate the time constant  that an RC circuit using a 0.1 [F] capacitor and a 1
[k resistor produce.
¿
3-2
You should choose a period of 20 (or rather a frequency of 1/(20) [Hz]) so that
there is plenty of time for the capacitor to discharge fully. Calculate this
frequency.
¿
3-3
Set up the RC circuit shown below powered by a 0 [V] to 3 [V] square wave at
the frequency you calculated in the previous question with R=1 [k and C=0.1
[F]. Use your function generator to create a square wave with a voltage
alternating between VMIN = 0 [V] and VMAX = 3 [V] by 1st setting the wave to
oscillate between +1.5 [V] and -1.5 [V] and then using the DC-offset to shift your
signal to have VMIN = 0 volts. The voltage across the capacitor should look like
'shark fins' on your oscilloscope as the capacitor exponentially charges and then
exponentially discharges. Do this now and check/discuss with students in other
groups to make sure you are getting it correct and understanding fully. Use the
same bottom ground setup as shown below:
20
¿
3-4
Based on the bottom ground setup shown previously, which oscilloscope
channel gives the voltage of the function generator source and which the
voltage across the capacitor?
¿
3-5
In this setup, is there any special feature of the oscilloscope that will allow you
to view the voltage on resistor? {The answer is 'yes', but explain.}
¿
3-6
During the time interval that the square wave source voltage is at +3 [V], is the
capacitor being charged or discharged? Which circuit below describes the
situation?
¿
3-7
During the time interval that the square wave source voltage is at +0 [V], is the
capacitor being charged or discharged?
21
¿
3-8
The following is an important reminder that you won’t need in today’s lab, but
is important to remember. Many times you may need to find the current in a
circuit. The component you must measure is the resistor because it is the only
ohmic device. if you want to determine the current of the circuit and use the
oscilloscope to measure the resistor's voltage amplitude, how could you then
turn this value into the current amplitude? {Hint: 'ohmic'.}
Observe the voltage across the capacitor and the total circuit voltage simultaneously using a
bottom ground configuration. You should see the “shark fin” pattern that is modulated by the
alternating square wave source voltage (turning on, then off).
¿
3-9
Use a double cursor measurement to find the time it takes for your charged
capacitor to decrease by half (in SI units so use scientific notation!).
¿
3-10
When a physical quantity decays exponentially, the time it takes for it to decay
to ½ its original value is called the half-life t½. Solve the half-life equation for t½
1
to find what t½ should be in this circuit in terms of R and C:

t half
1
Vo  Voe RC .
2
Show your work.

22
¿
3-11
Combine the results of the previous questions and calculate the experimentally
determined capacitance C of your capacitor using your half-life measurement in
3-9.
¿
3-12
Now use the double cursor method to find the time it takes for your capacitor to
discharge from ½ of its initial value to ¼ of its initial value.
The decaying exponential function has the unique property that each consecutive halving of its
value occurs in the same amount of time. Thus the half life is an important feature in
exponentially decaying systems because no matter when you begin measuring, you know that
each half life of time that passes, the value will have decreased by half.
¿
3-13
Using this knowledge, predict how long it should take for your capacitor to
discharge to 1/128 of its initial value (which is approximately 1% of its initial
value).
¿
3-14
Now use the cursors to collect voltage vs. time data for your decaying capacitor.
Then linearize your data, graph it on regular graph paper, and compute to the
capacitance C from the slope. Be sure your value is close to the labeled value.
Record your data below, make your graph on separate graph paper, linear your
data below, make your linearized graph on separate graph paper, find the slope,
and calculate C below.
23
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24
In-Lab Section 4: authentic assessment
Quickly set up a working circuit that simultaneously uses a capacitor and resistor in series
powered by a function generator. Use the concept of a half-life and a single measurement to
determine the capacitance of the capacitor. {Note using thalf=RC is much quicker than finding
the slope of a linearized graph with several data points, though much less accurate.} Sketch
your circuit and label the resistance of the resistor.
¿ 4-1
Show a student in a different group that you can successfully measure the
capacitance of a capacitor with only one measurement. Once you are successful,
have them sign below. Note: if someone is stuck, please give them advice!
"Yes, I have seen this student successfully find a capacitor's capacitance using
the measurement of the half-life. They are able to use this property of
exponential decay!"
Student Signature:___________________________________________________
25
In-Lab Section 5: open-ended / creative design
When capacitors are added in series, they have a combined capacitance determined by one of
the two following equations:
Ceffective  C1  C2 or Ceffective 
1
1
1

C1 C2
.
Design an experiment to determine which mathematical relationship is correct and which is
incorrect.

