Lab 4: Capacitance



Lab 4: Capacitance

Introduction

Doing some simple experiments, including making and measuring your own capacitor, will help you better understand the phenomenon of capacitance. In this lab, you will use a commercially available demonstration capacitor to investigate the basic principle of capacitance, expressed in the equation: C = q/V, where C is the capacitance of some system of conductors and insulators,

q is the charge associated with the system, and V represents the potential difference between the parts of the system.

Objectives

To verify the predictions of the capacitance equation

To make and measure the capacitance of a simple parallel plate capacitor

Equipment

Electrometer, electrostatics voltage source, insulated sphere, proof plane, and demonstration capacitor. The electrometer, the voltage source, insulated sphere and proof plane are the same devices that you used in the electrostatics lab. Be sure to review the operation of the electrometer and the voltage source before you proceed with this experiment.

The demonstration capacitor consists of two conductive discs, approximately 18 cm in diameter, mounted on a base. One disc is fixed to the base, the other disc is attached to a support which can be moved to change the spacing between the discs. Terminals are provided so that electrical connections can be made to the discs.

1. Investigating the capacitance equation

1a. Potential difference proportional to charge

The capacitance equation, C = q/V, seems to imply that the capacitance of a device depends on the change and the potential difference between the plates. But, in fact, the capacitance of a device depends only on its geometry and the insulating material between the plates. Rearranging the equation so that it reads: q = CV says that for a given capacitor the potential difference between the two plates of the capacitor will be directly proportional to the charge on the capacitor. That is, doubling the charge should double the potential difference, tripling the charge should triple the potential difference, and so forth. Let’s see if the demonstration capacitor obeys this prediction.

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Set up the apparatus as follows:

1. Adjust the demonstration capacitor so that the two plates are approximately 2.0 mm apart.. This can be most easily done using the scale printed on the base of the capacitor.

2. Connect the electrometer to the capacitor. Connect the black wire from the electrometer to the fixed plate of the capacitor and the red wire to the movable plate. Zero the electrometer (refer to the instructions in lab 1 if you don’t remember how) and select the 30 volt range on the function switch.

3. Connect the insulated sphere to the 1000 volt terminal of the electrostatics voltage source and connect the “com” terminal on the voltage source to the “gnd” terminal on the electrometer as you did in lab 1. Make sure that the insulated sphere is as far away from the electrometer and the capacitor as possible in order to avoid erroneous readings due to the electric field of the sphere. Now turn on the voltage source.

4. Re-zero the electrometer if necessary, charge the proof plane by touching it to the sphere, and transfer the charge to the movable plate of the capacitor. You only need to touch the proof plane to the edge of the plate, but be careful that you only touch the movable plate. Do not let the proof plane touch the fixed plate at any time. Note the voltage reading on the electrometer and record it on your worksheet. In the same way, transfer a second “unit” of charge and record the voltage reading. Do this five or six times.

Note: Depending on atmospheric conditions and other factors in the lab it is possible that the capacitor will “leak”. That is, some of the charge will not stay on the capacitor but will be conducted off by moisture in the air or dirt or other contaminants on the supporting structure of the capacitor. If this happens, the potential difference across the plates of the capacitor will immediately begin to decrease after the capacitor has been charged. Should this occur, you will have to transfer the charges rapidly and make quick voltage readings in order to minimize the effects of this leakage on your experiment.

Now let’s analyze your data and see if it supports the prediction. Make a graph of q (measured, for the moment, in “arbitrary units”) vs. V, plotting q on the vertical axis and potential difference horizontally. If this graph is reasonably straight (allowing, if necessary, for the effects of capacitor “leakage”) then you can conclude that the prediction of the capacitance equation has been verified. If the graph is not straight you need to think about whether that is due to capacitor leakage, or some other, unforseen, factor affecting what is going on. Specifically, what shape would you expect the graph to have if the capacitor were leaking as you transferred charge? On the worksheet, make a sketch of what you would expect under these circumstances and explain why that is the case.

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1b. Estimating the charge of one “arbitrary unit”

If your graph is reasonably straight, its slope should be equal to the capacitance of the capacitor. But you can calculate this capacitance. If the plates are not too far apart, the demonstration capacitor can be correctly modeled as a parallel plate capacitor, which obeys the equation:

C = (εo A)/d

Use this equation to calculate the capacitance of the demonstration capacitor. Show your work on the worksheet.

Now, using your calculated value of capacitance, and your graph of q vs V, calculate the charge (in coulombs) of one “arbitrary unit” of charge. In other words, you are calculating the actual amount of charge that the proof plane is picking up from the insulated sphere each time you touch the proof plane to the sphere. Show your work on the worksheet.

1c. Potential difference inversely proportional to capacitance

Rearranging the capacitance equation so that it reads: V = q/C gives us the second prediction. For a fixed value of charge, increasing the capacitance of the capacitor should decrease the potential difference across the plates, and vice-versa. To investigate this prediction, give the capacitor (which still has its plates about 2 mm apart) one or two “units” of charge and note the potential difference. (Again note that if the capacitor is “leaking” you will have to proceed fairly quickly in order to minimize the effects of losing charge.) Now move the movable plate farther away, say, to about 6.0 mm, and note what happens to the potential difference. Try moving the plate back and forth as you watch the electrometer reading. Does increasing plate separation increase or decrease the potential difference? Is this consistent with the prediction of the capacitance equation? Record your observations and conclusions on the worksheet.

