12 Aug Two conductors separated by an insulator can be electrically charged so that one conductor has a positive charge and the other conductor has an equal magnitude of negative charge.

The Notebook Paper Capacitor

Objectives

• To measure the capacitance of a parallel plate capacitor made out of sheets of
aluminum foil and plain white paper.
• To measure capacitance in series and parallel and see if it behaves as expected.
• To see if the relationship of capacitance to plate area behaves as expected
• To see if the relationship of capacitance to plate separation behaves as expected.
• To measure the dielectric constant of paper.
Background
Two conductors separated by an insulator can be electrically charged so that one conductor
has a positive charge and the other conductor has an equal magnitude of negative charge.
This arrangement is called a capacitor. Capacitors are sometimes called condensers, after the
name given them by Volta for their ability to “condense” an amount of electricity. The
capacitor was independently discovered in 1745 and 1746 by Ewald Georg von Kleist and
Pieter von Musschenbroek, of the University of Leyden, who invented a metal, water and
glass capacitor, the Leyden jar.
Capacitance is measured in Coulombs per Volt. The unit is the Farad (F), named after
Michael Faraday. 1 Farad = 1 Coulomb / 1 Volt. A 1 F capacitor would be very large!
More typically used are microfarads (1 μF = 10–6 Farads), nanofarads (1 nF = 10–9 F) or
picofarads (1 pF = 10–12 F). A parallel plate capacitor consisting of two parallel metal
conductors separated by vacuum has capacitance as given in equation 1.

(1)

D
AC0e=
Area = A
Separation = D
Figure 1. A diagram of a parallel
plate capacitor. The arrows
represent the electric field.

If a non–conducting material composed of polar molecules (which have intrinsic dipole
moments) is placed within a capacitor which is connected to a voltage source, the dipole
moments will be lined up parallel to the electric field. Such a material is called a dielectric.
The dipole moment will be aligned with the electric field, with the positive end of the
molecule in the direction of the field.

When aligned, the charges will cancel except at the edges of the dielectric, creating a surface
charge layer with charge opposite to that on the charged plate. This layer sets up an electric
field opposite to the applied field, weakening the original field. The weaker field is indicated
by a decrease in the number of field lines within the dielectric, as shown in figure 3.
The weaker field for a given charge results in a higher capacitance, given by eqn. 2.

(2) where κ is the dimensionless dielectric constant.

D
AC0ek=
Figure 2. Alignment of the molecular dipole moments of the dielectric.
Figure 3. Partial cancellation of the electric field within the dielectric.

Determination of the Properties of Parallel Plate Capacitors
In this lab you will construct parallel plate capacitors out of aluminum foil and paper. You
will put the capacitors in series and parallel and measure the capacitance and then compare
the result to the calculated equivalent capacitance. You will measure the capacitance with
different thicknesses of paper dielectric and determine the dielectric constant.
Equipment
• 20 sheets of paper
• Four sheets of aluminum foil of approximately equal size
• Some textbooks
• Multi–meter with capacitance gauge, wires, and alligator clips
• Micrometer
• 4 wires and paper clips
• Blocks of wood
• Ruler
• Scissors
Procedure: Part I
The data was collected for you. Here is a summary of how it was done.
1. Always record estimated uncertainties for each measured quantity.
2. Measure the length and width of your sheets of aluminum foil for both capacitors
3. Using the paper clips to connect the wires to the aluminum foil, make two capacitors with
6–page thick dielectrics and measure their capacitance. You can make the dielectric out
of notebook or printer paper and place the entire capacitor within the textbook. There
should be at least 200 pages (100 sheets) between the two capacitors. Make sure the
wires and aluminum foil do not short circuit! You can place each of your two
capacitors directly between blocks of wood, but you must be careful to avoid shorts.

**I made a video showing the setup
4. Measure the capacitance of each capacitor. When measuring the capacitance, place
the book on the floor with the block of wood on the cover, and have one member of the
group stand on it (or use another method of applying a uniform force).If the
capacitances are not approximately equal, adjust the sizes or alignment of the foil until
you have made two 5–page capacitors whose capacitances are within 10% of each other.
5. Measure the capacitance of the two 6–page capacitors connected in series.
6. Measure the capacitance of two 6–page capacitors in parallel.
7. Make sure you draw a circuit diagram clearly showing the connections for both
capacitors in series and for the capacitors parallel.
8. Take Capacitor #2 and cut each sheet in half to reduce the area to ½ its original area
and measure the capacitance of this 6–page capacitor of one half the area.

