PHYS112 : Labs

7 Capacitance

In addition to reading this assignment, you may need to refer to Appendix A on uncertainties and Appendix B on linear regressions.

Introduction

In Lab 1, you studied a spring - a device that can temporarily store mechanical energy when stretched or compressed. A capacitor is a device that can temporarily store electrical energy when charged. A capacitor consists of two conducting objects separated by an insulator. When a potential difference is applied to the two sides of a capacitor, a positive charge builds up on the side at higher potential, and a negative charge of equal magnitude builds up on the side at lower potential. Under an applied potential difference $V$, charge builds on a capacitor until it reaches a level

$$q = CV$$ (16)

where $C$ is the capacitance of the capacitor, and $V$ is the applied potential difference. Charging a capacitor is like compressing an electrical spring - the charges on each side of the capacitor repel each other. If the potential difference were removed, they would push the system back toward its uncharged equilibrium state. The work done in charging a capacitor is stored as electric potential energy.

Figure 14: An RC circuit. With switch A closed and switch B open, the capacitor is charged through the resistor by the power source. With switch A open and switch B closed, the capacitor discharges through the resistor.

Consider the RC circuit shown in Figure 14. Imagine that switch A is closed (connected) and switch B is open. Then, charge will move around the circuit until the capacitor is fully charged (i.e. until $q=CV$). If switch A is opened at this point and switch B is closed, the capacitor will discharge through the resistor until there is no net charge on either of its plates. The amount of time it takes to charge and discharge a capacitor through a resistor depends on both $R$ and $C$. The potential difference between the conductors of a capacitor charged through a resistor by a power supply as a function of time $t$ is given by,

$$V_+ = \mathcal{E} (1 - e^{-t/\tau}),$$ (17)

where $\mathcal{E}$ is the emf of the power supply. As the capacitor discharges through the resistor, the potential difference between the conductors is given by

$$V_- = V_o e^{-t/\tau}.$$ (18)

where $V_o$ is the initial potential difference across the capacitor.

Figure 15: The potential difference across a capacitor (a) charging and (b) discharging through a resistor.

The constant $\tau$ has units of time is called the time constant of the RC circuit, given by

$$\tau = RC$$ (19)

This is the time it takes the charge on a discharging capacitor to reach a fraction $1/e$ of its initial charge, where $e = 2.718 \ldots$ is the base of the natural logarithm. It is also the time it takes a charging capacitor to reach a fraction $1 - 1/e$ of the applied potential difference. One way of measuring capacitance is to measure the time constant of the charging/discharging of a capacitor through a known resistor. You will measure capacitances using this method. However, in practice, it is easier to measure the ``half time'' - the time it takes the voltage across the capacitor to fall/rise to half of its initial/final value - than the time constant itself. The half time $t_{1/2}$ is related to the time constant $\tau$ via

$$t_{\frac{1}{2}} = \ln(2) \tau$$ (20)

The capacitance of a capacitor is dependent upon the geometry of the capacitor and the insulating material between the conductors. To get sense of how capacitance varies with various geometrical characteristics, we will study a homemade parallel plate capacitor consisting of two parallel plates of equal area $A$ separated by a distance $d$. Electromagnetic theory beyond the scope of this course predicts that the capacitance of a parallel plate capacitor is given by

$$C=\varepsilon_o \kappa \frac{A}{d}$$ (21)

where $\kappa$ is the dielectric constant of the insulating material between the plates. You will construct your own parallel plate capacitor by sandwiching pages in a textbook between two sheets of aluminum foil. You will investigate the dependence of capacitance on $A$ and $d$ and extract the dielectric constant of the textbook paper from your data.

Experiments

A Square-wave Voltage Source

Figure 16: (a) The square-wave setting of a function generator behaves like two voltage sources and a switch connected as shown in the dashed box. The switch flips with constant frequency between positions A and B. (b) A plot of the output voltage of the function generator vs. time.a

For both of the experiments described below, we will use the square wave form setting of a function generator to charge and discharge a capacitor through a resistor. A sketch of how one could mimic the function generator using two power sources and a switch is shown inside the dashed box in Figure 16 along with a plot of the output voltage of the function generator vs. time.

Figure 17: (a) A circuit in which a function generator with internal resistance $r$ and a square wave form repeatedly charges and discharges a capacitor $C$ through a total resistance $r + R$. (b) A qualitative plot of the potential difference across the capacitor vs. time.

You will use circuits like the one shown in Figure 17 to measure capacitance in the experiments described below. You will use the LabPro interface to collect the voltage across the capacitor vs. time. You will then be able to use Logger Pro to measure half times ($t_{1/2}$) in order to deduce capacitances.

Measuring Capacitance

1. The internal resistance of your function generator is $50 \pm 5 \Omega$. (This can't be measured directly with the DMM.)

2. Construct the circuit shown in Figure 17 leaving out the external resistor $R$. The internal resistance $r$ of the function generator is sufficient to give an easily measurable time constant.

3. Connect the voltage probe to Channel 1 of the LabPro interface.

4. Make sure that the interface is on and connected to your laptop prior to running Logger Pro. If all goes well, you should automatically get a graph of Potential vs. time along with a table for numerical values. If not, ask for help.

5. Connect the voltage probe in parallel with the capacitor.

6. After opening Logger Pro, click on Data -> Column Options -> Time. Under ``Displayed Precision'' on the ``Options'' tab, click on the ``Significant Figures'' radio button and change the number to 5. This will ensure that the data table and the Examine button will display time values with adequate precision.

