EC-3  Capacitors and RC-decay

OBJECTIVES:

I.

Experiment with a parallel-plate capacitor to understand the behavior of capacitors.

II.

Scrutinize how well the parallel-plate model describes real capacitors.

III.

Experiment with the charge and voltage on parallel and series capacitors.

Preliminary Questions:

1. A capacitor plate holds a given charge Q. Why is the voltage of this plate small when a grounded plate is near, but large (for the same Q) when it is alone?
2. If the charge Q on a capacitor is doubled, what is the change in the voltage across the capacitor? (Remember that Q is the magnitude of the charge on each plate, both positive and negative.)
3. How could you double the positive charge on one capacitor plate without changing the negative charge on the other plate?

APPARATUS:

Conventional equipment: Parallel plate capacitor; Pasco electrometer & power supply; commercial capacitors & resistors on circuit board; aluminum paddle; low capacitance lead, insulated cup and shield; coaxial lead & test probe; BNC to bannana plug adaptor; heat gun.

Computer equipment: Computer, monitor, keyboard, mouse; PASCO interface module; PASCO Voltage Sensor (a pair of leads that plug into PASCO input A).

PRECAUTIONS: These are needed for reliable measurements:

1. The output BNC from the electrometer is calibrated to give $5$V when the meter reads full scale. This means you have to convert the voltage you read from the output in order to get the correct numerical value for the voltage at the input.

2. The input BNC from the electrometer has the outer conductor (the shield) connected to ground (this is true for all BNC connectors unless indicated otherwise on the connector.) This means that you run the risk of discharging your capacitors when you make measurements. Be careful with the polarity when you make measurements.

3. Large static charges (common in dry weather) if applied to electrometer input may damage the sensitive input field effect transistor (``FET''). Minimize this possibility by keeping the electrometer input grounded via the SHORT switch during the initial hook up and when you are done with the experiment. In otherwords: Keep the Input switch in the ``Short before making connections'' position whenever there is nothing connected to the input or while you are making a connection. (In older models, this switch may be labled ``lock.'' It is the left position in either case.) After the connection is made, you put the switch in the Input position. If you need to get rid of charge that may have been collected, you can move the switch to either of the short positions, the ``momentary'' position usually being more convenient.

4. If your clothing or hair has a net charge, the electrometer reading may change if you move around. Hence, during a given measurement, change position as little as possible and ground yourself (e.g. touch the a convenient groung during a measurement. Note that grounding yourself is usually a bad idea when working with electronics, because of the danger of electrocution. Be careful not to touch any voltage sources while you are grounded.)

5. To remove all charge from the cup, switch the electrometer momentarily to the SHORT position. (This connects the electrometer terminals to each other so that any charge flows from/to ground). If the meter does not read zero, notify your instructor.

6. Always discharge paddles and cup before starting an experiment. To test if an object is charged, put it into the cup and see whether the electrometer deflects. Conductors discharge easily by touching them to a grounded conductor. To discharge an insulator, you must create sufficient ions in the surrounding air. The insulator will then attract ions of the opposite charge until all charge is neutralized. An open flame is a simple source of ionized air; the ions in the flame convect upward with the hot gas. To avoid damage to the insulator, keep it at least 10 cm above the flame!

INTRODUCTION:

A capacitor consists of two electrodes separated by an insulator. An electrode is just a piece of metal that can be connected to a voltage or current source. The capacitance $C$ is a number that quantifies how much charge $Q$ is required to hold an electrode at a potential difference $\Delta V$ from a second electrode, $Q = C\Delta V$. The second electrode is usually either an electrode with opposite charge ($-Q$) located some distance $d$ away, or an imaginary surface at potential $V = 0$ located infinitely far away. In Part I you will study a parallel plate capacitor consisting of two circular metal plates separated by air. In Part II you will study simple circuits and the charging behavior of capacitors using commercial capacitors. These are made from two long strips of aluminum foil separated from each other by a thin sheet of plastic, all rolled into a cylinder.

Part I: THE PARALLEL PLATE CAPACITOR

EXPERIMENT A: Potential Difference vs Separation (for fixed charge)

INTRODUCTION:

For a fixed charge the voltage of a conductor, i.e., the potential difference between that conductor and ground, depends on what bodies are nearby. If you charge a parallel plate capacitor and then increase the plate spacing-leaving Q unchanged-you will find that the potential difference increases.

Q1.1)

How do you reconcile this with the fact that Q = CV remains constant? (Two good approaches are either using what you have learned about C, or using what you have learned about $\vec{E}$ for capacitors and the integral $\int\vec{E}\cdot d\vec{l}$).

