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O reaches in the limit t=infinity). Do the equations Teachi lil upplupim Write two paragraphs describing in your own words what is happening to the charge on the capacitor, the voltage on the capacitor, and the current in the circuit as the capacitor is 1) charging and 2) discharging. Comment on whether the time dependence shown by the equations agrees with the expected time dependence of the charge and current in the circuit as it charges or discharges. 5 Calculus Based Questions: Understanding the equations in more detail. Equations 1 and 3 describe the charging and discharging of a capacitor. The solutions to these equations are Equations 2 and 4, respectively. Equation 2(b) describes the charge as a function of time as the capacitor is charged. Find the currents for the charging capacitor by calculating the function f(t)=dQ/dt for this case. (Take the derivative of Equation 2 to find I(t).) Compare to Equation 2(a). Substitute the function Q(t) of Equation 2(b) and the function for I(t) which you have just found into Equation 1, show that Equation 1 does give zero. This proves that Equations 2(b) is a solution for this circuit. Equation 4 describes the charge as a function of time as the capacitor is discharged. Find the currents for the discharging capacitor by calculating the function (t) =dQ/dt for this case. (Take the derivative of Equation 4(b) to find I(t). By substituting the function Q(t) of Equation 4(b) and the function for I(t) which you have just found into Equation 3 show that Equation 4 is indeed a solution to the voltage equations for a discharging capacitor. (Show Equation 3 does give zero).

Charging and Discharging a Capacitor (approx. 2 h 20 min.) (5/16/12) Introduction A capacitor is made up of two conductors (separated by an insulator) that store positive and negative charge. When the capacitor is connected to a battery current will flow and the charge on the capacitor will increase until the voltage across the capacitor, determined by the relationship C-Q/V, is sufficient stop current from flowing in the circuit. Figure 1 shows a circuit that can be used to charge and discharge a capacitor. Equipment • PB-60 proto-board & • single-pole, double-throw . proto-board wires . alligator clips power adapter switch voltage sensor • multi-meter • 2 capacitors (100 F) 750 interface & Data Studio resistor (10 k.) For whole class: capacitance meter: spare fuses: Phillips screwdriver Circuit with two position switch, S, and S. Theory Charging the capacitor Before the switches are closed, there is no charge on the capacitor. When switch S, is closed, current will flow in the circuit as the capacitor is charged. According to Ohm's Law, Ve the voltage across the resistor will be VRIR while the voltage across the capacitor will be given by Vc=Q/C. Switch S, closed, charging the capaciter By Kirchhoff's Rule the voltage changes around the circuit s must add to zero so V -VR-Ve=Vhart-IR-Q/C=0 When the capacitor charges the charge, Q, starts at zero and Ver there is no voltage on the capacitor. This means the current initially flows at its maximum rate (I Max V R when Q=0). However, as the flowing current charges the capacitor, the voltage on the capacitor increases. This voltage opposes the Switch S. closed discharging the capacitor flow of more charge and the current begins to decrease. The rate at which the capacitor charges slows as the current R decreases -- as more and more charge builds up the current becomes smaller and smaller. The current decreases exponentially -- it asymptotically approaches zero for longer and longer times. Similarly the charge increases exponentially - it keeps growing but at a slower and slower rate and asymptotically approaches (but never actually reaches) its maximum value. It would take an infinite amount of time to actually reach the maximum value since the rate of increase - Figure 1: Circuit to charge and (current) becomes smaller and smaller as time goes by. If we discharge a capacitor take into account the fact that current and charge both vary with time, the equation obtained by applying Kirchhoff's Voltage Rule around the charging circuit becomes: Equation 1: Kirchhoff's Rule for a charging capacitor: V -V-V-V -1(0)R-

The exponentially decreasing current and increasing charge are described by: Equation 2(a): 160 for the charging capacitor 10) - (0%) Equation 2(b): 0(0 for the charging capacitor 00-c. (1-2%) The number e (Euler's Number also known as the base of natural logarithms) appears in any equation in which the rate of increase or decrease of a quantity depends linearly on the amount of that quantity. In this case the rate at which the capacitor charges depends on the current, which decreases as the charge (and hence the voltage) on the capacitor increases. Theoretically, it takes an infinite amount of time for the current to actually decrease to zero or for the capacitor to become fully charged. This is a property of exponential functions. The exponential behavior is characterized by the time constant. Ti Definition of time constant: T-RC. The time constant has units of seconds. The larger the product RC the longer it will take the capacitor to charge to any fraction of its maximum value. The time required to charge to l-e) of the maximum value is exactly one time constant, RC. Euler's number, e is an irrational number equal to 2.71828182845904523536028747...etc. Discharging the capacitor If we wait for until several time constants have passed, the capacitor will become nearly fully charged. At that time the current is nearly zero, the voltage on the capacitor is approximately equal to the voltage on the battery, Va, and its charge Qo is given by Q = CV Rat. Now we can change the switch from position S, to S. The current will flow through the resistor to ground, discharging the capacitor. Around this loop the sum of voltages is now given by Equation 3: Sum of voltages around the discharging circuit: V + V = 1(1) R+ 2 0 The voltage on the capacitor acts to "push" the current but as the current flows the capacitor discharges and the current slows down. Thus the rate of discharge slows as time goes by. Both the current and charge decrease exponentially with time: Equation 4a): It for discharging capacitor I(t) e-RC -VRC Equation 4(b): Olt) for discharging capacitor Q(t) = Q, UKC - CV VRC The same time constant, t = RC is used to characterize the discharging cycle but now the time RC describes the time it will take the charge to decrease to 1/e=e"=0.367879 times of its initial value. Procedure s Before building your circuit, flip the PB-60 proto-board over and carefully examine the pattern of connections, noting where gaps occur. Attach the power transformer to the adapter on the PB-60 proto-board and construct the circuit in Figure I using the +5 V output, a 100uF capacitor, a 10k