You are allowed to "cheat" by talking to other groups for ideas, but are not allowed to "cheat"
by just stating an answer you may already know, looking it up online or asking your TA.
Below you are given three prompts:
hypothesizing/planning, observations/data,
calculations/conclusion. Your job is to figure out the answer using these prompts as your
problem-solving model. In the event that you should run out of time, you may not discover the
correct answer, but you should make an attempt at each prompt. Grades are based on honest
effort.
Your open-ended solution should probably include some of the following items: sketches of
circuit diagrams, tables of data, calculations, recorded observations, random ideas, etc.
Write at the prompts on the next page.
26
¿
5-1
hypothesizing/planning:
¿
5-2
observations/data:
¿
5-3
calculations/conclusion
I, the physics 241 laboratory TA, have examined this student's Weekly Activity pages and found
them to be thoroughly completed.
!
TA signature: _______________________________________________________________
27
Post-Lab: RC Circuits - DC Source
!
You must complete this post-lab section after you attend your lab. You may
work on this post-lab during lab if you have time and have finished all the other
lab sections.
¿
X-1
A capacitor is charged by connecting it to a battery as shown below:
a) What is the direction of the current in the circuit as the capacitor charges,
clockwise or counterclockwise?
b) What is the current in the space between the capacitor plates as the
capacitor charges?
c) Which plate of the capacitor becomes positively charged, the upper or lower?
d) What happens to the magnitude of the current in the circuit as the capacitor
charges, increase, decrease or stay constant?
e) After the capacitor charges, which plate of the capacitor has a higher voltage,
the upper or lower?
f) After the capacitor charges, what is the direction of the electric field in
between the plates, upward or downward?
28
¿
X-2
A capacitor is charged with a battery and then placed in series with a resistor to
discharge.
a) What is the direction of the current in the discharging RC circuit as the
capacitor discharges, clockwise or counterclockwise?
b) Compare the currents at points a, b, c, d, e, and f in the discharging RC
circuit. How are they related?
c) For the charging capacitor circuit, compare the magnitude between point
pairs a-b and c-d.
d) For the discharging capacitor circuit, compare the magnitude between point
pairs a-b and c-d.
¿
X-3
Why do capacitors in series have the same magnitude of charge on each plate.
Draw a diagram to help explain. {Hint: focus on the charge separation of the
conducting 'island'.}
29
¿
X-4
What affect does resistance have on the time is takes for a capacitor in an RC
circuit to discharge?
¿
X-5
If it takes 10 seconds for a cully charges capacitor to discharge though a 10 [k
resistor, how long would it take to discharge through two 10 [k resistors
connected in series?
¿
X-6
If it takes 10 seconds for a cully charges capacitor to discharge though a 10 [k
resistor, how long would it take to discharge through two 10 [k resistors
connected in parallel?
¿
X-7
If it takes 10 seconds for a cully charges capacitor to discharge though a 10 [k
resistor, how long would it take to discharge through two 10 [k resistors
connected in series?
30
Weekly Activity 8: RC Circuit - AC Source
Pre-Lab
!
You must complete this pre-lab section before you attend your lab to prepare
for a short quiz. Be sure to complete all pages of the pre-lab.
Continue until you see the stop pre-lab picture:
Subsection 0-A
Last week you studied RC circuits, examining the exponential time dependence of the capacitor
voltage as you charged and discharged the capacitor with a constant source voltage. To do this
you used a square wave with a DC offset. Now you will examine the behavior of a capacitor
when a sinusoidal source voltage is applied: Vsource(T)  Vsource sin( D t) , where D is called the
amplitude
angular driving frequency of the circuit.

