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1 (ta initials) first name (print) last name (print) brock id (ab17cd) (lab date) Experiment 1 Capacitance In this Experiment you will learn the relationship between the voltage and charge stored on a capacitor; how to compensate for the effect of a measuring instrument on the system being tested; to visualize data in different ways in order to improve the undwerstanding of a physical system; to extend your data analysis capabilities with a computerbased fitting program; to apply different methods of error analysis to experimental results. Prelab preparation Print a copy of this Experiment to bring to your scheduled lab session. The data, observations and notes entered on these pages will be needed when you write your lab report and as reference material during your final exam. Compile these printouts to create a lab book for the course. To perform this Experiment and the Webwork Prelab Test successfully you need to be familiar with the content of this document and that of the following FLAP modules ( Begin by trying the fasttrack quiz to gauge your understanding of the topic and if necessary review the module in depth, then try the exit test. Check off the box when a module is completed. FLAP PHYS 11: Introducing measurement FLAP PHYS 12: Errors and uncertainty FLAP PHYS 13: Graphs and measurements FLAP MATH 11: Arithmetic and algebra FLAP MATH 12: Numbers, units and physical quantities WEBWORK: the Prelab Capacitance Test must be completed before the lab session! Important! Bring a printout of your Webwork test results and your lab schedule for review by the TAs before the lab session begins. You will not be allowed to perform this Experiment unless the required Webwork module has been completed and you are scheduled to perform the lab on that day.! Important! Be sure to have every page of this printout signed by a TA before you leave at the end of the lab session. All your work needs to be kept for review by the instructor, if so requested. CONGRATULATIONS! YOU ARE NOW READY TO PROCEED WITH THE EXPERIMENT! 1
2 2 EXPERIMENT 1. CAPACITANCE The capacitor A capacitor is a device that stores electric charge (electrons). It consists of two electrically conductive parallel metal plates separated by an insulating layer of air or other dielectric material, a material that can support an electric field. The total amount of charge q stored is proportional to the electric potential difference, or voltage, V C between the plates, so that q = CV C. (1.1) The capacitance C of a parallel plate capacitor is proportional to the plate area and the dielectric constant of the medium between the plates, and inversely proportional to the plate separation. Capacitance is measured in units of Farads (F), microfarads (µf = 10 6 F) and picofarads (pf = F). Figure 1.1: Basic capacitor circuit A series circuit provides only one path for movement of charge. Figure 1.1 shows a series circuit consisting of a source of electric potential energy V of voltage V, a switch S, a current limiting resistor R and a capacitor C. Kirchoff s Voltage Law (KVL) states that the algebraic sum of the voltages in any closed circuit loop is zero, V = 0. With voltage sources considered positive and voltage drops considered negative, we establish that the source voltage V will be equal to the voltage drops V R across R and V C across C, so that V = V R + V C. (1.2) The capacitor plate material consists of a conductive lattice of atoms. The positive nucleus is fixed in place while some of the negative electrons surrounding an atom are free to migrate from the atom when subjected to an external force. Let us assume that initially there is no charge stored on the capacitor plates so that the plates are electrically neutral and the voltage across the capacitor V C = 0. When the switch S is closed, the positive terminal of the voltage source attracts electrons away from the upper plate of the capacitor, leaving the upper plate with a net positive electric charge. The positive charge now present on the upper plate attracts electrons from the voltage source negative terminal to the lower capacitor plate, giving it a net negative charge. This flow of charge dq through the circuit during a time interval dt defines the electric current i = dq/dt. The current i is directly proportional to the voltage V R across the resistor R and inversely proportional to the circuit resistance R. This relationship between current, voltage and resistance is known as Ohm s Law: i = V R /R. (1.3)
3 3 The charge separation q at the two capacitor plates establishes a voltage, or potential difference, V C = q/c across the capacitor. As V C increases, the difference V R = V V C across R decreases, as does the current i = (V V C )/R flowing through R. This process continues until the voltage V C across C is equal to the voltage V of the source, at which time charge no longer flows and the current i = 0. From Kirchoff s Voltage Law, we know that the sum of all the voltage sources minus all the voltage drops in a circuit equals to zero. To examine the capacitor charging process, we traverse the circuit of Figure 1.1 clockwise from the negative () terminal of the battery, adding each voltage source and subtracting the voltage drop across each component: V V R V C = 0 V ir q C = 0 (1.4) A current i that varies as a function of time t is symbolized by i = i(t). Substituting this relationship in Equation 1.4 and rearranging yields the charging equation for the circuit: i(t) = dq dt = V ( ) 1 R q (1.5) RC The solution of this differential