Procedure: Part II (this uses Capacitor #1 from Part I)
1. Determine the size of each tick mark on the micrometer scale.
2. Determine the zero point on your micrometer by closing it gently, noting where it starts
to slip. If it closes below zero, you must add the offset to your measurements.
3. For this portion of the lab, you need to measure the thickness of the paper dielectric.
Measure each set of pages that you used; do NOT simply multiply the thickness of one
page by “n”. When measuring the thickness of paper, close the micrometer on the paper,
but not so tightly that the paper is squeezed and cannot slip out.
4. Measure the capacitance (Cm) for capacitors of 4, 6, 8, 10, 12, and 14 pages thickness.
The data table should include at least the following. Make sure to note somewhere what
kind of paper you used for the dielectric: notebook, printer, or textbook pages.

Description Foil Length
L
(use σL=1mm)
Foil Width
W
(use σW=1mm)
Thickness
D (use σD = 0.02mm)
Capacitance
(Measured)
Cm
(σCm will be provided)
Part I
6–page #1
5–page #2
series Leave black boxes
parallel empty
half size
Part II (Part ii uses Capacitor #1)
4–page
6–page
8–page
10–page
12–page
14–page

Data Analysis: Part I
1. Compute the predicted capacitance for two capacitors in series and in parallel of the
two 6–page capacitors and determine whether the measured values follow the predicted
relationships to the capacitances of the individual capacitors.
2. Determine whether the capacitance of the one half size capacitor obeys the predicted
relationship to a full size capacitor.
Note: in Part 1, you do not need the dielectric constant.You are just using the relationships
for series & parallel capacitors and examining the relationship of capacitance to plate area.

Data Analysis: Part II
1. Plot Cm (y–axis) as a function of (x–axis) for the full size capacitors of part II. You
will need to add error bars as in the Error Analysis section below.
2. Inspection of eqn. 2 shows that this graph should be a straight line with slope equal to κ
and y–intercept equal to the internal capacitance of the meter: Cinternal. Using Excel
(or another spreadsheet), determine the slope and intercept of this line.
3. Use steepest and shallowest lines (as described in Error Analysis section) to get a range
of values of the slope and intercept (which equals a range of of κ and Cinterntal)

Error Analysis
1. Plot error bars in x and y. The uncertainty in y is that which you determined for the
capacitance measurement. The uncertainty in x depends on your measurements of
thickness and area, and is computed as given below. This is different for each thickness.
(3) This is different for each measurement.
where (4) This is the same for each measurement.
IMPORTANT: You may need to print the graph out (as large as you can) and draw error
bars manually. This is because error bars are different for each data point and some
software (like Google Sheets) does not allow you to graph those kinds of error bars

2. Plot the steepest and shallowest lines that are
visually consistent with the data points. They
should deviate as much as possible from the best–
fit line without going outside the plotted error
bars (if possible). They should NOT be parallel
lines. **See example at right from a different lab

IMPORTANT: If your estimated σ for each point are too small, use the scatter of the
points themselves to estimate a maximum and minimum slope.
D
A0e
2
2
2
2
0
ADD
AAD
x
sses+ ́=
2
2
2
2
WLAWL
A
sss+ ́=

3. Calculate the slopes of these lines. Use the difference between the best fit slope and the
maximum and minimum slopes to determine the uncertainty in the dielectric constant, .
Remember, the maximum and minimum slopes give you the maximum and minimum
values of κ.
4. Calculate the y–intercepts of these lines or read them off the graph. Use the maximum
and minimum values of the intercept to determine the uncertainty in the internal
capacitance,

Conclusions
Answer the following in your Conclusion:
Part I: Compare the measured and computed values of the series, parallel, and half size
capacitors to predictions (predictions made by equations we developed).
Part II: Look up a value for the dielectric constant of paper. Does your value of κ agree
with this to within the uncertainty? Is this plot consistent with a straight line? Is the internal
capacitance significant compared to the capacitance(s) of the notebook paper capacitors
which you made?
Overall: What are the major sources of error in this lab?

Summary Table I [Not the same as the data table!]
C5 series C5 parallel C5 half size
Measured value
Predicted value

Summary Table II
Measured Dielectric
Constant (from slope)
Reference Dielectric
Constant
Internal Capacitance of
the Meter (from intercept)
κ ± σκ Cinternal ±

Remember to clearly state where you found your reference value.