RC Circuits: Measuring Capacitance and Resistance, Study notes of Physics

Download RC Circuits: Measuring Capacitance and Resistance and more Study notes Physics in PDF only on Docsity! Lab 9. RC Circuits Goals • To appreciate the capacitor as a charge storage device. • To measure the voltage across a capacitor as it discharges through a resistor, and to compare the result with the expected, theoretical behavior. • To use a semilogarithmic graph to verify that experimental data is well described by an exponential decay and to determine the decay parameters. • To determine the apparent internal resistance of a digital multimeter. Introduction A diagram of a simple resistor-capacitor (RC) circuit appears in Figure 9.1. A power supply is used to charge the capacitor. During this process, electrons accumulate on one side of the capacitor and repel electrons from the other. This gives the appearance of continuous charge flow through the capacitor, but "flow" will stop when the power supply is incapable of forcing any additional electrons onto the negative side of the capacitor. A digital multimeter set to measure voltage behaves in a circuit like a large (in ohms) resistor (Represented here as a resistor and ideal meter in parallel). When the power supply is disconnected from the capacitor, charge “leaks” from one side of the capacitor, through this resistor, back to the other side of the capacitor, until no voltage appears across the terminals of the capacitor. This leakage is the repulsion of all the electrons on one side of the capacitor, and the attraction of the electron deprived opposite side. 5 V 3 V 12 ȍ 20 ȍ30 ȍ V + RPower Supply Voltmeter Capacito Switch + 5 V V + R Power Supply Voltmeter Capacitor Switch + Figure 9.1. Diagram of RC circuit and power supply. 77 CHAPTER 9. RC CIRCUITS 78 The power supply in Figure 9.1 is represented by a battery. Note that the positive output of the power supply is connected to the plate of the capacitor marked with a plus sign. The capacitors used in this experiment are polarized, meaning charge accumulation only works properly in one direction. Electrolytic capacitors (the polarized type we are using) can be made inexpensively and are widely used in power supplies. As you may remember from chemistry, the sign of the voltage is critical in electrolytic reactions. Make sure that the positive end of the capacitor is connected to the positive output of the power supply in your circuit. The voltmeter in Figure 9.1 is enclosed by a dashed line. The voltage sensing circuit is represented by a circle with a “V” inside. All voltmeters have resistance, and this resistance is represented by the resistor symbol inside the box. Our goal is to measure the value of this resistance, R. Theory We plan to monitor the voltage across the capacitor as a function of time after the switch is opened. The functional form of this dependence can be derived by circuit analysis using Kirchhoff’s loop law. A simplified diagram of the circuit after the switch is opened is shown in Figure 9.2. For the purposes of analysis, we indicate the positive direction of current by an arrow. This choice defines the sign of positive charge, Q on the capacitor. (Q is positive when the arrow points toward the plate with positive charge.) It also defines the positive direction of ∆V . (∆V is positive when the arrow points in the direction of increasing potential.) RC i Figure 9.2. Diagram of RC circuit and power supply. Because our circuit contains no source of emf, the only potential differences in the circuit appear across the capacitor and across the resistor. By Kirchhoff’s loop rule, the total potential change as you go all the way around the loop (∆Vloop) must be zero. Let the potential difference across the capacitor be ∆VC and the potential difference across the resistor be ∆VR. Then ∆Vloop = ∆VC +∆VR = 0 . (9.1) In the presence of a positive charge Q on the capacitor, ∆VC must be negative, as the potential drops as one moves from a positively charged plate to a negatively charged plate. The capacitance, C, of a capacitor is defined so that the magnitude of ∆VC is Q/C. Therefore ∆VC = −Q/C. Likewise, potential drops as charge passes through a resistor in the direction of positive current, I. The magnitude of this drop is given by Ohm’s law, so that ∆VR =−IR. Substituting these relations into Kirchhoff’s loop rule yields CHAPTER 9. RC CIRCUITS 81 A graph with the logarithm of one quantity on one axis versus a non-logarithmic quantity on the other axis is called a semilog graph. (The logarithm appears on only one of the two axes.) Plot a semilog graph of your data. Again include the error bars with each plotted point. Does your graph support the hypothesis that the relationship between the voltage and time is an exponential function? Using the value of C marked on your capacitor, compute the value of the R of the voltmeter and compare it to the value from the manufacturer’s specification. Internal resistance of an inexpensive voltmeter Repeat your measurements of ∆VC(t) versus time using the relatively inexpensive (smaller, red or black) digital voltmeter at your lab station. Repeat the analysis above to determine its internal resistance. How does it compare with the internal resistance of the relatively expensive Fluke digital voltmeter? The internal resistance of a voltmeter is one measure of its quality. To measure the potential difference across a component with a high resistance, the internal resistance of your voltmeter should be much higher than the resistance of the component. A voltmeter with a high internal resistance can be used in applications where the measurement error of a meter with a low internal resistance would be unacceptably high. Summary Begin by “filling in the blanks” of the argument for a simple exponential function being a straight line when plotted semi-logarithmically. Then