The capacitor voltage will no longer exhibit exponential time behavior. Instead the capacitor
voltage will oscillate sinusoidally with the same frequency as the source driving frequency.
(This can be proven by writing the differential equation for the circuit, finding its solution, and
checking the solution. However, this requires knowledge of solving inhomogenous differential
equations. This will not be done in this course.)
Instead, the most useful results of that calculation are provided: the time dependent voltages
across each component. Thus, you are not required to be able to derive the solutions to the
AC-driven RC circuit, but you must memorize, understand and be able to use these results.
31
Each component of the sinusoidally driven RC circuit has a sinusoidally varying voltage across it,
but each peaks at a different time determined by a phase shift. Did you catch that? Different
components reach their maximum voltages at different times than other components. The
solutions for the time dependent voltages of each component are given by the equations:
Vsource (t )  Vsource sin(D t  shift   )
amplitude
R
VResistor(t )   Vsource sin(D t )
 Z  amplitude

VCapacitor (t )   C
 Z



V
sin

t

 source
 D

2
 amplitude 
There are several new parameters to discuss. First notice that the source voltage is now
written with a source phase shift shift, the capacitor voltage has a phase shift of –/2, and the
resistor voltage has no phase shift. This can be seen in the graph below.
32
What this means in practice is that we will use the resistor voltage as a reference for all other
components in the circuit: i.e. we will measure the phases of each component in relation to
what is happening inside the resistor. This is because the resistor is ohmic and can always
provide the time dependent current via Ohm’s law simply by dividing the resistor voltage by
V (t )
resistance, I circuit (t )  R .
R
!
This choice of measuring voltage phase shifts from the perspective of the
resistor's voltage is not a uniform convention, and other teachers and texts may
make a different choice (so be sure you understand the principles).
In the previous graph, the source voltage can be seen as the negative sum of the components'
voltages VS (t )  VR (t )  VC (t ) . This is due to the conservation of energy. If you add the
electrical potential energy (per unit charge) in a circle, the sum should be zero. This is just like
making a loop on a staircase; if you end up at the same point, then you will have gained as
much gravitational potential energy as you have lost. Thus
VS (t )  VR (t )  VC (t )  0 ,
or reminding you that component voltage is the change in voltage across the component,
VS (t )  VR (t )  VC (t )  0 .
The Source Voltage Equation:
Vsource (t )  Vsource sin(D t     )
amplitude
The source voltage equation is straightforward. It oscillates sinusoidally, i.e. it is a sine function
of time. The maximum voltage applied across the whole circuit is Vsource . The source oscillates
amplitude


with an angular driving frequency D  2f D (which you will set later with your function
generator). The source voltage is phase shifted from the resistor voltage by an amount
 
of the capacitor given by the equation
  arctan C  where XC is the reactive capacitance 
 R 

1
C 
(more on this later). Note that this x-like variables are really the capital Greek letter
 DC
Chi (pronounced kai). The  is also included as an additional phase shift, but it is equivalent to
multiplying by -1, sin(t   )   sin(t ) , which emphasizes that electric potential in the circuit is
conserved VS (t )  VR (t )  VC (t ) .
33
If you look at the equation for resistor voltage, you will see no phase shift. Again, what this
means is that we measure all phases in relation to the resistor not the source. The resistor will
have its maximum voltage at a different time than when the source voltage is maximum.
¿
0-A-1
Imagine a sinusoidally driven RC circuit. If the source voltage has an amplitude
of Vsource =1.8 [volts], a linear driving frequency fD=555 [Hz], a resistance R=150
amplitude
[, and a capacitance C=1.5x10-5 [F], find the phase shift of the source voltage
compared to the resistor.

The Resistor Voltage Equation:
R 
VR (t)   Vsource sin( Dt)
Z  amplitude
The resistor voltage oscillates sinusoidally without a phase shift while R is simply the resistance.
Z is the impedance of the whole circuit. Z acts like the “total resistance” of the circuit. Z is
measured in SI units of Ohms and is given by the equation Z  R 2   L  C  .
2

This definition has new stuff, too. XL and XC are like the “resistances” of the inductor and
capacitor, respectively. We won’t study inductors until later in the semester, but it is easier to

memorize the complete equation. Since we don’t have an inductor (coil) in the circuit, you can
set this to zero. So we have Z  R 2  C2 . C is called the reactive capacitance and is
measured in ohms [].
¿
0-A-2

Now examine the resistor
equation as VR (t)  Vresistor sin( Dt). The maximum and
amplitude
R
Z
minimum voltage would oscillate across the resistor is Vresistor  Vsource .
amplitude
amplitude
Imagine a sinusoidally driven RC
circuit. If the capacitance is increased, explain
what happens to the amplitude of the resistor voltage?

