1:
2:
(ta initials)
first name (print)
last name (print)
brock id (ab13cd)
(lab date)
Experiment 1
Capacitance
In this Experiment you will learn
• the relationship between the voltage and charge stored on a capacitor;
• how to compensate for the effect of a measuring instrument on the system being tested;
• to visualize data in different ways in order to improve the undwerstanding of a physical system;
• to extend your data analysis capabilities with a computer-based fitting program;
• to apply different methods of error analysis to experimental results.
Prelab preparation
Print a copy of this Experiment to bring to your scheduled lab session. The data, observations and
notes entered on these pages will be needed when you write your lab report and as reference material
during your final exam. Compile these printouts to create a lab book for the course.
To perform this Experiment and the Webwork Prelab Test successfully you need to be familiar with the
content of this document and that of the following FLAP modules (www.physics.brocku.ca/PPLATO).
Begin by trying the fast-track quiz to gauge your understanding of the topic and if necessary review the
module in depth, then try the exit test. Check off the box when a module is completed.
FLAP PHYS 1-1: Introducing measurement
FLAP PHYS 1-2: Errors and uncertainty
FLAP PHYS 1-3: Graphs and measurements
FLAP MATH 1-1: Arithmetic and algebra
FLAP MATH 1-2: Numbers, units and physical quantities
WEBWORK: the Prelab Capacitance Test must be completed before the lab session
☛✟
Important! Bring a printout of your Webwork test results and your lab schedule for review by the
!✠
✡
TAs before the lab session begins. You will not be allowed to perform this Experiment unless the
required Webwork module has been completed and you are scheduled to perform the lab on that day.
☛✟
Important! Be sure to have every page of this printout signed by a TA before you leave at the end
!✠
✡
of the lab session. All your work needs to be kept for review by the instructor, if so requested.
CONGRATULATIONS! YOU ARE NOW READY TO PROCEED WITH THE EXPERIMENT!
1
2
EXPERIMENT 1. CAPACITANCE
The capacitor
A capacitor is a device that stores electric charge (electrons). It consists of two electrically conductive
parallel metal plates separated by an insulating layer of air or other dielectric material, a material that
can support an electric field. The total amount of charge q stored is proportional to the electric potential
difference, or voltage, VC between the plates, so that
q = CVC .
(1.1)
The capacitance C of a parallel plate capacitor is proportional to the plate area and the dielectric constant
of the medium between the plates, and inversely proportional to the plate separation. Capacitance is
measured in units of Farads (F), microFarads (µF = 10−6 F) and picoFarads (pF = 10−12 F).
Figure 1.1: Basic capacitor circuit
A series circuit provides only one path for movement of charge. Figure 1.1 shows a series circuit
consisting of a source of electric potential energy V of voltage V , a switch S, a current limiting resistor R
and a capacitor C.
P Kirchoff’s Voltage Law states that the algebraic sum of the voltages in any closed circuit loop is zero,
V = 0. With voltage sources considered positive and voltage drops considered negative, we establish
that the source voltage V will be equal to the voltage drops VR across R and VC across C, so that
V = VR + VC .
(1.2)
The capacitor plate material consists of a conductive lattice of atoms. The positive nucleus is fixed in
place while some of the negative electrons surrounding an atom are free to migrate from the atom when
subjected to an external force.
Let us assume that initially there is no charge stored on the capacitor plates so that the plates are
electrically neutral and the voltage across the capacitor VC = 0.
When the switch S is closed, the positive terminal of the voltage source attracts electrons away from
the upper plate of the capacitor, leaving the upper plate with a net positive electric charge. The positive
charge now present on the upper plate attracts electrons from the voltage source negative terminal to the
lower capacitor plate, giving it a net negative charge.
This flow of charge dq through the circuit during a time interval dt defines the electric current i = dq/dt.
The current i is directly proportional to the voltage VR across the resistor R and inversely proportional
to the circuit resistance R. This relationship between current, voltage and resistance is known as Ohm’s
Law:
i = VR /R.
(1.3)
3
The charge separation q at the two capacitor plates establishes a voltage, or potential difference,
VC = q/C across the capacitor.
As VC increases, the difference VR = V − VC across R decreases, as does the current i = (V − VC )/R
flowing through R. This process continues until the voltage VC across C is equal to the voltage V of the
source, at which time charge no longer flows and the current i = 0.