equation in terms of q is given by ( ) dq V dt = e t/rc (1.6) R Here, e = is the base of the natural logarithms (ln), not the elementary charge. We note in Equation 1.6 that at time t = 0 the exponential term is e 0 = 1 and i = V/R does not have any dependence on time. Let this initial constant current be I 0. Then the current i(t) flowing in the circuit at any time t is given by Capacitors in parallel i(t) = I 0 e t/rc (1.7) The charge stored on a capacitor is directly proportional to the surface area of the capacitor plates. Referring to Figure 1.2 we note that putting two capacitors in parallel results in an equivalent plate surface area that is the sum of the individual plate areas. This qualitative result can be expressed mathematically. The voltage across each capacitor is V. Applying Equation 1.1 to the charge stored in each capacitor: q 1 = C 1 V, q 2 = C 2 V. (1.8) The total charge q stored in the parallel arrangement of capacitors is the sum of the charges stored in each capacitor, q = q 1 + q 2 = (C 1 + C 2 )V (1.9) The equivalent capacitance C p with the same total charge q and voltage V is then Figure 1.2: Capacitors in parallel C p = q V = C 1 + C 2 (1.10)
4 4 EXPERIMENT 1. CAPACITANCE The relationship can be extended to any number of capacitors in parallel by simply adding the contributions from the charge stored in each of the capacitors: C p = C 1 + C C N = N C i (1.11) i=1 Capacitors in series When a voltage V is applied across several capacitors connected in series, a charge separation q = C 1 V 1 = C 2 V 2 =... will be induced across each capacitor. Using KVL, the sum of the voltages across each capacitor is equal to the applied voltage: V = V C1 + V C2 +..., then the equivalent capacitance C s of two or more capacitors in series is given by 1 C s = 1 C C = N i=1 1 C i (1.12) Figure 1.3: Capacitors in series Procedure Figure 1.4: Schematic diagram of experimental setup, connected to measure a single capacitor Manufacturer s values of components used in the experimental circuit: R d ± (R d ) = (100 ± 5)Ω, R c ± (R c ) = (1.00 ± 0.05) 10 5 Ω, C ± (C) = (2.2 ± 0.2) 10 6 F. Figure 1.4 shows the schematic diagram of the electrical circuit used in this experiment. The circuit uses one or two removable jumper wires to electrically arrange the capacitors in series, parallel, or to only include a single capacitor as in Figure 1.4. The capacitors C 1 = C 2 = C have the same nominal value.
5 5 A computer USB port provides the voltage source V =5V used to power the circuit. This connection is already made across terminals A and P. With the USB ground, or zero voltage reference point, connected to the input terminal A, there are 5V present at the terminal P. To measure voltage properly, the LabPro voltage probe R p should be connected across R c so that it s zero reference point, the black clip, is also connected to the A terminal. The red clip is connected to the B side of R c as shown in Figure 1.4. The LabPro unit acts as a resistance of R p = 10 7 Ω in parallel with resistance R c. The effective circuit resistance of these two resistors in parallel is given by 1 1 R = 1 R c + 1 R p (1.13) Calculate R and R using the given values of R c, R c, and R p. Assume R p = ±0.005 R p. R =... =... =... ( Rc ) 2 ( ) 2 R = R 2 Rp + =... =... R 2 c R 2 p R =... ±... Ω When the normallyopen switch S is pressed, R d connects across C and any charge present on the capacitor plates discharges very quickly through R d so that V C 0. Since R and R d are in series, V = V R +V Rd. With R R d, the voltage drop across R d is nearly zero and can be ignored. Now, V R = V AB = V and by Ohm s Law, a steady current I 0 = V AB /R flows through R. When the switch S is released, the timedependent current i(t) = V AB /R decreases exponentially with time as a voltage V C develops across the test capacitor(s), and hence V AB decays exponentially to approximately zero. Replacing i(t) in Equation 1.7 we get an expression for the voltage V AB across R: V AB = I 0 R e t/rc. (1.14) Part 1: Single capacitor Assemble the circuit board as shown in Figure 1.4, with a jumper wire connecting the terminal P to terminal P 2. Verify the connection of the board to a USB port and the LabPro voltage probe. Close any open Physicalab programs, then start a new PhysicaLab session and enter your address in the entry box. Your graphs will be sent there for later inclusion in your online lab report. yourself a copy of all graphs. Check the Ch1 box and choose to collect 50 points at 0.05 s/point. Select scatter plot. Press and hold the switch S to discharge the capacitor. Click Get data. As soon as data appears, release the switch. 1 for a derivation of the parallelresistor equation, refer to the Resistance experiment
6 6 EXPERIMENT 1. CAPACITANCE Your graph should display a horizontal line at V 5 V followed by an exponential decay to V 0 V from the time that the switch was released. The LabPro converts the continuously varying analog input voltage into a digital representation consisting of discrete and equally spaced increments in V, so that the input voltage is linearly quantized. The voltage difference between two adjacent voltage levels defines the internal scale and hence the resolution of the LabPro. Zoom in on the nearzero region of data at the end of the decay curve by unchecking Autoscale and adjusting the Xaxis scale values, then click Draw. Your graph should display these points placed on one of several horizontal equally spaced lines that represent the the discrete voltage steps V s of the converter output. From the graph and the