Name: Lab Partner(s): Date lab performed:

Dr. Julie J. Nazareth Physics 123L Section:

Capacitors Parts A & B: Measurement of capacitance – single, series, and parallel combinations Table 1: Voltage and capacitance for individual capacitors and capacitors in series and parallel Power Supply Voltage, Vps (volts) Average Known Measured Measured Experimental Known Capacitor, Voltage, Voltage, Capacitance, Capacitance, Percent C = 10 µF Vm (volts) V (volts) Cexp ( ) (µF) Difference Capacitor, C1

C1 = 22

Capacitor, C2

C2 = 47 Cpcalc =

Capacitors in Parallel

Cscalc =

Capacitors in Series

Calculations: Using equation 3, calculate the experimental capacitance of the unknown capacitor for C1, C2, C1 and C2 in parallel, and C1 and C2 in series. Calculate the percent difference between the known capacitance values and your experimental values. SHOW THE CALCULATIONS for C1 (the one marked 22 µF), Cpcalc, and Cscalc. THIS MEANS SHOW WHAT NUMBERS YOU PUT INTO THE FORMULA, AS WELL AS THE DECIMAL ANSWER WITH UNITS. Use the average measured voltage for V in equation 3. Also, use your experimental capacitance values for C1 and C2 to determine the known (calculated) capacitance for the capacitors in series and capacitors in parallel. Don’t forget units! Experimental capacitance and percent difference for capacitor C1 (the one marked 22 µF) ⎡V ps ⎤ C1 exp = C ⎢ − 1⎥ = ⎣ V ⎦ C1 % diff. = [(C1exp – 22 µF)/(22 µF)] x 100% = Calculating the “known” capacitance for series and parallel capacitors Cpcalc = C1exp + C2exp = Cscalc =

C1 exp C 2 exp C1 exp + C 2 exp

Lab: Capacitors

=

Updated 04/18/15

Data & Reporting score:

Part C: Measurement of internal resistance, R Table 2: Time for a RC circuit to fall to ½ maximum voltage One half Maximum maximum Time for Voltage, voltage, V = ½ Vmax, Discharge Trial Vmax (volts) ½Vmax (volts) t1/2 ( ) 1 (10 µF)

Experimental Capacitance, C (µF) 10

Internal Resistance of Voltmeter, Rint ( )

2 (22 µF) 3 (47 or 50 µF) Calculations: Using equation 6, calculate the in internal resistance of the voltmeter for the three trials. SHOW THE CALCULATION for the first trial. THIS MEANS SHOW WHAT NUMBERS YOU PUT INTO THE FORMULA, AS WELL AS THE DECIMAL ANSWER WITH UNITS. [Remember: micro-Farads, µF = 10-6 F, Farads.] Use units of resistance for your final answer, NOT seconds and microfarads. t Trial 1: Rint = 1 / 2 = (C ln 2) Without numbers, substitute/cancel units to SHOW how seconds/Farads equals Ohms (s/F = Ω). Some of the following relationships may be helpful. You won’t need to use all of them. Electrical charge (Coulombs): C = A s Electric Field Strength: N/C Voltage (volts): V = J/C Resistance (Ohms): Ω = V/A Capacitance (Farads): F = C/V Electrical current (Amperes): A = C/s Part D: Measurement of a time constant ***You must set the power supply so that the voltmeter reads exactly 4.0 volts before the switch is opened and the capacitor begins to discharge. Use the 22 µF capacitor !!!!! *** Table 3: Time for the voltage to fall to specified amounts for an RC circuit Voltage, V (volts) 4.00 3.75 3.50 3.25 3.00 2.75 natural log voltage, 1.39 1.32 1.25 1.18 1.10 1.01 ln V Time, t (s)

2.25

0.916

0.811

0

Voltage, V (volts) 2.00 natural log voltage, 0.693 ln V

1.75

1.50

1.25

1.00

0.75

0.50

0.560

0.405

0.223

0

-0.288

-0.693

Time, t (s)

Lab: Capacitors

2.50

Updated 04/18/15

Graph: Plot the natural logarithm of the voltage versus time (lnV(t) vs. t). Spread the data out use most of the sheet of graph paper. Draw a best-fit straight line to your data points and calculate the slope. SHOW YOUR SLOPE CALCULATION ON YOUR GRAPH PAPER IN AN UNUSED PORTION OF THE PAPER. Draw a small box around the points (not data points) you used to calculate the slope. Don’t forget to title and label your plot appropriately. Note: the natural log of the voltage, ln V(t), has no units, but the time, t, does. One graph per lab group is allowed if the graph is completed, slope calculated and recorded on data sheets, and the graph is signed off by the instructor by the end of the lab period. Calculation: Using all three Rint values from Table 2, determine a sample uncertainty for the internal resistance of the voltmeter. Be sure to round your final answer properly and include units. Rave = ( ±

)

Calculation: SHOW the calculation of the “theoretical” time constant, using the average resistance from the previous calculation, including uncertainty, and the experimental capacitance of the capacitor used in the Part D procedure. Be sure to round your final answer properly and include units. Your final answer units should be units of time, NOT Ohms and Farads. [Remember: micro-Farads, µF = 10-6 F, Farads.] τtheo = RaveC = ( =( ±

±

)

(

)

=

±

)

