Workshop Physics 111-112: Unit 14: Circuits and Capacitance Unit Author: John Luetzelschwab V 5.0 --

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INSTRUCTOR UNIT 14: CIRCUITS and CAPACITANCE

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Objectives:

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1. To understand the relationship between charge, voltage, and capacitance. 2. To work with capacitors connected in parallel and in series. 3. To understand the relation between charge and voltage for combinations of capacitors.

Workshop Physics 111-112: Unit 14: Circuits and Capacitance Unit Author: John Luetzelschwab V 5.0 --

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DAY 1{Fri} Hand in Homework Assignment 1. Chapter 18 problems 39 and 43 2. Turn in Activity Guide Unit 13 (pp. 1-11) Discussion of Homework Assignment (5 minutes) Charging and Discharging a Capacitor (45 minutes)

+

+ -

S

V -

- - - - - - - - - - - - -

In the section on batteries and bulbs, you observed the discharging of a capacitor. You will repeat the process, but now will do some quantitative work. Connect the power supply, 0-10V voltmeter, 0-1mA ammeter, 10kΩ resistor, switch, and capacitor as shown in Figure 14-1. The dashed line on the switch indicates that the two switches are connected: use adjacent sides of the double-pole-double-throw switch. Set the power supply voltage at 10V (as measured by the voltmeter). The positive and negative sides of the capacitor must be connected as shown, if they are reversed, the capacitors do not work properly. The positive side of the capacitor has a small indentation in it. Close the switch. If everything is connected properly, you should read 1mA on the ammeter. When you are ready, open the switch and record the current as a function of time. Replace the capacitor with the other capacitor at your table and record the current as a function of time. +

A

+ C

-

R

-

Figure 14-1 DATA: Capacitor #1 (listed value:_______) Capacitor#2 (listed value:_______) Time Current Time Current

Workshop Physics 111-112: Unit 14: Circuits and Capacitance Unit Author: John Luetzelschwab V 5.0 -June 16, 2002

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GE14-1: Where is the Charge Going? From your knowledge of resistors and capacitors, answer the following questions. 1. If you have a capacitor with a fixed capacitance C and some charge on it, what determines the voltage across this capacitor? (Assume that no external source of voltage is connected to the capacitor; it just has the charge.)

2. What determines the voltage across any resistor R?

3. In the circuit of Figure 14-1, what is the relation between the voltage across C and the voltage across R?

4. After the switch is opened, what two quantities are decreasing in the capacitor?

5. How do these decreasing quantities affect the voltage across R, and therefore, how do they affect the current in the circuit? Does this conclusion agree with what you observed?

------------------------------------------------------------------------------------------The actual equations for the charge on the capacitor and the current in the circuit are: Q = Qo e-t/RC = Qo e-t/τ

and

I = Io e-t/RC = Io e-t/τ

where Qo = the initial charge on the capacitor, Io = the initial current through the circuit (= Vo /R = Qo /RC ), and τ = RC = the time constant for the RC circuit. These are the equations for an exponential decrease as shown in Figure 14-2.

Workshop Physics 111-112: Unit 14: Circuits and Capacitance Unit Author: John Luetzelschwab V 5.0 -June 16, 2002

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Ιo

-t/RC

Ι = Ιo e

.37 Ιo - - - - -

- - - - -

Ι

τ = RC

t

Figure 14-2 The current reaches a value of Io e-1 = 0.37Io , when t = τ = RC. Therefore, you can calculate the value of C by finding the time at which the current has a value of 0.37 of its initial value. -------------------------------------------------------------------------------------------

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GE14-2: Checking the Value of Your Capacitor Using your current data, calculate the value of C for your capacitors. You do not need to plot the data, just interpolate if needed to get a rough value for the time the current reaches 0.37Io . (Note: keep your data (you will turn in your activity guide before you use this data). You will plot it for the capacitor laboratory.) The nominal capacitance value listed on the capacitor is generally only accurate to within 50% (the actual value is generally higher than the listed nominal value), so expect your calculated value to differ from the listed value.

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Workshop Physics 111-112: Unit 14: Circuits and Capacitance Unit Author: John Luetzelschwab V 5.0 --

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Capacitors in Parallel and Series (50 minutes) For the next exercise, you will replace the single capacitor in Figure 14-1 with two capacitors, first connected in parallel and then in series. But first, let's do some predicting of what should happen. -------------------------------------------------------------------------------------------

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GE 14-3: Parallel and Series Capacitors 1. Draw a circuit diagram for two capacitors in parallel connected across a resistor. Put a switch in the circuit.

2. If the capacitors are charged and then the switch is opened, predict if the capacitors will discharge faster or slower than one by itself? Explain. Do not use equations, give a physical argument.

3. Draw a circuit diagram for two capacitors in series connected across a resistor. Put a switch in the circuit.

Workshop Physics 111-112: Unit 14: Circuits and Capacitance Unit Author: John Luetzelschwab V 5.0 -June 16, 2002 GE 14-3 continued

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4. If the capacitors are charged and then the switch is opened, predict if the capacitors will discharge faster or slower than one by itself? Explain. For this situation, think about the voltage across each, the charge on the two plates that are connected together, and therefore the relation between the charge on each capacitor. Again, do not use equations, just physical arguments.

5. Now you can check your predictions. Replace the single capacitor in Figure 141 with two in parallel then close and open the switch. Record the current as a function of time (take good date - you will use this data in your capacitance lab on Monday); be sure to get data down to 0.2 mA. Repeat for the two capacitors in series. How do your results compare with your predictions? If you predicted wrong, give the correct explanation.