7. Click on the Setup Collection button ( ) in the toolbar. Change sampling rate to 1000 samples/second and the experiment length to 2 seconds.

8. Turn on the function generator, and make sure that it is set to produce a square wave form on the 0.1-1 V scale.

9. Collect a voltage vs. time measurement using Logger Pro, and click on the autoscale button ( ). If the graph does not look something like Figure 17b with a peak-to-peak variation of several volts ...
   1. Check all of your connections to make sure that they are secure.

   2. Try changing your voltage probes to see if that works.a

   3. Ask for help.

10. Use the ``Examine'' button ( ) to measure the half time $t_{1/2}$ - the time it takes the voltage to drop from its maximum value to 0 V as shown in Figure 17b. Record it and its uncertainty.

11. Record the capacitance value written on the capacitor.

Homemade Capacitors

1. Use the DMM to precisely measure the resistance of your resistor. It should be in the $1-3 \mathrm{M}\Omega$ range. Ask for help if it isn't.

2. Use the Vernier caliper to measure the thickness of a large number of pages (at least 100) of the textbook. If you are unsure about how to read the caliper, ask for help. Calculate the thickness of one page and its uncertainty.

3. Place your two sheets of aluminum foil on either side of 20 pages in the book. Place them so that they overlap completely and a small amount of one edge of each sheet emerges from the book. Clip an alligator lead to each sheet without bringing the sheets or leads into (electrical) contact.

4. Set up the circuit shown in Figure 17 using your M$\Omega$ resistor for $R$. (For the small capacitances you will be measuring here, a large resistance is needed in order to slow down the charging/discharging enough to measure it with the LabPro interface.) Use a breadboard to mount the resistor and connect it with the capacitor. Use the plastic compression nuts to connect the wire leads to the function generator. When you connect the bottom lead of the capacitor to the black terminal of the function generator, leave enough bare wire exposed so that you can clip the black lead of the voltage probe to it.

   In Figure 17, the bottom end of the function generator corresponds to the black output terminal. According to the diagram, this terminal should be connected to one end of the capacitor. It is important to get this right, because this terminal is grounded, and so is the black lead of your voltage probe. If you switch the order of the capacitor and resistor in this circuit and try to measure the voltage across the capacitor, you will ground both ends of the resistor and destroy the effect you are trying to measure.

5. Click on the ``Setup Collection'' button ( ), and set the sampling rate to 50000 samples/second and the experiment length to 0.05 seconds.

6. Set the frequency of the function generator to about 50 Hz.

7. Apply pressure to the book so that your homemade capacitor is held firmly together while you collect a voltage vs. time graph. If the resulting graph looks like a saw-tooth wave form, the frequency is much too high. If it looks close to a square wave form, the the frequency is much too slow.

8. If necessary, adjust the frequency of the function generator and keep collecting, until you have a graph similar to Figure17b.

9. Use the ``Examine'' button ( ) to measure the half time $t_{1/2}$. Record it and its uncertainty.

10. Repeat this process for 30, 40, 50, and 60 pages. As you increase the number of pages, the time constant will decrease. You may want to adjust the frequency of the function generator accordingly.

    Caution! If you use page numbers to figure out how many pages you've got, remember that each piece of paper in the book corresponds to two numbered pages.

Analysis

Measuring Capacitance

Use Eqs. 19 and 20 to calculate the capacitance of the capacitor from your measured $t_{1/2}$ value and the given internal resistance of the function generator. Propagate the uncertainties in the time and resistance to determine the uncertainty in the capacitance.

Homemade Capacitors

1. Calculate the area of your capacitor and its uncertainty.

2. Determine the plate separation $d$ in meters of each of your homemade capacitors and the associated uncertainty.

3. Determine the capacitance of each of your homemade capacitors along with its uncertainty using the method you used with your ``store-bought'' capacitor. Remember that in these cases, both the M$\Omega$ resistor and the internal resistance of the function generator were connected in series with the capacitor.

4. Make a graph of $C$ vs. $1/d$ with vertical and horizontal error bars reflecting your uncertainties. If your graph appears to be compatible with a linear model, use the LINEST function to calculate the parameters of the best linear fit to your graph (see Appendix B).

5. Write the area of your capacitor and your 20-page capacitance, with uncertainties, on the board.

6. Record all of the 20-page capacitances and areas from your lab section, and put them in your spreadsheet.

7. Make a graph of $C$ vs. $A$ for all of the 20-page measurements from your lab section with vertical and horizontal error bars reflecting the uncertainties. If your graph appears to be compatible with a linear model, use the LINEST function to calculate the parameters of the best linear fit to your graph.

8. Devise a method of extracting, if possible, a best value of the dielectric constant of paper and its uncertainty taking all of your capacitance measurements into account.

Before You Leave Lab

Discuss with your instructor preliminary answers to the questions below.

Hand In ...

... your calculations, a printout of your spreadsheet, and answers to the following.

1. Give your result for the capacitance of the ``store-bought'' capacitor. Does your result agree with the value written on the capacitor within uncertainty?

2. Are your observations compatible with Eq. 21? (Your response should involve a discussion of your graphs of $C$ vs. $1/d$ and $C$ vs. $A$.)

3. Give your best value of the dielectric constant $\kappa$ of paper, and describe your method of extracting it from your data. If extracting a best value is not feasible, explain why not.

4. If you changed the aluminum foil to foil made of another metal, do you think it would affect your results? Explain.

Copyright © 2006-2009, L.A. Riley, T. J. Carroll, J.S. Scott

Updated Sun Apr 26 23:00:14 2009

This work is licensed under a Creative Commons License.