Use the slope of the graph to determine the experimental time constant. Record the results in Table 4. You can just do the experimental time constant calculation in your calculator – you don’t have to write it out for your report. Table 4: Comparing experimental and theoretical results for the time constant, τ Slope Experimental time constant, Theoretical time constant, ( ) τexp ( ) τtheo ( ) ± Calculations: Verify equation 4 directly from your data twice, once using your experimental time constant, τexp, and the other time, using the “theoretical” time constant, τtheo. Use the time you recorded in Table 3 for 3.50 volts. SHOW BOTH CALCULATIONS. Don’t forget your units! Time from Table 3 at 3.50 volts: t = ______________ seconds Vmax = 4.00 volts (if you followed directions) Verify eq. 4 using τexp V(t) = Vmax e–t/ τexp = Verify eq. 4 using τtheo V(t) = Vmax e–t/ τtheo = Lab: Capacitors

Updated 04/18/15

Extra credit questions (extra credit will be granted only for correct answers, not for effort): 1. Use algebra to figure out what would happen to the time constant of an RC circuit if you used two identical capacitors of capacitance, C, connected in parallel, instead of just one. (Resistance = R). Be specific/precise and show your work or state your reasoning to receive credit. [Note: saying it increases or decreases is not precise enough.]

2. Use algebra to figure out what would happen to the time constant of an RC circuit if you used two identical capacitors of capacitance, C, connected in series, instead of just one. (Resistance = R). Be specific/precise and show your work or state your reasoning to receive credit. [Note: saying it increases or decreases is not precise enough.]

3 Use variables to find what percentage of the initial potential remains after one time constant has passed (t = τ)? (Do NOT use your data to calculate this.) Show your work to receive credit.

Don’t forget to write your conclusion paragraph! If you feel that you need more than one paragraph to discuss the various parts of this lab, that’s ok – it’s a long multipart lab Before writing your conclusion paragraph(s), you might want to consider the following “hint” questions. I have phrased the following “hints” as individual questions to focus your critical thinking skills and point out the major results of your experiment that should be discussed in your conclusion paragraph(s). Once you have answered the “hint” questions in your head, you can then work on turning those answers in your head into a coherent, connected paragraph where sentences transition from one idea to the next. - Continued on next page -

Lab: Capacitors

Updated 04/18/15

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Without plagiarizing the lab manual or another student, state the purpose(s) of this lab. (Write the sentence(s) like it was the introductory sentence of a conclusion paragraph.) In part A, how well did your experimental capacitance values match the given values? If the values are more than 10-15% off from each other, who do you think is off – you or the manufacturer – and why? In part B, did you verify the equations for capacitors in series and capacitors in parallel? If your answer is yes, then what about your data or results verifies this? If your answer is no, what thing or things beyond your control may have affected your experiment(s)? [Things beyond your control do NOT include mistakes. You fix mistakes.] Note: It is OK to have verified one equation, but not the other – just discuss them separately. What is the internal resistance of the voltmeter in Ohms, with uncertainty? (Part C) How does the experimental time constant you got from the slope of your graph compare to the theoretical time constant you calculated from the resistance and capacitance? Do the values agree within the uncertainty of the “theoretical” time constant? If they do not agree within uncertainty, are the values “close” (tell me your definition of “close”)? What things beyond your control might have affected the determination of one or both time constants? [If the values are quite different, see the instructor right away!] Did the voltage across the capacitor decay at the rate expected as the capacitor discharges (i.e., did you directly verify equation 4 for either τexp or τtheo or both)? [If you don’t know how to answer this question, think about the voltages you calculated using equation 4 and the time you measured for 3.50 volts in part D? Were those voltages “pretty close” to 3.50 volts? From experience with this particular experiment, I would say “pretty close” is ± 0.10 volts.] Which time constant, τexp or τtheo is the most accurate? Consider the voltages you calculated when verifying equation 4. [Hint: Which value is closer to 3.50 volts]. Make sure you state your reason or reasons for choosing that particular time constant.

Now, write out your conclusion paragraph(s) on lined paper or type it (not the back of your graph). To turn the answers to the “hint” questions into a conclusion, you would merely write the answers in paragraph form with transitions from one idea to the next. This means that you CANNOT answer each question individually - you would NOT say “Yes, in part A my experimental capacitance values match the given values.”) Instead, ideas need to flow smoothly from one topic to the next. For example: “The Capacitors lab was done to learn about the capacitance of several capacitors, to explore how the capacitance of the circuit changes when you put capacitors in series or parallel, and to measure the time constant of a RC circuit created by putting a voltmeter in series with a capacitor. In part A of the experiment, our experimental capacitance values were fairly close to the values given for each capacitor, with the percent differences ranging from 4% for the “22” µF capacitor to 16% for the “47” µF capacitor. As we had very consistent measurements of the voltage for the “47” µF capacitor, and we were very careful to discharge all of the capacitors between each experimental run, I think the manufacturer may have marked the capacitor incorrectly. This possible mismarking of the “47 µF capacitor didn’t affect our ability to verify the equations for capacitors in series or parallel. In part B, our percent differences were less than 15%, supporting the series and parallel capacitor combination equations. In part C, we used the time dependent decrease in voltage across a discharging capacitor to determine the very large internal resistance of a Fluke 87 multimeter set to measure voltage. We determined that the internal resistance was (11.7 ± 0.6)x106 Ohms. We then used this result to calculate a theoretical time constant of ….”. Hopefully, you get the idea. Lab: Capacitors

Updated 04/18/15