state your findings clearly, succinctly, and com- pletely. No Effort Progressing Expectation Scientific SL.A.a Is able to analyze the experiment and recommend improvements Labs: 1-3, 5, 7, 9, 11, 12 No deliberately identified reflection on the efficacy of the experiment can be found in the report Description of experimental procedure leaves it unclear what could be improved upon. Some aspects of the experiment may not have been considered in terms of shortcomings or improvements, but some are identified and discussed. All major shortcomings of the experiment are identified and reasonable suggestions for improvement are made. Justification is provided for certainty of no shortcomings in the rare case there are none. SL.B.b Is able to explain patterns in data with physics principles Labs: 1-3, 5, 7, 9-11 No attempt is made to explain the patterns in data An explanation for a pattern is vague, OR the explanation cannot be verified through testing, OR the explanation contradicts the actual pattern in the data. An explanation is made which aligns with the pattern observed in the data, but the link to physics principles is flawed through reasoning or failure to understand the physics principles. A reasonable explanation is made for the pattern in the data. The explanation is testable, and accounts for any significant deviations or poor fit. CHAPTER 9. RC CIRCUITS 82 No Effort Progressing Expectation Scientific CT.B.a Is able to describe physics concepts underlying experiment Labs: 1-3, 5, 7, 9-12 No explicitly identified attempt to describe the physics concepts involved in the experiment using student’s own words. The description of the physics concepts underlying the experiment is confusing, or the physics concepts described are not pertinent to the experiment for this week. The description of the physics concepts in play for the week is vague or incomplete, but can be understood in the broader context of the lab. The physics concepts underlying the experiment are clearly stated. QR.A Is able to perform algebraic steps in mathematical work. Labs: 3, 6, 9-12 No equations are presented in algebraic form with known values isolated on the right and unknown values on the left. Some equations are recorded in algebraic form, but not all equations needed for the experiment. All the required equations for the experiment are written in algebraic form with unknown values on the left and known values on the right. Some algebraic manipulation is not recorded, but most is. All equations required for the experiment are presented in standard form and full steps are shown to derive final form with unknown values on the left and known values on the right. Substitutions are made to place all unknown values in terms of measured values and constants. QR.B Is able to identify a pattern in the data graphically and mathematically Labs: 1-3, 5, 7, 9-11 No attempt is made to search for a pattern, graphs may be present but lack fit lines The pattern described is irrelevant or inconsistent with the data. Graphs are present, but fit lines are inappropriate for the data presented. The pattern has minor errors or omissions. OR Terms labelled as proportional lack clarity - is the proportionality linear, quadratic, etc. Graphs shown have appropriate fit lines, but no equations or analysis of fit quality The patterns represent the relevant trend in the data. When possible, the trend is described in words. Graphs have appropriate fit lines with equations and discussion of any data significantly off fit. QR.C Is able to analyze data appropriately Labs: 1-4, 6, 7, 9-12 No attempt is made to analyze the data. An attempt is made to analyze the data, but it is either seriously flawed, or inappropriate. The analysis is appropriate for the data gathered, but contains minor errors or omissions The analysis is appropriate, complete, and correct. IL.A Is able to record data and observations from the experiment Labs: 1-12 "Some data required for the lab is not present at all, or cannot be found easily due to poor organization of notes. " "Data recorded contains errors such as labeling quantities incorrectly, mixing up initial and final states, units are not mentioned, etc. " Most of the data is recorded, but not all of it. For example measurements are recorded as numbers without units. Or data is not assigned an identifying variable for ease of reference. All necessary data has been recorded throughout the the lab and recorded in a comprehensible way. Initial and final states are identified correctly. Units are indicated throughout the recording of data. All quantities are identified with standard variable identification and identifying subscripts where needed. CHAPTER 9. RC CIRCUITS 83 No Effort Progressing Expectation Scientific WC.B Is able to draw a graph Labs: 3, 6, 9, 11 No graph is present. A graph is present, but the axes are not labeled. OR there is no scale on the axes. OR the data points are connected. "A graph is present and the axes are labeled, but the axes do not correspond to the independent (X-axis) and dependent (Y-axis) variables or the scale is not accurate. The data points are not connected, but there is no trend-line. " The graph has correctly labeled axes, independent variable is along the horizontal axis and the scale is accurate. The trend-line is correct, with formula clearly indicated. WC.D Is able to draw a circuit diagram Labs: 3, 4, 9, 10 No circuit diagram is drawn. Components of the circuit are missing, or connected incorrectly. Components are not clearly labelled. "Circuit diagram is missing key features, but contains no errors. It may be difficult to follow electrical pathways, but it can be determined which components are connected with sufficient scrutiny. " Circuit diagram contains minimal connecting lines, components are neatly arranged to ensure labels are readily identified to appropriate components.