Lab 08: Capacitors Last edited March 5, 2018 Learning Objectives: 1. Understand the short-term and long-term behavior of circuits containing capacitors. 2. Understand the mathematical relationship between the current in the circuit as a function of time, resistance, capacitance, and potential difference. 3. Understand the voltage drop across a capacitor. 4. Calculate the equivalent capacitance of a network of capacitors. Student Expectations: 1. Participation - Up to 10 points for participation and full involvement are earned during each lab class. 2. Lab Records - Take notes for all activities and record items as indicated in the lab instructions. At the end of each lab session, turn in the original Lab Records and upload digital copies to Bb. This assignment is graded for the required elements (out of 14 points). 3. Outside of Class a. For full Homework credit, complete the Lab 08 Homework on Blackboard within one hour before Lab 09. b. For full Quiz credit, read the Pre-Lab Notes and complete the Pre-Lab Quiz for Lab 09 at least one hour before Lab 09. I. Understanding the Models for the Behavior of a Circuit with a Capacitor The capacitor as a circuit element is introduced in the Lab 08 Pre-Lab Notes. To summarize: A capacitor stores electrical energy. A capacitor is modeled as a pair of parallel plates of conductive material, separated by a nonconducting gap. The current through any capacitor is zero. There are circumstances and time scales where the current in the rest of the circuit is non-zero, even with a capacitor in the circuit branch. The schematic symbol is two equal length parallel lines, perpendicular to the wires. The net charge of a capacitor is always zero. Consider the circuit in Fig. 1. h l 1. At the instant just before the switch is closed, what is the potential difference between the plates which are separated by a distance d? 2. What is the magnitude of the electric field between these capacitor plates at this time? Recall that the strength of the electric field can be expressed as a potential difference divided by a distance. Figure 1 A capacitor C in a circuit with battery B and SPST switch S. Consider the circuit divided in half, with one half containing the positive terminal of the battery and plate h and the other half containing the negative terminal of the battery and plate l. When the switch closes, current flows through each half circuit and opposite charges build up on both plates of the Lab 08 Capacitors 1
capacitor. Eventually the potential difference between the plates will be the same as the potential difference of the battery since the potential of plate h is the same as the potential of the positive terminal of the battery and the potential of plate l is that of the negative terminal. If there is a total charge +Q on plate h, there is a total charge Q on plate l. 3. A capacitor in this state is said to be fully charged. Note that this description is really a misnomer because, as noted above, the net charge of the capacitor is zero. Fully charged would be better to describe a single capacitor plate, but this is the terminology that is used for the full capacitor. a. When the capacitor is fully charged, what is the electric field in the part of the circuit between the positive terminal of the battery and plate h? b. What about the electric field between the negative terminal of the battery and plate l? c. Based on your responses, what can you say about the magnitude of current flowing through the circuit? The charge magnitude Q on the plates and the potential difference V between the plates are directly proportional to each other and can be expressed as Q = CV. The constant of proportionality C is called the capacitance of the capacitor. Note that capacitance is the ability of an object to store separated electrical charges and that any object that can be electrically charged exhibits capacitance. Because capacitors are able to store electrical energy, they act like small batteries and can store or release the energy as required. The capacitance depends only on the geometry of the plates and not on charge or potential difference between the plates. The unit of capacitance is the farad (denoted as F), which is defined as 1 F = 1 C/V or 1 coulomb/volt. Although determining the capacitance appears to be a matter of measuring the charge, such devices are expensive. Instead, a circuit modified from that in Fig. 1 is used to measure the capacitance (see Fig. 2). Adding a resistor in series with the capacitor and battery allows investigation of the time behavior of the battery current and voltage across the capacitor. Figure 2 Series RC circuit with SPDT switch to charge and discharge capacitor. The conceptual and mathematical models of voltage, current, and resistance (e.g., V = IR and Kirchoff s Rules) that were developed in earlier labs for a steady current flow hold for flow that is not steady. Using these models, it can be shown that the time dependence of the current flow through the battery or resistor when the switch is first connected to pole a at t = 0 is given by equation (1). The models can also be used to show that during this charging, the capacitor voltage V cap (the potential difference between or across the plates) is given by equation (2). Lab 08 Capacitors 2
