University Physics II Experiment 5: RC Circuits Objective: We have assumed that emf and resistances are constant, so that the potentials, currents, and powers are time independent. When looking at charging or discharging a capacitor, the currents, voltages, and powers do change with time. In this experiment, you will look at charging and discharging a capacitor through a resistor. Background & Equations: Capacitors can be defined as circuit elements that store charge. The size of the capacitor determined how much charge it can hold for some given voltage Q = CV- (eq. 1) where C is the capacitance, Q is the charge, and Ve is the voltage across the capacitor. VR = IR (eq. 3) Ohm's Law, where Ve is the voltage across the resistor. C R VO Part 1: Charging a Capacitor: When the capacitor is fully charged, the current decreases to zero, the potential difference across the resistor, I's becomes zero, and the entire voltage supplied, Vo appears across the capacitor, so that, Vo = V- Applying Kirchhoff's Loop Rule to the diagram above to determine the capacitor's charge as a function of time, for a charging capacitor, we obtain, Vo - Vc - VR = 0 (eq. 3) where Vo is the supplied voltage, I, is the voltage across the capacitor, and Va is the voltage across the resistor. Substituting for VR (eq. 2) and Ve (eq: 1) in eq.3, we obtain, 1Vo - 2- IR = 0 (cq. 4) Solving eq. 4 for current, I, RC (eq. 5) When the charge, Q increases, ~- becomes larger and the capacitor's charge reaches its final value, Q, and RC the current decreases and finally becomes zero, then, Of = CV- (eq. 6) (eq. 7) current is defined as the rate of flow of charge, using this equation in cq. (5) we obtain, (cq. 8) a differential equation that can be solved to find charge as a function of time for a charging capacitor, Q(t) = Q/ (1 -e me) (eq. 9) RC Circuit, charging capacitor, where RC is the time constant of the circuit, and has units of seconds. The instantaneous current of charging capacitor, is just the time derivative of eq. 9, I(t) = = be m (cq. 10) Part 2: Discharging a Capacitor: Similarly, we can obtain an equation for charge as a function of time for a discharging capacitor, except in this case V. (the final voltage of the capacitor once it is fully charged) eventually reaches zero as the current, /, and charge Q approach zero, then the Kirchhoff's Loop Rule becomes, - - IR = 0 (eq. 11), solving for the current, I, gives, dit (eq. 17) another differential equation that can be solved to find charge as a function of time for a discharging capacitor, 2(t) = Qqe-t/RC (cq. 13) RC circuit, Q as a function of time, discharging capacitor, and, RC circuit, current as a function of time of a discharging capacitor.Procedure Part 1: Charging a Capacitor through a Resistor: Click here to open an online simulation and click "begin". Play with the simulation to see how it works and get familiar with changing the values of the circuit's elements etc. Set the resistor value to 670 Ohms, the capacitor value to 50 micro-Farads, and the voltage of the battery voltage to 8 Volts. Close the switch and observe the capacitor charging. Record the time needed for the capacitor to fully charge below, 1 = Observe what happens to the voltage across the resistor and the current in the circuit as the capacitor is reaching its final charge, state your observations below. Click on "Neutralize Capacitor" to reset the simulation. Using the same values for the resistor, capacitor, and battery as above, record the values of the Capacitor Voltage, Resistor Voltage, Capacitor Charge, Current, and Charging Time for the various Capacitor Target Voltages of the capacitor in the process of charging. To do this you will need to open the switch when the Capacitor Voltage is close to the corresponding Target Voltage, record all values, and then close the switch to resume. Table 1: Charging a Capacitor Capacitor Target Capacitor Resistor Voltage, V- Voltage, V. Voltage, VR Capacitor (V) (V) Charge, Q (C) Current, I (A) Charging Time, t (s) (V) 0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 1. Add Ve and V, at any given time during the charging process. What does it equal to? Is Kirchhoff's Loop Rule satisfied? 2. Using the data from Table 1, make the following graphs in excel: a. VR vs. time, tb. V. vs. time, t c. Current, I vs. time, t d. Capacitor charge, Q vs. time, t Make sure to plot these using the scatter plot option in excel, title your graphs, title the axes, include the units, etc. Choose the correct fit for your graphs. Discuss the graphs. Part 2: Discharging a Capacitor through a Resistor: Click here to open the simulation for this part of the lab and click "begin". Look at the circuit and locate all elements. By clicking on the individual elements, you can change their values, and then come back to the full view of the circuit. Note that the value of the resistor remains unknown for this part. There are two meters connected to this circuit that you will use to read the voltage and the current. The Voltmeter is connected in parallel to the resistor, you will use it to read the voltage. The Ammeter is connected in series and reads the current. The simulation must be paused to read and measure voltage and current. 1. Click on the Resistor and set it to a random value by clicking "Change resistor". 2. Click on the Capacitor and set it to 2200 micro-Farads. 3. Click "Charge Cap" to charge the capacitor. 4. Click on the voltmeter, read the voltage, and record in the first row of the table below. 5. Click on the ammeter, read the current, and record in the first row of the table below. 6. Click "Resume" to start discharging the Capacitor. Click pause after about 15 seconds, and read and record the current, voltage, and elapsed time in the table below. 7. Repeat step 6 until you have filled the table below. Calculate charge, Q. Table 2: Discharging a Capacitor Measurement Capacitance, Voltage, V Discharging Charge, Q number: CF) (V) Current, I (A) time t ($) 8. Using data from Table 2, plot the following two graphs in excel: a. Voltage, V vs. time, t b. Charge, Q, vs. time, t c. Current, I vs. time, t Make sure to plot these using the scatter plot option in excel, title your graphs, title the axes, include the units, etc. Choose the correct fit for your graphs. Discuss the graphs