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Sample Homework Problems (10 minutes) Discuss the charging and discharging of capacitors.

Homework Assignment #15 •Read section 18.3 in your text •Work Chapter 18 problem 49, 50 and 51

Workshop Physics 111-112: Unit 14: Circuits and Capacitance Unit Author: John Luetzelschwab V 5.0 --

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Day 2{Mon} Hand in Homework Assignment Chapter 18 problems 49, 50 and 51 Discussion of Capacitors in Parallel and in Series; Problems (25 minutes) We will discuss your observations for the parallel and series capacitors and your instructor will work some problems. NOTES: Go over the voltage and charge on two parallel and series capacitors and work a problem on a combination of parallel-series capacitors.

Homework Assignment #16 •Reread section 16.5 in your text •Work Chapter 16 problem 59 and 62 (due Friday) Review for Exam II (85 minutes)

Day 3{Wed} Test II (110 minutes) (Activity Guide Units 12 and 13)

Workshop Physics 111-112: Unit 14: Circuits and Capacitance Unit Author: John Luetzelschwab V 5.0 --

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Day 4{Fri} Hand in Homework Assignment Chapter 16 problems 59 and 62 Discussion of Exam and Homework (15 minutes) Capacitance Laboratory (95 minutes) Objective:

To observe the properties of a capacitor To understand the principles of charge on a capacitor and its relation to voltage Equipment: Two capacitors, switches, charging voltage, digital voltmeter This laboratory is similar to the Electrostatics Laboratory; you will charge capacitors in various configurations and observe the results. You will need to explain what you observed. Therefore, take careful notes of what you do, how you do it, and what happens. Quantitative information is needed, so take careful measurements. 1. You have the data for this part: gather your data from the discharging of the individual capacitors and of the parallel and series capacitors. 2. Connect your two capacitors as shown in Figure 14-3. The switches should be open. (a) Charge up each capacitor with 10 volts from the voltage supply (simultaneously touch the wire from the positive voltage to the + side of the capacitor and the wire from the negative voltage to the - side ). Remove the wires, then close switch S2 and then close switch S1 and observe the maximum voltage on the DVM. (The value you read will probably be slightly less than the actual voltage on the capacitors: the meter takes a little time to read the voltage, time during which the capacitor can discharge slightly.) S1

S2 + -

+ C1

-

C2

V

Figure 14-3 (b) After the capacitors have discharged (you can help this along by shorting out the voltmeter), open the switches and charge up C1 only. Close the switches S2 then S1 and record the initial voltage.

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(c) Repeat part (b), except charge up C2 only. (d) Reverse the connection of the smaller of the two capacitors (i.e. put the+ side of the capacitor at the bottom of Figure 14-3). With both switches open, charge each with 10V according the polarity of each capacitor (i.e one is charged opposite to the other). Then close switch S2 then switch S1. Record the initial votage. 3. Connect the two capacitors as shown in Figure 14-4. S1

(a) + (b)

C1

-

V

S2 (c)

+ -

C2

(d)

Figure 14-4 (a) Close switch S2 and charge up both C1 and C2 in series (i.e. place the charging leads across points (a) and (d); positive at (a), negative at (d)). Open S2, then close S1 and read the initial voltage across C1. (b) Short out each capacitor, connect the voltmeter and switch across C2 ( or just exchange C1 and C2) and repeat part (a) and measure the initial voltage across C2. 4. Connect the circuit shown in Figure 14-5(a) and set the power supply to 10 V. S

Power supply

S +

10kΩ

+

Power supply

+ -

C

5kΩ

+ -

+ -

(b)

Figure 14-5

1

C

2

10 kΩ

(a)

C

5kΩ

Workshop Physics 111-112: Unit 14: Circuits and Capacitance Unit Author: John Luetzelschwab V 5.0 --

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4(continued). Connect your DVM across the top resistor and close the switch S. Record the intial and final (after the voltage stops changing) voltage across the resistor. Open the switch, discharge the capacitor, connect the DVM across the other resistor, close S and record the initial and final voltage across the resistor. Repeat this process for the capacitor. Do the same for each resistor and capacitor in the circuit shown in Figure 14-5(b). Analysis 1. On Cricketgraph plot the I vs time data you took for each capacitor alone, the two in series and the two in parallel. Fit each curve with an exponential function and find the time constant and the value of C; compare with what you would expect theoretically. When you plot the graphs and fit them with an exponential curve, the fit should give an initial value of about 1.0 mA. If the fit does not give an initial value close to 1.0, look at your data and if a point or two are throwing the curve off, delete those points. Cricketgraph fits the curve with a log10 curve, so you will need to convert from log 10 to natural logs; the equation will be in the form 10-bx, and you want e-ax. To make the conversion (i.e. to find a if you are given b): a = 2.3b. Once you have found a (= 1/RC), you can find C. For example, if the curve equation is 0.985*10^-0.065x, then b = 0.065 and a = (2.3)(0.065) = 0.150. The values you find for the individual capacitors will probably differ by 20 - 50% from the listed values: the listed values are only nominal values. Use the values you found for the individual capacitors as the "correct" values and use these values to calculate the expected equivalent capacitance of the two in parallel and in series and for any quantitative explanations below. 2. Explain what you observed in procedure parts 2, 3 and 4. You need to use the values of C found earlier, then develop equations and calculate values to verify what you measured. You therefore need both words and numbers. For your explanation, remember that if a resistor has voltage across it then a current must be flowing through it and if a capacitor has voltage across it, then a charge must be on it. Oral Laboratory Report: Make an appointment for the week of March 16 - 20 Homework Assignment •No homework assignment; work on your lab.