II(tt) = VV batt RR ee tt/rrrr (1) VV cap (tt) = VV batt 1 ee tt/rrrr (2) 4. Using these equations, what is the magnitude of the current through the battery and the voltage across the capacitor just after the circuit is connected (i.e. t = 0)? 5. Now assume that enough time has passed that the capacitor is fully charged. a. What is the magnitude of the current flow and the potential difference across the capacitor at just before the switch is flipped to position b in Figure 2 (i.e., tt = for Eqs. (1) and (2))? b. How do these values compare to your predictions made earlier in Step 3? c. What is the potential difference across the capacitor a long while later when the switch remains at position b in Figure 2? 6. During the process in step 5c, the capacitor is said to be discharging. The current flowing to the capacitor and the voltage across the capacitor can be expressed as the following functions of time: II(tt) = QQ 0 RRRR ee tt/rrrr (3) VV(tt) = VV 0 ee tt/rrrr (4) where tt is now the time after the switch has been thrown, QQ 0 is the initial charge on the capacitor when it begins to discharge, and VV 0 is the voltage across it. QQ 0 is calculated from QQ 0 = CCVV 0. a. What does the negative sign in front of the term QQ 0 indicate in Eq. 3? RRRR b. What is the magnitude of current flowing in the circuit a long time after the switch was thrown? 7. In the four equations above, the factor RRRR in the denominator of the exponent is called the time constant. This is the time that it takes for a charged capacitor to discharge to 36.8% (= 1/ee) of its charged value or for an uncharged capacitor to reach 63.2% = 1 1/ee of full charge. The time constant is denoted by the symbol τ or Tau, where τ = RC. The units of τ are seconds when RR is in ohms (Ω) and CC is in farads (F). Checkpoint 1! Check your answers with your instructor before proceeding. Be sure that you can explain how equations 1, 2, 3, and 4 describe the charging and discharging of the capacitor in Figure 2. II. Measuring Capacitance in an RC Circuit In this experiment the PASCO interface box output serves as both the battery and switch for the RC circuit. It provides a voltage output in the form of a square wave which is a periodic wave that varies abruptly in amplitude between two fixed values (such as 0 volts and 5 volts), spending equal times at each. In doing so, the square wave simulates continually flipping the switch in Figure 2 between points a and b. Note that this interface box also serves as a voltmeter using a second set of wires in the analog port B (see photo below). Lab 08 Capacitors 3
1. Measure and record the resistance (and estimated uncertainty) of the resistor on the breadboard (with nothing connected to it but the Wavetek multimeter; manufacturer s specified accuracy is 2% of reading.) 2. Construct an RC circuit similar to Figure 2. The ScienceWorkshop interface takes the place of the elements in the dashed gray box and also measures VV cap. 3. Figure 3 shows an example of the change of VV cap with time in units of RRRR (i.e. the time constant). Sketch this graph in your Lab Records and identify and describe what is happening in the circuit for tt < 7RRRR and for tt > 7RRRR. Figure 3 Example of voltage reading across a capacitor Checkpoint 2! Have your RC circuit checked before continuing. Be sure you can explain Figure 3. 4. On the lab computer desktop, click on Data Studio Experiments> Capacitors to open the DataStudio program for this experiment. To take the data, follow the instructions on the splash page only so far as, Press Start to begin Data Acquisition. a. Sketch and label the resulting graph in your Lab Records. b. Compare your sketch with Figure 3. How does it differ? Why might that be the case? 5. Use the DataStudio curve fitting tool to determine the time constant for the circuit. Hint: Remember that the time constant term shows up in the exponential part of all equations related to the charging and discharging of the capacitor; since the charge is proportional to the voltage and voltage is what Lab 08 Capacitors 4
is measured, consider equation 4. Use the selection tool to include data points from the decay portion of the curve. Confusion alert! The constant that DataStudio gives you that you need to use to find the capacitance is C. This constant is NOT the capacitance. 6. Use the values of the time constant and resistance to determine the capacitance CC time constant. 7. Disconnect everything on the breadboard. Use the digital capacitance meter (on the 2000 μf scale) to find the capacitance of the capacitor CC meter that was in the circuit and compare this value to that obtained in step 5. Assume that the only uncertainty comes in the capacitance meter reading. Recall from Physics 1 that the criterion for equivalence (two uncertain numbers represent the same quantity) in this case would be CC time constant CC meter 2uu(CC meter ). The manufacturer s stated precision for the capacitance meter is 5% + 10. E.g., if the reading is C = 100 µf, the uncertainty is u(c)=(5%)(100 µf) + 10 µf = 15 µf. Checkpoint 3! Check your findings with your instructor. Lab 08 Capacitors 5
III. Determining equivalent capacitance Earlier you worked with two equations that separately calculate the equivalent resistance of a circuit involving resistors in series and resistors in parallel. This section will help you determine the two equations for the equivalent capacitance for capacitors connected in series and capacitors connected in parallel. 1. Measure and record the capacitance of the second capacitor with the Protek meter. 2. Design an experiment to address the research question, How is the equivalent capacitance of two capacitors connected in parallel affected by one of the capacitors? 3. Connect the two capacitors in parallel. Connect the meter as if you were measuring the voltage drop across the combination. Record the measured equivalent capacitance of the pair of capacitors. 4. Using the values of the individual capacitors, how would you combine them to come up with a mathematical rule for the equivalent capacitance of two capacitors in parallel? 5. Design an experiment to address the research question, How is the equivalent capacitance of two capacitors connected in series affected by one of the capacitors? 6. Connect the two capacitors in series. Measure the equivalent capacitance of the pair of capacitors. 7. What is the mathematical rule for the equivalent capacitance of two capacitors in series? Hint: this form is not as obvious, but there are similarities between the equations for equivalent capacitance in series/parallel and resistors in series/parallel. Checkpoint 4! Check the relationships you determined for equivalent capacitance with your instructor. Lab 08 Capacitors 6