34
¿
0-A-3
If the frequency is increased what happens to the amplitude of the resistor
voltage?
¿
0-A-4
Explain what happens to the current through the circuit if the resistor voltage
amplitude decreases?
¿
0-A-5
Explain what happens to the power lost through heating the resistor if the
resistor voltage amplitude decreases? (Remember that PR=IRVR.)
¿
0-A-6
Imagine a sinusoidally driven RC circuit with source voltage amplitude VS,
resistance R, and capacitance C. Explain whether the resistor will become
hotter if you increase the driving frequency? Use the concept that Z = total
impedance of the circuit.
35
The Capacitor Voltage Equation:
C 

 
VC (t)   Vsource sin Dt  
 Z  amplitude 
2 
The capacitor voltage oscillates sinusoidally and lags behind the resistor voltage by 90o. The
reactive capacitance C is like the resistance of the capacitor and is measured in SI units of
ohms [].

The “resistance” of the capacitor is related to the capacitance of the capacitor and the driving
1
frequency. This relationship C 
can be derived from the differential equation modeling
 DC
the circuit, but you must memorize it. The larger the capacitance, the less “resistance” in the
capacitor. But just as importantly if the driving frequency is increased, the “resistance” of the
capacitor decreases. This is why a capacitor is often used as a high pass filter in electronics: the

capacitor has less resistance to more quickly oscillating currents. BE SURE TO REMEMBER THIS
DURING THE LAB!

 
If we rewrite the capacitor equation as VC (t)  Vcapacitor sin Dt  , the capacitor voltage

2 
amplitude
C
amplitude is given by Vcapacitor 
Vsource . That means that the ratio of the capacitive
Z amplitude
amplitude
reactance and the total circuit impedance
times the source amplitude gives the amplitude of

the voltage across the capacitor.

In a previous equation, you found that the resistor voltage amplitude increases when the
frequency is increased. Since the voltage across the resistor and capacitor must add to the
voltage across the source, if the resistor voltage amplitude increases, then the capacitor voltage
must decrease. Therefore, as you increase the driving frequency, the resistor voltage
amplitude increases while the capacitor voltage amplitude decreases.
36
Subsection 0-B
Work though an example before beginning. Remember the equations below as you work:
Vsource (t )  Vsource sin( D t     )
amplitude
R 
VR (t)   Vsource sin( Dt)
Z  amplitude
C 

 
VC (t)   Vsource sin Dt  
 Z  amplitude 
2 

¿
0-B-1
If your
circuit
has
Vsource
 2 [V] ,
R  10,000 []
,
amplitude

 D  1,500 [radians/s ec] find the following values (in SI units):
C  1x10 -7 [F] ,
and
XC =
Z=
=
VR,amplitude. =
VC,amplitude. =
¿
0-B-2
Now compute VR,amplitude + VC,amplitude =
Your answer to this question has a sum that is greater than Vsource
amplitude!!! No, you didn’t make a mistake. Since the voltages are out of
phase, their maximums do not add together at the same time.
¿
0-B-3
At any instant of time, explain what should the component voltages add to,
V R (t )  VC (t )  ?
37
Now let’s try and visualize this circuit’s behavior.
¿
0-B-4
Write the functions for VS (t) , VR (t) and VC (t) using the numerical solutions to the
previous questions.



¿
0-B-5
Quickly sketch VR (t) and VC (t) on the oscilloscope screen below using a graphing
calculator. Don’t worry about providing the scale of the time axis. Then sketch
VR (t) + VC (t) onto the screen using a dotted line. This should equal the function
check it using
VS (t) so
 your graphing calculator.



