Pre-Lab Quiz / PHYS 224 R-C Circuits Your Name Lab Section 1. What do we investigate in this lab? 2. For the R-C circuit shown in Figure 1 on Page 3, RR = 100 ΩΩ and CC = 1.00 FF. What is the time constant of the R-C circuit? 3. Consider that the capacitor is initially not charged and the voltage of the battery is 3.00 V. The switch is closed at tt = 0. Using the appropriate given formulas, write down below the voltage VV cc (tt) on the capacitor and the current II(tt) in the circuit. 1
4. Consider that the capacitor is initially fully charged to 3.00 V. The switch is opened at tt = 0. Using the appropriate given formulas, write down below the voltage VV cc (tt) on the capacitor and the current II(tt) in the circuit. 2
Instructor s Lab Manual / PHYS 224 R-C Circuits Your Name Lab Section Objective In this lab, you will study an electric circuit involving a resistor (R) and a capacitor (C), and measure the time constants for basic R-C circuits. Background When establishing an electric potential difference of VV between the two terminals of a capacitor of capacitance CC, the two terminals will accumulate charges of the opposite signs but of the same magnitude, qq = VV/CC. In an idealized scenario of a completely isolated capacitor, its charge qq changes instantaneously when VV changes, i.e., one can instantaneously charge or discharge a capacitor! In practice, one can never test on an isolated capacitor. Moreover, an isolated capacitor is not thrillingly useful! Now consider a basic R-C circuit (Figure 1) consisting of one capacitor of capacitance CC and one resistor of resistance RR. Because of the resistor, the current (II) in the circuit cannot be infinitely large. Therefore, discharging/charging the capacitor does not occur instantaneously (note: discharging/charging must proceed through current). For the same voltage, a larger RR leads to a smaller II and 3
thus leads to a longer discharging/charging time. To change the voltage across the capacitor by the same magnitude, a larger CC leads to a larger change of qq and thus also leads to a longer discharging/charging time. Therefore, larger RR or CC leads to larger time constant for the R-C circuit. Such relationships make the time constant a controllable and useful parameter! For the R-C circuit shown in Figure 1, the sum of the voltage across the resistor, VV RR (tt) = II(tt) RR, and the voltage across the capacitor, VV CC (tt) = qq(tt)/cc, is: VV(tt) = II(tt) RR + 1 CC qq(tt). (1) The current carries the electric charge to or from the two terminals of the capacitor. Because of the conservation of charge, the current must equal the rate of change of the charges on the terminals of the capacitor. Using calculus, it means that II(tt) = dddd(tt), dddd and Equation (1) can be rewritten as: VV(tt) = dddd(tt) dddd RR + 1 CC qq(tt). (2) Charging a Capacitor: Consider an initially discharged capacitor. When we turn on the switch at tt = 0, the total voltage changes from 0 to VV 0 at tt = 0, such that VV(tt) = 0 at tt < 0 and VV(tt) = VV 0 for a time immediately after we close the switch. Under those conditions, the solution of Equation (1) for tt 0 is: qq(tt) = CC VV 0 1 ee tt/(rrrr) (3) and 4
II(tt) = VV 0 RR ee tt/(rrrr) (3 ) where we used the natural exponential ee 2.71828. The voltage across the resistor and the voltage across the capacitor are respectively VV RR (tt) = VV 0 ee tt/(rrrr) (4) and VV CC (tt) = VV 0 1 ee tt/(rrrr). (5) Discharging a Capacitor: When the switch is on for a time period much longer than the RC-product, the capacitor becomes approximately fully charged. Consider an initially fully charged capacitor. Turn off the switch at tt = 0. The total voltage changes from VV 0 to 0 at tt = 0, namely, VV(tt) = VV 0 for tt < 0 and VV(tt) = 0 for a time immediately after the switch is opened. The solution of Equation (1) for tt 0 is: qq(tt) = CC VV 0 ee tt/(rrrr) (6) and II(tt) = VV 0 RR ee tt/(rrrr). (6 ) The voltage across the resistor and the voltage across the capacitor are respectively VV RR (tt) = VV 0 ee tt/(rrrr) (7) 5