resistor and a double-pole, single throw knife switch. W ole, single throw knife switch. With these values, the product of R times Cis ond. (Note: If there is an arrow on the capacitor, it should point from m the lower voltage connection.) Use the banana-to-alligator cords to co switch terminals. As you set-up the circuit, use a he banana-to-alligator cords to connect the proto-board to the continuity in your circuit. Then use it as a voltme you set-up the circuit, use a multimeter, set as an ohmmeter, to check cut. Then use it as a voltmeter and make sure you understand how the circuit works. ASK YOUR INS S. ASK YOUR INSTRUCTOR TO CHECK YOUR CIRCUIT BEFORE PROCEEDING. "She data acquisition hardware: On pour computer desktop, click on Data Studio, select Create txperiment, and choose Voltage sensor. Connect the sensor to the PASCO 750 interace Shown. Select Graph from the display menu. Connect the voltage leads across the capacitor. By measuring the voltage across the capacitor you can measure the charge since QCV. 10 collect data you should start with the switch in the grounded position (S2 in Figure 1). You can use an extra wire to ground both sides of the capacitor to make sure it is fully discharged before you start. CHICK Start on the program. You should see a horizontal line at about zero volts. After a couple of seconds flip the switch to the charging position. You should see the voltage continually increase. If the voltage increase is too fast or too slow stop and re-adjust the time scale by clicking and dragging on the time axis. Repeat until you are sure you can make a good, readable data plot. Optimize the graph: right click on the graph and you see several functions: "scalewill auto-scale the graph, "in/out" zooms, "measure" will create a cross-hair cursor. You can select or delete certain graph. In addition you can click and drag on the number on the axis and can zoom in/out in this way. Once you have adjusted the time scale and data collection rate you can begin taking data. Start with the switch in the S, position and ground the capacitor to discharge it fully. Click start on the program and let it run for a while to establish a flat zero volt baseline. Flip the switch to S. Let the program run as the voltage rises. When the voltage reaches the maximum let the program run so that there is a flat line at the maximum voltage -- the charge on the capacitor has now reached its maximum and the current has stopped flowing. Keep the program running and flip the switch back to S2. Now the capacitor will discharge. Let the program run long enough to again establish a flat zero volt baseline. If you have recorded data for several cycles, choose the best looking charge/discharge graph (you can delete the other graphs, if you like). From your graph, you will determine the time constant of the circuit for both the charging and the discharging portions of the curve. From the time constant and the measured value for your resistance, you will derive the capacitance. (See Data Analysis section NOW for detailed instructions) IMPORTANT: STOP and analyze your data before proceeding to the next section! You will need to have the data visible on the computer in order to analyze it. Capacitors in parallel and in series Repeat the data collection for two capacitors in series and for two capacitors in parallel. Demonstrate that the sum of the capacitance conforms to the formulae derived in class: Capacitors in Parallel: Copuler =C+ +C+... 1 1 1 1 1 Capacitors in Series: C E27- -CGCS

the sheby time to discharge called the time color these times. Data Analysis Be sure and complete your analysis as you complete each part of the lab Understanding the time constant. The time to charge" or the time to discharoe is theoretically infinite. The charging and discharging are therefore characterized by a timer RC. called the time constant. From the equations for voltage as a function of time we can figure out what the voltage actually is at these times. When the capacitor is charged its charge increases according to Equation 2 and therefore its voltage as a function of time is given by the equation: Vc(t) = 20 = c (1-e-% (Charging) The time constant, RC, is defined as the time when the charge reaches the value given by setting the time t equal to the value RC. At this time the voltage reaches a certain fraction of the battery voltage: Vatt-RC) = V... (1-e')=0.632 V (Charging) To find the time constant from the graph simply find the time it takes the voltage and hence charge) to increase from zero to (1-e") times its final value. You will find this time by analyzing the graph in the following way: • Right click on graph and select measure. You should see a cross-hair cursor, which gives you the time and voltage (CV); • Move cursor to the exact point where the voltage starts to rise. Note the time t. • Move the cursor to the point where the voltage is leveled off and maximal. Note V max. • Move the curser along the graph to the level where the voltage reads 0.632Vmax. Note the time • Calculate the time interval Att-, between V-0 and 0.632 Vmes, which is the time constant. When the capacitor is discharging, the charge decreases according to Equation 4 and therefore its voltage obeys the equation: Vo = Vue RC (Discharging) The time constant is: T RC. At this time the voltage reaches a certain fraction of the battery voltage given by: VattRC) = Vle) -0.368 V (Discharging) To find the time constant from the graph simply find the time it takes the voltage to decrease from its maximum value to (el) times that value, by using a similar procedure as described above. Once you have found the time constant, complete the calculations in the data table to compare the measured time constant to the theoretical value. You can use the known value of the resistor to calculate an experimental value for the equivalent capacitance for the capacitors in parallel and series. To calculate the theoretical Ccouly use the theoretical equations for the series and parallel circuits on page 3. For your report For your report, write a cover page with brief introduction and include data table, graphs and analysis. All graphs should have axes labeled with units. The analysis, which gives you your time constant, should be presented clearly. Calculations given in the data table should be shown below the graphs or on a separate page. Summarize your experimental results and discuss errors or discrepancies and their possible sources.