corresponding data table values, select two adjacent voltage steps V 1 and V 2, then determine the voltage resolution dv = V 2 V 1 of the Labpro converter scale and the error V AB of a reading: V 1 =..., V 2 =..., dv =..., V AB =... Display the complete data set by checking Autoscale, then check the X grid and Y grid boxes to display a grid on your graph. Select scatter plot, then click Draw. You will be simultaneously fitting two separate equations to your data. The first equation is given by Y = A and will fit a straight line at Y = V AB to the initial portion of your data, from time t = 0 to the release of the switch at time t 0 = C, where A,C represent parameters of the fitting equations. The second equation will attempt to fit the exponential portion of the data, from the time t 0 = C and amplitude V AB = A to a final value of V AB 0 at some later time. This equation is given by Y = A exp( B (x C)). The fitting parameter B determines the decay rate of the exponential curve, and comparison with equation 1.14 shows that B = 1/RC and x= t. To summarize, we can express the time dependence of the voltage V AB across R by the expression I 0 R, t < t 0 V AB (t) = I 0 R exp( B (t t 0 )), t t 0 To fit your data, check Fit to: y= and select from the dropdown list or enter the following string, without spaces, in the fitting equation box: A*(x<C)+A*(exp(B*(xC)))*(x>=C). The fit parameters A, B and C are initially set to one. These values may be too distant from the actual fit values to allow the fitting algorithm to converge and provide a valid result. If you attempt to perform a fit and get an error message, review the scatter plot of your data to estimate some more appropriate values for A and C. To estimate the parameter B, you can use the fact that a capacitor discharge curve decays to 1/e = 1/ of the original voltage level after a time t = RC, defined as the time constant of the circuit. Choose a time t 1 along the exponential portion of the curve and a time t 2 at the point where the curve has decreased to approximately 1/3 of the level at t 1. Since t = t 2 t 1 RC then B = 1/RC 1/ t. Label the axes and title the graph with your name and circuit arrangement used. Click the Send to: button to yourself a copy of your graph for the exponential decay of a single capacitor. Check the Y log box to display the voltage in logarithmic units, then uncheck autoscale and set Y= 3 to 1 in 4 steps to range y from 10 3 to Redraw and your graph.
7 7? What does the exponential decay look like and why is this so?? What feature of the graph does the fit parameter B represent? How would you prove it? Recall that you have taken the log of the voltage.? Why are the points at the bottom right corner of the graph scattering? Use values from your data set to support your conclusions.? Do any negative voltage points from your data set appear on the graph? Can a logarithmic plot display values of y 0? Record below the initial voltage A and decay parameter B B = 1/RC =... ±...1/s A = I 0 R =... ±...V Calculate the experimental value C and C for the capacitor: C =... =... =... ( B ) 2 ( ) R 2 C = C + =... =... B R Calculate the initial current I 0 and its error I 0 : C =... ±...F I 0 =... =... =... I 0 = I 0 ( A A ) 2 + ( ) R 2 =... =... R I 0 =... ±... The manufacturer s value of the capacitance C used in this part of the experiment is C =... ±...F Part 2: Capacitors in parallel Remove all jumper wires. Connect a jumper wire from P 1 to P 3 and another from P 2 to P. Acquire a data set, then fit and send the exponential decay graph for two capacitors in parallel. Calculate the following parameters, then enter the results below.
8 8 EXPERIMENT 1. CAPACITANCE B = 1/RC p =... ±...1/s A = I 0 R =... ±...V C p I 0 =... ±...F =... ±...A Calculate the effective capacitance C p and the error, or tolerance C p of the two capacitors in parallel using the manufacturer s values of C 1 and C 2. C p =... =... =... C p =... =... =... C p =... ±...F Part 3: Capacitors in series Remove all jumper wires. Connect a jumper wire from P 3 to P. Acqiure a data set, then fit and print the exponential decay graph for two capacitors in series. Use a separate sheet to calculate the following parameters, then enter the results below. B = 1/RC s =... ±...1/s A = I 0 R =... ±...V C s I 0 =... ±...F =... ±...A Calculate the effective capacitance C s and the error, or tolerance C s of the two capacitors in series using the manufacturer s values of C 1 and C 2. C s =... =... =... C s =... =... =... C s =... ±...F
9 9 Part 4: Circuit time constants For the three circuits, use your experimental C value to calculate the t c time constant of each charging circuit. Also calculate the t d, the time constant of the discharging circuit, when the switch S is closed and the charge stored in the capacitor discharges through R d.? How do these time constants vary with C, R and R d? t c (1) =... =... =... t d (1) =... =... =... t c (2) =... =... =... t d (2) =... =... =... t c (3) =... =... =... t d (3) =... =... =... IMPORTANT: BEFORE LEAVING THE LAB, HAVE A T.A. INITIAL YOUR WORKBOOK! Lab report Go to your course homepage on Sakai (Resources, Lab templates) to access the online lab report worksheet for this experiment. The worksheet has to be completed as instructed and sent to Turnitin before the lab report submission deadline, at 11:00pm six days following your scheduled lab session. Turnitin will not accept submissions after the due date. Unsubmitted lab reports are assigned a grade of zero.
10 10 EXPERIMENT 1. CAPACITANCE
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