From Kirchoff’s Voltage Law, we know that the sum of all the voltage sources minus all the voltage drops
in a circuit equals to zero. To examine the capacitor charging process, we traverse the circuit of Figure 1.1
clockwise from the negative (-) terminal of the battery, adding each voltage source and subtracting the
voltage drop across each component:
V − VR − VC
q
V − iR −
C
= 0
= 0
(1.4)
A current i that varies as a function of time t is symbolized by i = i(t). Substituting this relationship
in Equation 1.4 and rearranging yields the charging equation for the circuit:
1
V
dq
=
−
q
(1.5)
i(t) =
dt
R
RC
The solution of this differential equation in terms of q is given by
V
dq
=
e−t/RC
dt
R
(1.6)
Here, e = 2.718 . . . is the base of the natural logarithms (ln), not the elementary charge. We note in
Equation 1.6 that at time t = 0 the exponential term is e0 = 1 and i = V /R does not have any dependence
on time. Let this initial constant current be I0 . Then the current i(t) flowing in the circuit at any time t
is given by
i(t) = I0 e−t/RC
(1.7)
Capacitors in parallel
The charge stored on a capacitor is directly proportional to the surface area of the capacitor plates.
Referring to Figure 1.2 we note that putting two capacitors in parallel results in an equivalent plate surface
area that is the sum of the individual plate areas. This qualitative result can be expressed mathematically.
The voltage across each capacitor is V . Applying Equation 1.1 to the charge stored in each capacitor:
q1 = C1 V,
q2 = C2 V.
(1.8)
The total charge q stored in the parallel arrangement of capacitors is the sum of the charges stored
in each capacitor,
q = q1 + q2 = (C1 + C2 )V
(1.9)
The equivalent capacitance Cp with the same total
charge q and voltage V is then
Cp =
q
= C1 + C2
V
(1.10)
Figure 1.2: Capacitors in parallel
4
EXPERIMENT 1. CAPACITANCE
The relationship can be extended to any number of capacitors in parallel by simply adding the contributions
from the charge stored in each of the capacitors:
Cp = C1 + C2 + . . . + CN =
N
X
Ci
(1.11)
i=1
Capacitors in series
When a potential difference V is applied across several capacitors connected in series, a charge separation q = Cs V will be induced across each capacitor.
The sum of the potential differences across the capacitors is equal to the applied potential difference
V . The equivalent capacitance of two or more capacitors in series is given by
N
X 1
1
1
1
=
+
+ ... =
Cs
C1 C2
Ci
i=1
(1.12)
Figure 1.3: Capacitors in series
Procedure
Figure 1.4: Schematic diagram of experimental setup, jumpered to measure a single capacitor
Manufacturer’s values of components used in the experimental circuit:
Rd ± ∆(Rd ) = (100 ± 5)Ω,
Rc ± ∆(Rc ) = (1.00 ± 0.05) × 105 Ω,
C ± ∆(C) = (2.2 ± 0.2) × 10−6 F.
Figure 1.4 shows the schematic diagram of the electrical circuit used in this experiment. The circuit
uses one or two removable jumper wires to electrically arrange the capacitors in series, parallel, or to only
include a single capacitor as in Figure 1.4. The capacitors C1 = C2 = C have the same nominal value.
5
A power supply is connected across the input terminals A and G of the circuit, with the positive (red)
terminal of the power supply connected to A and the negative (black) terminal connected to G. The
LabPro voltage probe contacts are connected (see Figure 1.4) across Rc with the red wire on the A
side when using the 5 V power supply, or on the B side when the board is USB-powered.
The LabPro unit acts as a resistance of Rp = 107 Ω in parallel with resistance Rc . The effective circuit
resistance of these two resistors in parallel is given by
1
1
1
=
+
R
Rc Rp
(1.13)
• Calculate R and ∆R using the given values of Rc , ∆Rc , and Rp . Assume ∆Rp = ±0.005Rp .
R = .................... = .................... = ....................
s
∆Rp 2
∆Rc 2
∆R = R2
+
= .................... = ....................