38
In-Lab Section 1: examining the components
Set up the sinusoidally driven RC circuit with R  10,000 [] , and C  1x10 -7 [F] . Set your
function generator to create a sin wave with a voltage amplitude of a nice round number like 3
[V]. You may want to adjust your frequency later, but start at about 400 [Hz]. Set up a middle
ground to view the voltage across both the resistor and the capacitor simultaneously making
sure to invert the correct channel (a necessary step when using a middle ground). Check your
setup with other students in the lab.
¿
1-1
Make a sketch of the oscillating resistor and capacitor voltages on the
oscilloscope screen below. Label the signals VR (t) and VC (t) on your sketch


39
¿
1-2
Explain which signal is phase shifted to lag by 90o using calculus (derivatives).
¿
1-3
Find the amplitudes of each signal by measuring the peak-to-peak voltage of
each signal.
¿
1-4
Use the labeled values to determine the impedance of your circuit for this


2
2
driving frequency. Remember Z  R  C . .

¿
1-5
Use your previous answer to determine what the signal amplitudes should be
and then compare these predicted (calculated) amplitudes to your measured
amplitudes in the other previous question (they should be close).
¿
1-6
Find the frequencies f of each signal using oscilloscope measurements.
40
¿
1-7
Use your answers to the previous questions to write equations for VR (t) , VC (t)
and VS (t) entirely with numerical values (no free parameters). (Don’t forget the
phase shift.)



¿
1-8
Set your oscilloscope to plot VR (t) on the x-axis and VC (t) on the y-axis (an XY
plot). Sketch the result on the oscilloscope screen below.


41
¿
1-9
In an XY plot, if the signal on the y-axis oscillates twice as fast as the signal on
the x-axis and the signals are 90o out of phase, then sketch what will appear on
the oscilloscope screen below.
42
In-Lab Section 2: experimentally finding the capacitance
1
by observing a sinusoidally driven RC circuit using
 DC
many different driving frequencies. Use the same circuit set up as in the previous part of the
lab.
Next you will test the relationship C 
¿

2-1
As you increase the driving frequency, the amplitude of the resistor voltage will
R
Z
increase because the total circuit impedance is decreasing, i.e. Vresistor  Vsource .
amplitude
amplitude
Work through this logic so that you are sure you understand it.

Meanwhile, as the driving frequency increases, the capacitor amplitude decreases. This makes
sense because the resistor and the capacitor are the only two components in the circuit other
than the source. Since the voltages across both must add up to the source voltage at any
instant in time, if the voltage amplitude of one increases, then the other must decrease.
Therefore, there must be some specific driving frequency when the amplitude of the resistor
voltage is the same as the capacitor voltage: Vresistor  Vcapacitor for a specific angular driving
amplitude
amplitude
frequency D.

R

Realizing that Vresistor  Vsource and Vcapacitor  C Vsource , setting these two voltages equal
Z amplitude
Z amplitude
amplitude
amplitude
assuming the circuit is being driven at some specific angular driving frequency D,equal you get
C
R
Vsource  Vsource , which simplifies to  C  R . In other words, the voltage across the
Z amplitude
Z amplitude


capacitor equals the voltage across the resistor if their "resistances" are equal.

The first method for finding the capacitance of an unknown capacitor makes use of the
previous equation. All you need to do is adjust the driving frequency of your circuit until the
1
capacitor voltage amplitude and the resistor voltage amplitude are equal. Then use C 
 DC
and  C  R for the specific D,equal to find the capacitance.

43
¿
2-2
Substitute C 
1
into  C  R , and then solve for C. Be sure to realize that
 DC
this equation is only true when the circuit is being driven at the specific
frequency D,equal that makes the resistor and capacitor voltages equal.

¿
2-3
Set up a 3 [volts], 400 [Hz] sinusoidally driven RC circuit with R  10,000 [] , and
C  1x10 -7 [farads] . Set up a middle ground to view the voltage across both the
resistor and the capacitor simultaneously making sure to invert the correct
channel. Adjust the driving frequency until the resistor and capacitor voltages
are equal. Then use your formula from 2-2 to find your experimentally
determined value for capacitor's capacitance. Remember that the function
generator reads the linear frequency.
44
The second method for finding an unknown capacitance is more involved, but more accurate as
it involves multiple measurements. The voltage amplitudes of the sinusoidally driven RC are:
Vresistor 
amplitude
and
Vcapacitor 
C
Z
amplitude

R
Vsource
Z amplitude
Vsource
.
amplitude
Dividing these two equations gives

 C  Vsource 
amplitude 


Vcapacitor 
Z

  C
amplitude




R

V
Vresistor
R
source
amplitude 

amplitude


Z


Therefore,
C  R

.
Vcapacitor
amplitude
Vresistor
.
amplitude
In order to experimentally determine C for your capacitor, simply combine the last equation
1
with the definition C 
and rearrange:
 driveC