1. Making and measuring a capacitor

Let’s now make a simple parallel-plate capacitor and use a basic property of capacitor networks to measure its capacitance. To make the capacitor, proceed as follows:

1. From a roll of aluminum foil, cut a piece about 15 cm long.

2. Cut this piece into two equal parts as shown in figure A on the next page. Discard the shaded pieces. The goal is to make two approximately square pieces with tabs on them which look like figure B.

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[pic]

3. Inspect them for any rough or sharp edges which might puncture the thin plastic that you will put between them. Smooth any rough edges with your finger.

4. Place a large sheet of mylar plastic on the worktable and tape it down so it won’t move. Wipe the plastic with a damp (not wet!) paper towel to rid it of any dirt or stray charge which might affect the results of your experiment. Make sure there is no stray moisture left on the plastic.

5. Place one of your foil pieces in the center of this sheet of plastic.

6. Take a small sheet of mylar, clean it on both sides with a damp paper towel, and place it on top of the foil piece. Make sure the mylar completely covers the foil piece except for about two or three centimeters of the tab.

7. Place the other foil piece on top of the mylar, with the tab on the opposite side from the first tab.

8. Clean a third sheet of mylar and place it on top of the second piece of foil, making sure that only the two tabs protrude from under this sheet. Tape the sheet down so it doesn’t move.

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9. When finished, the stack should look like the figure below:

[pic]

10. Now place a book on top of the stack, leaving the tabs exposed so you can make connections to them.

Connect the electrometer to your capacitor. Be careful not to tear the tabs. Now connect two wire leads to a 6 Volt battery and charge your capacitor by briefly touching the leads from the battery to the tabs of the capacitor. Watch the electrometer to see whether the capacitor “leaks”. If there is little or no voltage drop after about 30 seconds or so, then the capacitor is holding charge satisfactorily. Consult your instructor if this is not the case. Now discharge your capacitor by simultaneously touching both tabs with your fingers.

You are now ready to measure the capacitance of your capacitor. At your lab station, there should be a small (2.0 nF) capacitor mounted on a plastic carrier. Connect one terminal of this capacitor to the negative tab of your capacitor. Connect another wire to the other terminal of the 2 nF capacitor, but do not yet connect it to your capacitor. We’ll call this the ”test wire”. At this point, the circuit should look like the figure below:

[pic]

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To measure the capacitance of your capacitor, proceed as follows:

1. Discharge both capacitors by briefly touching their terminals with your fingers.

2. Charge your capacitor by briefly touching its terminals with the leads from the 6 Volt battery. (Remember to touch the negative battery lead to the negative tab of your capacitor.) Note and record the electrometer reading. We’ll call this reading Vbefore.

3. Now briefly touch the test wire to the positive tab of your capacitor. The electrometer reading should drop to a new, lower value. Note and record this reading. We’ll call it Vafter.

4. Repeat the above procedure four or five times.

Consult the appendix to learn how to use these data to calculate the capacitance of your capacitor. Show your work and record your results on the worksheet.

Questions for speculation

Take a look at the commercially available capacitor on the instructor’s desk. It has a value of

0.47 μF.

1. This capacitor is also made of plastic and foil, in much the same way as you made yours. Assuming that the plastic in this capacitor is the same thickness as the plastic you used to make yours (it’s not, but assume that), what would have to be the plate area of this capacitor in order for it to have the capacitance it has? If these plates were square, what would be the length of one side?

2. In view of your answers to question 1, how do you think the manufacturers managed to make their capacitor as small as they did?

Answer these questions in the space provided on the worksheet. Use extra sheets of paper if necessary.

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Appendix

Measuring capacitance by the method of shared charge.

When two capacitors are connected in parallel, as shown in the figure, they act as a single capacitor which has a capacitance equal to the sum of the individual capacitances.

Ctotal = C1 + C2

[pic]

If a charged capacitor, C1, having an unknown value, is connected in parallel with an uncharged capacitor, C2, whose value is known, some of the charge from C1 is transferred to C2. As this transfer occurs, the potential difference across C1 decreases and the potential difference across C2 increases until they arrive at a common final value, which we can call Vafter. (Note that this process takes place almost instantly.) The total charge of the system remains the same, it has just been redistributed. In equation form we would write:

Vbefore = Q/C1 (call this equation 1) and: Vafter = Q/(C1 + C2) (call this equation 2). Note that both Q and C1 are unknown quantities in this system of equations. However, if we divide equation 1 by equation 2 we can eliminate Q between them and arrive at a single equation involving only one unknown, C1. This equation can then be solved for C1 in terms of the known quantities C2, Vbefore, and Vafter.

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Capacitance

Worksheet

Name_________________________

Name_________________________

1. Investigating the capacitance equation

1a. Potential difference proportional to charge

Charge (arbitrary units) Potential Difference (Volts)

0 0

1

2

3

4

5

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Make your graph using “Graphical Analysis” or by any other convenient method. Attach your graph to your completed lab report.

In the space below, sketch and explain the graph you would expect if the capacitor were “leaking:

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1b. Estimating the charge of one “arbitrary unit”

In the space below, show your calculation of the capacitance of the demonstration capacitor.

In the space below, show your calculation of the charge of one “arbitrary unit”.

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1c. Potential difference inversely proportional to capacitance

Record your observations and conclusions below.

2. Making and measuring a capacitor

Vbefore Vafter

_________________ ________________

_________________ ________________

_________________ ________________

_________________ ________________

_________________ ________________

Use your data to calculate the capacitance of your capacitor. (Note: Remember that the average of several experimental values is generally more reliable than any individual value.) Show your work below.

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Questions for speculation

Answer questions 1 and 2 below. Use the back of this page if you need additional space.

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