Rc2
Rp2
R = ................. ± ................. Ω
When the normally open switch S is pressed, any voltage VC present across the capacitor discharges
very quickly through Rd and the voltage at point B decreases to approximately zero volts. Note that as R
and Rd are in series, the same current I0 flows through both resistors.
Using Ohm’s Law and the fact that R has a resistance some three orders of magnitude greater than
that of Rd , the voltage drop across Rd is nearly zero, the voltage VAB across R is approximately equal to
the power supply voltage V , and a steady current I0 = VAB /R flows through R.
When the switch S is released, the time dependent current i(t) = VAB /R decreases exponentially with
time as a voltage develops across the test capacitor(s), and hence VAB decays exponentially to approximately zero. Replacing i(t) in Equation 1.7 we get an expression for the voltage VAB across R:
VAB = I0 R e−t/RC .
(1.14)
Part 1: Single capacitor
• Turn off the power supply. Assemble the circuit board as shown in Figure 1.4, with a jumper wire
connecting the common terminal G to terminal P2 .
Have the instructor check your circuit before you proceed.
• Close any open Physicalab programs, then start a new PhysicaLab session and enter your email
address and that of your partner, if any, in the email entry box. Your graphs will be sent there for
later inclusion in your online lab report. Email yourself a copy of all graphs.
• Check the Ch1 box and choose to collect 50 points at 0.05 s/point. Select scatter plot. Press and
hold the switch S to discharge the capacitor. Click Get data . As soon as data appears, release the
switch.
6
EXPERIMENT 1. CAPACITANCE
Your graph should display a horizontal line at V ≈ 5 V followed by an exponential decay to V = 0 V from
the time that the switch was released.
The LabPro converts the continuously varying analog input voltage into a digital representation consisting of discrete and equally spaced increments in V , so that the input voltage is linearly quantized. The
voltage difference between two adjacent voltage levels defines the internal scale and hence the resolution
of the LabPro.
• Zoom in on the near-zero region of data at the end of the decay curve by unchecking Autoscale and
adjusting the X-axis scale values, then click Draw .
Your graph should display these points placed on one of several horizontal equally spaced lines that represent the the discrete voltage steps Vs of the converter output. From the graph and the corresponding data
table values, select two adjacent voltage steps V1 and V2 , determine the voltage resolution dV = V2 − V1
of the Labpro converter scale and then the error ∆VAB of a reading:
V1 = ............., V2 = ............., dV = ............., ∆VAB = .............
• Display the complete data set by checking Autoscale, then check the X grid and Y grid boxes to
display a grid on your graph. Select scatter plot, then click Draw .
You will be simultaneously fitting two separate equations to your data. The first equation is given by
Y = A and will fit a straight line at Y = VAB to the initial portion of your data, from time t = 0 to the
release of the switch at time t0 = C, where A,C represent parameters of the fitting equations.
The second equation will attempt to fit the exponential portion of the data, from the time t0 = C
and amplitude VAB = A to a final value of VAB = 0 at some later time. This equation is given by
Y = A exp(−iB (x−C)). The fitting parameter B determines the decay rate of the exponential curve, and
comparison with equation 1.14 shows that B = 1/RC and x= t. To summarize, we can express the time
dependence of the voltage VAB across R by the expression

 I0 R,
t < t0
VAB (t) =
 I R exp(−B (t − t )), t ≥ t
0
0
0
• To fit your data, check Fit to: y= and select or enter the following string, without spaces, in the
fitting equation box: A*(x<C)+A*(exp(-B*(x-C)))*(x>=C).
The fit parameters A, B and C are initially set to one. These values may be too distant from the actual
fit values to allow the fitting algorithm to converge and provide a valid result. If you attempt to perform
a fit and get an error message, review the scatter plot of your data to estimate some more appropriate
values for A and C.
To estimate the parameter B, you can use the fact that a capacitor discharge curve decays to 1/e =
1/2.718 . . . of the original voltage level after a time ∆t = RC, defined as the time constant of the circuit.
Choose a time t1 along the exponential portion of the curve and a time t2 at the point where the curve
has decreased to approximately 1/3 of the level at t1 . Since ∆t = t2 − t1 ≈ RC then B = 1/RC ≈ 1/∆t.
• Label the axes and title the graph with your name and circuit arrangement used. Click the Send to:
button to email yourself a copy of your graph for the exponential decay of a single capacitor.