Q1.2)

Preliminary calculation: Assuming air has a dielectric constant $\kappa=1$ estimate how many excess electrons exist on one plate of the capacitor in front of you when V is set to 15 V and the plates are separated by 0.5 cm.

EXPERIMENT A: Potential Difference vs Separation for Fixed Charge on a Capacitor

1)

Referring to Fig. 1a. and b., connect the electrometer across the capacitor, but use the special low capacitance lead and a separate ground instead of the shielded coaxial cable.

Use the movable plate as the grounded one, and turn the apparatus so the fixed plate faces away from you. The ground removes any new excess charge that might accidentally come in contact with the plate, in order to keep the voltage the same. As a result, it provides some shielding of the system from charges on your hand or clothing. Also, with one plate at ground, you will only have to touch the other plate with the supply in order to charge the capacitor in step 2) below.

Figure 1a: Setting the supply and charging the plate

Figure 1b: Reading the voltage

2)

Start with an initial plate separation of $d = 0.5\,$cm. This is large enough to keep charge from leaking across the spacers on dry days. Use the 30 V scale and output of the PASCO DC power supply. Connect the negative terminal to the ground terminal of the electrometer. (Standby switch must be in proper position or no voltage results even though meter reads). Set the output of the DC power supply to 15 Volts using the Electrometer (not the meter on the supply; it is not as accurate. Watch out, the Electrometer can acquire and keep a voltage bias because of its very high input impedance...how do you avoid this before setting the supply output?) Charge the capacitor to 15 V by touching the appropriate plate with the positive voltage supply lead.

Now, change the plate spacing and observe the change of the voltage across the capacitor. Record your qualitative answer in your lab book, and then record readings from the electrometer meter of the voltage for different plate spacings. Does it make sense to make small changes to the plate separation $d$ when the plates are close? How about when they are far appart? Span 10cm in your measurements. Of course, zero on the cm scale will not be zero separation). Devise and report a way to make sure no charge has leaked off or been acquired by the plates during your measurements.

NOTE: In dry weather stray static charge on your body can adversely affect the charge on the parallel plate capacitor. Keep body movement to a minimum. There is an optional shielding screen which you may place in front of your body to minimize this effect. In addition there is an optional extension handle that attaches to the moving plate which will increase your arm to plate distance.

In humid weather the charge may leak too rapidly off the plate to get reasonableresults. Use the heat gun to gently warm up the parallel plate capacitor and eliminate some moisture.

3.)

Analysis:
Plot the voltage on the capacitor $V$ vs. the distance $d$ between the plates using the graphical analysis for windows software GAX. Click on ``title'' and enter your name, to avoid confusion at the printer. Double-click on the graph, go to ``column appearance,'' and change them to $V$ and $d$.

Q 1.3

Which parts of the plot are consistent with our model of a parallel plate capacitor, and which are not? Make sure only the graph is selected (highlighted), and print your plot by printing ``selected display.'' Show your answer on the plot.

Q 1.4

Explain the deviation from ideal behavior

Now, create two new columns in GAX from your existing data by making a column of 1/V and a column of 1/d. Plot 1/V vs. 1/d.

Q1.5

If your capacitor were behaving as a single ideal capacitor, your data would follow a curve $1/V = (\epsilon A/Q)\times 1/d$. Derive this formula.

Q1.6

This formula will pass through the origin when d gets large (1/d goes to zero). What do you think is causing your data not to go through the origin? (Hint: if the capacitance of your cables is important, then it adds in parallel to the capacitance of the plates: C(plates) + C(cables) = Q/V.)

Q1.7

Why do we want the lead to have low capactiance?

EXPERIMENT B: (OPTIONAL after completing PART II)

Surface Charge Distribution on a Parallel Plate Capacitor
(at fixed Potential Difference)

SUGGESTIONS:

1)

Ground yourself, the electrometer and the movable plate of the capacitor. Turn the parallel plate capacitor so that you are behind the movable plate. Set up electrometer and cup as in E1 but not close to the capacitor. (Why?)

2)

Connect 500 volts to the fixed plate. Do NOT apply this voltage to electrometer directly.

3)

Use the aluminum paddle (as in experiment E1) to probe the charge density on the capacitor's surfaces, and then use the electrometer and cup to measure the charge on the paddle. To avoid spurious effects from charges on the paddle's handle, touch the paddle to the bottom inside of the cup and remove the paddle before taking the reading.

4)

Record the relative charge density (sign and magnitude) on both the inner and outer surfaces of the two plates for three radial positions: center, halfway out, near edge of plate. Use plate separations of 2.5, 5, and 10 cm.

QUESTIONS for Experiment B:

1)

Why are measurements for separations $<$ 2.5 cm not very meaningful?

2)

How does relative charge density, $\sigma$, inside and outside the capacitor depend on plate spacing? On distance from center of the plate? Explain.