V
1 resistor
amplitude
 C drive
.
R Vcapacitor
amplitude

45
V
1 resistor
amplitude
 C drive
R Vcapacitor
amplitude
looks like a weird arrangement for this equation, but if you think of y=mx, then you see that if
you graph
V
1 resistor
amplitude
vs.  drive, you should obtain a linear graph with a slope equal to C.
R Vcapacitor

amplitude
¿
2-4
Find C by collecting data for multiple driving frequencies, making a graph on

separate graph paper and finding the slope.
46
In-Lab Section 3: authentic assessment
¿
3-1
Quickly set up a working circuit that simultaneously uses a random capacitor
and a 1000  resistor in series powered by a sinusoidal source voltage on your
function generator. Then make the necessary measurements to determine the
capacitance of the capacitor. Be sure your experimentally determined
measurements give the correct capacitance. Show your results to a student in a
different group:
"Yes, I have seen this student find an unknown capacitance 'the easy way'."
Student Signature:___________________________________________________
47
In-Lab Section 4: open-ended / creative design
Make a capacitor from the square cardboard pieces covered in conductive aluminum foil.
Sandwich a non-foil square of cardboard between the foiled boards, and be sure your makeshift
capacitor is not shorted out by accident. Measure the capacitance of your homemade
 A
capacitor. The equation for the capacitance of two parallel plates is given by C  o . Use
d
this equation to report the dielectric constant  of the sandwiched cardboard between the
C2
plates with correct units. Note: o  8.85x1012
. Design an experiment to determine the
N  m2

capacitance of your cardboard capacitor and the dielectric constant of the cardboard.
You are allowed to "cheat" by talking to other groups for ideas, but are not allowed to "cheat"

by just stating an answer you may already know, looking it up online or asking your TA.
Below you are given three prompts:
hypothesizing/planning, observations/data,
calculations/conclusion. Your job is to figure out the answer using these prompts as your
problem-solving model. In the event that you should run out of time, you may not discover the
correct answer, but you should make an attempt at each prompt. Grades are based on honest
effort.
Your open-ended solution should probably include some of the following items: sketches of
circuit diagrams, tables of data, calculations, recorded observations, random ideas, etc.
Write at the prompts on the following page.
48
¿
4-1
hypothesizing/planning:
¿
4-2
observations/data:
¿
4-3
calculations/conclusion
I, the physics 241 laboratory TA, have examined this student's Weekly Activity pages and found
them to be thoroughly completed.
!
TA signature: _______________________________________________________________
49
Post-Lab: RC Circuits - AC Source - REVIEW PRACTICE OF GAUSS'S LAW
!
You must complete this post-lab section after you attend your lab. You may
work on this post-lab during lab if you have time and have finished all the other
lab sections.
¿
X-1 Gauss’s Law applied to systems with spherical symmetry:
A Cartesian representation of the electric vector field, E  E x xˆ  E y yˆ  E z zˆ , is useless. Try
ˆ  E ˆ . Because of the
using a spherical coordinate system representation, E  E r rˆ  E 

symmetry of the system, E  0 and E  0 . So E  E r rˆ . Thus the problem of solving for a
vector field reduces to a problem of solving for 
a scalar field quantity that only depends on the
radial distance, Er.



50
a. Is the electric field constant in magnitude for a fixed radius? Explain.
b. In which direction does the electric field point, and how does this depend on
the sign of ? Explain.
c. Explain why the sphere is the appropriate Gaussian shape to draw?
d. What must the units of the constant A be?
e. Write the integral that defines the total electric flux through a Gaussian
sphere. Solve it for each of the three Gaussian spheres shown.
51
f. Write and solve the integrals to find the total charge enclosed within each
Gaussian sphere.
g. Use Gauss’s law to equate your answers to e and f to solve for the electric
field for all three places: a radial distance r inside the hollow region, inside the
charged region and outside the sphere. Be sure to express your answers as
vectors.
h. Explain what the resultant electric field would be like if the charge density 
had turned out to be “balanced”, i.e. the total charge found by integrating over
the entire sphere is equal to zero.
52