• Check the Y log box to display the voltage in logarithmic units, then uncheck autoscale and set
Y= -3 to 1 in 4 steps to range y from 10−3 to 101 . Redraw and email your graph.
7
? What does the exponential decay look like and why is this so?
? What feature of the graph does the fit parameter B represent? How would you prove it? Recall that
you have taken the log of the voltage.
? Why are the points at the bottom right corner of the graph scattering? Use values from your data
set to support your conclusions.
? Do any negative voltage points from your data set appear on the graph? Can a logarithmic plot
display values of y ≤ 0 ?
• Record below the initial voltage A and decay parameter B
B = 1/RC = ................. ± .................1/s
A = I0 R = ................. ± .................V
• Calculate the experimental value C and ∆C for the capacitor:
C = ................... = .................... = ....................
s
∆B 2
∆R 2
∆C = C
+
= .................... = ....................
B
R
C = ................. ± .................F
• Calculate the initial current I0 and its error ∆I0 :
I0 = ................... = .................... = ....................
s
∆R 2
∆A 2
+
= .................... = ....................
∆I0 = I0
A
R
I0 = ................. ± .................
• The manufacturer’s value of the capacitance C used in this part of the experiment is
C = ................. ± .................F
Part 2: Capacitors in parallel
• Turn off the power and remove all jumper wires. Connect a jumper wire from P1 to P3 and another
from P2 to G.
• Acquire a data set, then fit and send the exponential decay graph for two capacitors in parallel.
Calculate the following parameters, then enter the results below.
8
EXPERIMENT 1. CAPACITANCE
B = 1/RCp = ................. ± .................1/s
A = I0 R = ................. ± .................V
Cp = ................. ± .................F
I0 = ................. ± .................A
• Calculate the effective capacitance Cp and the error, or tolerance ∆Cp of the two capacitors in parallel
using the manufacturer’s values of C1 and C2 .
Cp = ...................
= .................... = ....................
∆Cp = ...................
= .................... = ....................
Cp = ................. ± .................F
Part 3: Capacitors in series
• Turn off the power and remove all jumper wires. Connect a jumper wire from P3 to G.
• Acqiure a data set, then fit and print the exponential decay graph for two capacitors in series. Use
a separate sheet to calculate the following parameters, then enter the results below.
B = 1/RCs = ................. ± .................1/s
A = I0 R = ................. ± .................V
Cs = ................. ± .................F
I0 = ................. ± .................A
• Calculate the effective capacitance Cs and the error, or tolerance ∆Cs of the two capacitors in series
using the manufacturer’s values of C1 and C2 .
Cs = ...................
= .................... = ....................
∆Cs = ...................
= .................... = ....................
Cs = ................. ± .................F
9
Part 4: Circuit time constants
• For the three circuits, use your experimental C value to calculate the tc time constant of each charging
circuit. Also calculate the td , the time constant of the discharging circuit, when the switch S is closed
and the charge stored in the capacitor discharges through Rd .
? How do these time constants vary with C, R and Rd ?
tc (1) = ................... = .................... = ....................
td (1) = ................... = .................... = ....................
tc (2) = ................... = .................... = ....................
td (2) = ................... = .................... = ....................
tc (3) = ................... = .................... = ....................
td (3) = ................... = .................... = ....................
IMPORTANT: BEFORE LEAVING THE LAB, HAVE A T.A. INITIAL YOUR WORKBOOK!
Lab report
Go to your course homepage on Sakai (Resources, Lab templates) to access the online lab report worksheet
for this experiment. The worksheet has to be completed as instructed and sent to Turnitin before the lab
report submission deadline, at 11:55pm six days following your scheduled lab session. Turnitin will not
accept submissions after the due date. Unsubmitted lab reports are assigned a grade of zero.
................................................................................
................................................................................
................................................................................
................................................................................
................................................................................
................................................................................
................................................................................
10
EXPERIMENT 1. CAPACITANCE
................................................................................
................................................................................
................................................................................
................................................................................
................................................................................
................................................................................
................................................................................
................................................................................
................................................................................
................................................................................
................................................................................
................................................................................
................................................................................
................................................................................
................................................................................
................................................................................
................................................................................
................................................................................
................................................................................
................................................................................
................................................................................
................................................................................