EXPERIMENT C: (OPTIONAL)

For a fixed spot inside the capacitor, find how $\sigma$ varies with voltage.

Part II:
CAPACITORS IN PARALLEL, IN SERIES, AND CONNECTED TO RESISTORS

SUGGESTIONS:

1)

Use the electrometer to test voltages in Part II: A and B experiments-review the Electrometer Precautions under the apparatus section at the beginning of the lab!. Use the lucite circuit board containing different capacitors and resistors. Although a push-button switch (plus connectors) permits applying 30 V momentarily to any capacitor, you may prefer just to touch the voltage supply leads directly to the capacitor being charged.

EXPERIMENT A: CAPACITORS IN SERIES

1) Discharge all capacitors first by momentarily shorting leads with a banana plug connector. 2) Connect 20 V across two capacitors in SERIES as the circuit and schematic at shows. (The schematic is only a suggested configuration.) 3) Calculate the potential difference expected across each capacitor. 4) Now measure the voltage across the individual capacitors. OPTIONAL: If that's too easy, try this: charge C$_1 = 0.5 \mu$F to 30 V; discharge C$_2 = 1.0 \mu$F and then connect 10 V across C$_1$ + C$_2$. Now measure the voltage across C$_1$ and across C$_2$. Is this consistent with the idea of charge conservation in the region between the two capacitors?

EXPERIMENT B: CAPACITORS IN PARALLEL

1) Discharge C$_2 = 1.0 \mu$F and remove from circuit. 2) Charge C$_1 = 0.5$F to 20 V and then disconnect the D.C. supply. 3) Measure voltage on C$_1$. 4) Connect C$_2$ in PARALLEL to C$_1$ and measure the final voltage across the capacitors (with the supply still disconnected.) 5) Calculate the voltage you would expect.

EXPERIMENT C: DISCHARGING OF CAPACITOR THROUGH A RESISTOR

In this experiment you will use the electroscope and computer interface to observe the discharging of a capacitor through a resistor. So far you have only observed the ``steady state'' or DC behavior of capacitors. Now you will charge the capacitor to a given voltage and then remove the voltage supply. Because the two plates of the capacitor are actually connected through the resistor, the charge on the capacitor will ``drain,'' or move to the opposite plate through the resistor. You will measure the voltage drop across the capacitor, which tells you the amount of charge $Q$ remaining on the capacitor. This is related to the flow of charge, or current $I$ by the following relations as a function of time $t$ during the discharge:

$$\frac{V(t)}{R} = I =-\frac{dQ(t)}{dt} \mbox{   and   } Q(t)=C V(t).$$

Solving the above equation gives the relation: $V=V_0 e^{-t/\tau}$ where $\tau \equiv RC$ is the time constant for a particular circuit. $R$ is the value of the resistor, which is measured in Ohms, or $\Omega$. Note that the input resistance of the Electrometer is much larger than the values of $R$ you will be using, so the charge flow into the electrometer is negligable.

SUGGESTED PROCEDURE:

1)

Connect the $10^{7} \Omega$ resistor (marked as $10 Mohm$ or 10 M$\Omega$) across the large capacitor ($1.0 \mu F$) and set up the electrometer to measure the voltage on the capacitor. Fig. 4 shows the nominal circuit configuration and a possible wiring diagram for the circuit. Depressing the switch connects the power supply to the circuit, which will rapidly charge the capacitor, and releasing the switch will initiate the discharge.

Figure 4

2)

Connect 30 V across the capacitor and observe the voltage V$_{C}$ on the capacitor as a function of time after you disconnect the 30 volts and switch in the resistance $R$. Qualitatively describe the discharge behavior observed by watching electrometer display.

3)

After making sure the PASCO interface is connected to the electrometer output (through Channel A), CLICK on the ``Launch RC III'' icon below to initiate the PASCO interface software. There should be a panel, a table and a graphing display for V vs t.

4)

CLICK the REC button, charge the capacitor, and record data while the capacitor is discharging. CLICK on the STOP button. Record six (or so) evenly spaced data points into your lab book.

5)

Move the leads to the $1 M\Omega$ resistor and repeat the experiment. (Configure the interface to show multiple data sets.)

6)

Move the leads to the $100 k\Omega$ resistor and repeat the experiment.

QUESTIONS:

1)

Do the curves have the expected functional behavior?

2)

By moving the curser over the inital voltage and time, and then over the voltage at $1/e$ of the inital value, compare the nominal product $R \times C$ to the time required for the voltage to drop to $1/e$ of the initial value ( $e = 2.72 \ldots$). You may need to adjust the sampling rate to check this for the 100$$k$\Omega$ resistor.