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EC-3  Capacitors and RC-decay

OBJECTIVES:

I.
Use the text-book model of capacitance to understand measurements of the behaviors of capacitors.
II.
Scrutinize how well this model describes real 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; digital multimeter (DMM); heat gun.

Computer equipment: Computer, monitor, keyboard, mouse; PASCO interface module; PASCO voltage test leads (plug into PASCO input A).



PRECAUTION: Remember electrometer precaution on page [*].


INTRODUCTION:

A capacitor consists of two electrodes separated by an insulator. An electrode is just a fancy word for a piece of metal that can be connected to a voltage or current source. In part I you will study a parallel plate capacitor consisting of two circular metal plates held on lucite insulators. 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 plastic foil, all rolled into a cylinder.

Part I: THE PARALLEL PLATE CAPACITOR

Introduction:

For a fixed charge the potential of a conductor 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 electrons exist on one plate of the capacitorin front of you when V is set to 30 V and the plates are separated by 1cm.

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

1)

Referring to Fig. 1, 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, thereby shielding the other plate from charges on your hand or clothing. Because this plate is at ground, you will only have to touch the other plate in order to charge the capacitor in step 2) below. This also reduces the effects of your presence in the configuration of conductors.

Figure 1
\includegraphics[height=3.2in]{figs/e3-01.eps}

2)
Start with the minimum plate separation. This is set by the spacers on the fixed plate, which are 1 mm thick. Use the 30 V scale and output of the PASCCO 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 20 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 it's very high input impedance...how do you avoid this before setting the supply output?) Charge the capacitor to 20 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 of the voltage for different plate spacings. Does it make sense to make small changes to 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 reasonable results. Use the heat gun to gently warm up the parallel plate capacitor and eliminate some moisture.

3.)
Analysis:
Q 1.3.
How do you expect the voltage between a parallel plate capacitor with a fixed charge to change with plate separation? (Use a formula.)
3.a)
Plot the voltage on the capacitor vs. the distance d between the plates using the graphical analysis for windows software GAX. Double-click on the axis labels on the graph and change them to V and d.
Q 1.4
Which part of the plot is consistent with a parallel plate capacitor, and which is not? Print your plot and show your answer on the plot.
Q 1.5
Is the deviation from ideal behavior what you expect from what you know about capacitance? (Hint: a conductor has a lower capacitance when it is solitary than when a second conductor is nearby, and V = Q/C, so should you expect V to deviate above or below the value predicted by this equation?)

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.

Q1.6
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.

Q1.7
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.)

Q1.8
Why do we need the low capacitance lead?)


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. (If properly configured the electrometer switch must be in the proper position or no voltage results!)
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 insulator, touch the paddle to the bottom 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 digital multimeter to test voltages in Part II: A and B experiments. 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 left 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$_1$ to 30 V; discharge C$_2$ 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?

\includegraphics[height=3.45in]{figs/e3-2.eps}

EXPERIMENT B: CAPACITORS IN PARALLEL

1)
Discharge C$_2$.
2)
Next charge C$_1$ to 20 V and
3)
Measure voltage on C$_1$.
4)
Disconnect D.C. supply,
5)
Connect C$_2$ in PARALLEL to C$_1$ and measure the final voltage across the capacitors.
6)
Calculate the voltage you would expect.

\includegraphics[height=2.in]{figs/e3-3.eps}

EXPERIMENT C: DISCHARGING OF CAPACITOR THROUGH A RESISTOR

In this experiment you will use the electroscope and computer interface to observe discharging of a capacitor through a resistor. So far you have only observed the ``steady state'' behavior. In this case, after charging the capacitor, the transient flow of charge (current) through the resistor will be monitored by measuring the voltage drop across the capacitor.

The basic relationships (as a function of time $t$ during the discharge) are:

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

which upon solving gives the relation: $V=V_0 e^{-t/\tau}$ where $\tau \equiv RC $ is the time constant for a particular circuit. latex

SUGGESTED PROCEDURE:

1)
Connect the $10^{7} ohm$ 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. Figure 4 show the nominal circuit configuration and a possible wiring diagram while will achieve this desired circuit. Depressing the switch will rapidly charge the capacitor and releasing the switch will initiate the discharge.

Figure 4
\includegraphics[height=3.2in]{figs/e3-04.eps}

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. (The interface should be configured to shown 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 function 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 product $R \times C$ to the time required for the voltage to drop to $1/e$ of the initial value ( $e = 2.72 \ldots$).


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Next: About this document ... Up: Physics 201/202  &  207/208 Lab Manual Previous: Physics 201/202  &  207/208 Lab Manual
Physics Laboratory 2000-09-28