Name: Dr. Julie J. Nazareth Lab Partner(s): Physics: 133L Date lab performed: 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, V ps ( ) Average Known Measured Measured Experimental Known Capacitor, Voltage, Voltage, Capacitance, Capacitance, Percent C = 10 µf V m ( ) V ( ) C exp ( ) (µf) Difference Capacitor, C 1 C 1 = 22 Capacitor, C 2 Capacitors in Parallel Capacitors in Series C 2 = 47 Or 50 C pcalc = C scalc = Calculations: Using equation 11, calculate the capacitance of the unknown capacitor for C 1, C 2, C 1 and C 2 in parallel and C 1 and C 2 in series. Calculate the percent difference between the known capacitance and your experimental values. SHOW THE CALCULATIONS for C 1 (the one marked 22 µf), C pcalc, and C scalc. 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 11. Also, use your experimental capacitance values for C 1 and C 2 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 C 1 (the one marked 22 µf) Vps C1exp = C 1 = V C 1 % diff. = [(C 1exp - 22 µf)/(22 µf)] x 100% = Calculating the known capacitance for series and parallel capacitors C pcalc = C 1exp + C 2exp = C1ex pc2 ex p C scalc = = C + C 1ex p 2 ex p Data & Reporting score:
Part C: Measurement of internal resistance, R Table 2: Time for a RC circuit to fall to ½ maximum voltage Maximum Voltage, V max One half max voltage, ½V max Time for V = ½ V max, Experimental Capacitance, Discharge Trial ( ) ( ) t 1/2 ( ) C (µf) 1 (10 µf) 10 2 (22 µf) 3 (47 or 50 µf) Average internal resistance of voltmeter, R ave ( ): Internal Resistance of Voltmeter, R int ( ) Calculations: Using your EXPERIMENTAL CAPACITANCE values and equation 14, calculate the 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 DECIMAL ANSWER WITH UNITS. [Remember: micro-farads, µf = 10-6 F, Farads.] Use units of resistance for your final answer, NOT seconds and microfarads. Trial 1: = t R 1/ 2 in t C ln 2 ( ) = Without numbers, substitute/cancel units to SHOW how seconds/farads equals Ohms (s/f = Ω). Some of the following definitions/relationships may be helpful. You won t need to use them all. Electrical charge: Coulombs = C Electrical 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 voltage reads exactly 4.0 volts before the switch is opened and the capacitor begins to discharge. Use the capacitor marked 22 µf!!! *** 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 2.50 2.25 natural log voltage, ln V 1.39 1.32 1.25 1.18 1.10 1.01 0.916 0.811 Time, t (s) 0 Voltage, V (volts) 2.00 1.75 1.50 1.25 1.00 0.75 0.50 natural log voltage, ln V 0.693 0.560 0.405 0.223 0-0.288-0.693 Time, t (s)
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 (a shared graph) 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. Alternatively, students may (individually NOT shared) use a computer program like LINEFIT or EXCEL to graph the data AND calculate the best fit line. (If you don t know how to make the program calculate the best fit line and display it with the data points, then you must hand draw the graph). Make sure the computer graph print-out is properly titled, and axes labeled (either using the computer or writing by hand) and the units for the slope are shown (either next to the formula displaying the equation of the best fit line, or next to the numerical results for slope and intercept if the slope results are in table form). Calculations: You can do the following calculations in your calculator you don t have to write it out for the lab report. Record the results in Table 4. Your final answers should be in units of time, NOT Ohms and Farads (or µf). Calculate the theoretical time constant, using the average resistance of the three trials in Part C and the experimental capacitance of the capacitor used in the Part D procedure. Use the slope of the graph to determine the experimental time constant. Table 4: Comparing experimental and theoretical results for the time constant, τ Slope Experimental time constant, Theoretical time constant, ( ) τ exp ( ) τ theo ( ) Calculations: Verify equation 12 directly from your data twice, once using your experimental time constant, τ exp, and the other time, using the theoretical time constant, τ theo. Use the time you recorded in Table 3 for 3.50 volts. SHOW BOTH CALCULATIONS. Don t forget units! Time from Table 3 at 3.50 volts: t = seconds Vmax = 4.00 volts (if you followed directions) Verify eq. 12 using τ exp V(t) = V max e -t/ τ exp = Verify eq. 12 using τ thep V(t) = V max e -t/ τ theo =
Questions: [Modified from Phy 123L/Phy 133L lab manual, Capacitors Experiment] 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 precise/specific 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 precise/specific and show your work or state your reasoning to receive credit. [Note: saying it increases or decreases is not precise enough.] Extra credit questions (Extra credit will be granted only for correct answers, not for effort) EC1. 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 or state your reasoning.) EC2. Considering all three R int values from Table 2, determine a sample uncertainty for the internal resistance of the voltmeter. SHOW WORK. Then, using the uncertainty in the average internal resistance, calculate the uncertainty in the theoretical time constant. SHOW WORK. Your final answer should be in units of time, NOT Ohms and Farads (or µf), and properly rounded. Knowing the uncertainty, does your theoretical time constant agree with your experimental time constant, within uncertainty? 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. - Continued on next page -
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. Without plagiarizing the lab manual or another student, state the purpose(s) of this lab. This is the introductory sentence of your 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? (Part C) Is the experimental time constant you got from the slope of your graph identical or close to the theoretical time constant you calculated from the resistance and capacitance? On what basis did you decide whether the two values agreed or were or were not 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!] Which time constant, τ exp or τ theo is the most accurate? Consider the voltages you calculated when verifying equation 12. [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)x10 6 Ohms. We then used this result to calculate a theoretical time constant of.. Hopefully, you get the idea.