VV CC (tt) = VV 0 ee tt/(rrrr). (8) Note the opposite signs in Equations (4) and (7), owing to reversing the direction of the current. As explicitly shown by the above equations, one cannot instantaneously charge or discharge the capacitor in a real R-C circuit. Instead, the voltage across the capacitor increases/decreases according to a specific time constant of the circuit, which for the shown basic R-C circuit is τ0 = RC. EXPERIMENT Apparatus Figures 2(a) and 2(b) show the two R-C circuits to be investigated in this lab. 6
Procedures 1. Set up the RC circuit with RR = 111111 ΩΩ (Figure 2(a)) i) Connect the lead S1 (charging) of the double-throw switch to the positive lead of the two connected batteries (approximately 3.0 V will be used as a DC power supply), and ii) connect the lead S2 (discharging) of the switch to the ground lead of the batteries, iii) connect the center lead S0 (neutral) of the double-throw switch to the positive terminal of the ammeter, iv) connect the negative terminal of the ammeter to the positive terminal of the 1.0-F capacitor, and v) connect the negative terminal of the capacitor to the open lead of one of the two 100-Ω resistors. The other lead of the resistor is connected to S2 and thus to the ground of the battery. Finally, vi) connect the voltmeter and the capacitor in parallel with the correct polarities. Ask your TA to check the circuit. 2. Charging the capacitor connected to the 100-Ω resistor Set the appropriate sensitivities for the Ammeter (200 ma) and the Voltmeter (20 V). One student will read the watch and the other student will read and the ammeter and the voltmeter and record the readings in Table 1. When ready, start the watch at tt = 0 and simultaneously flipping the double-throw switch from S0 to S1. This starts the charging process. TABLE 1 Charging the capacitor connected to RR = 111111 ΩΩ t (s) I (A) V (V) t (s) I (A) V (V) 5 210 30 250 60 300 7
90 350 120 400 150 500 180 600 3. Discharging the capacitor connected to the 100-Ω resistor Wait for another 2 minutes to have the capacitor fully charged from the previous Step 2. Again, one student will read the watch and the other student will read and the voltmeter and record the readings in Table 2. When ready, start the watch at tt = 0 and simultaneously flipping the double-throw switch to S2. This starts the discharging process. TABLE 2 Discharging the capacitor connected to RR = 111111 ΩΩ t (s) V (V) t (s) V (V) 5 210 30 250 60 300 90 350 120 400 150 500 180 600 4. Set up the RC circuit with RR = 5555 ΩΩ (Figure 2(b)) Repeat Step 1 except replacing RR = 100 Ω with RR = 50 Ω (by connecting two 100-Ω resistors in parallel) and removing the ammeter from the circuit. Ask your TA to check the circuit. 5. Charging the capacitor connected to the 50-Ω resistor One student will read the watch and the other student will read and the voltmeter and record the readings in Table 3. When ready, 8
start the watch at tt = 0 and simultaneously flipping the double-throw switch to S1. This starts the charging process. TABLE 3 Charging the capacitor connected to R=50 Ω t (s) V (V) t (s) V (V) 5 105 15 120 30 135 45 150 60 180 75 210 90 250 Analysis For the analysis below, there is an MS Excel file that you can download from the instructor s website which can be helpful for plotting the data and fitting your results. 1. Charging the RC circuit with RR = 111111 ΩΩ By using data in Table 1, plot VV CC -versus-tt. Use Equation (5), VV CC (tt) = VV 0 1 ee tt/(rrrr), and find the best parameters VV 0 and RRRR-product to fit your experimental data. Record the fitting parameters: VV 0 = ττ 01 = RRRR = 9
Similarly, by using data in Table 1, plot II-versus-tt. Use Equation (3 ), II(tt) = VV 0 RR ee tt/(rrrr), and find the best parameters VV 0 RR and RRRRproduct to fit your experimental data. Record the fitting parameters: VV 0 RR = ττ 01 = RRRR = 2. Discharging the RC circuit with RR = 111111 Ω By using data in Table 2, plot VV CC -versus-tt. Use Equation (8), VV CC (tt) = VV 0 ee tt/(rrrr), and find the best parameters VV 0 and RRRR-product to fit your experimental data. Record the fitting parameters: VV 0 = ττ 01 = RRRR = 3. Charging the RC circuit with RR = 5555 Ω By using data in Table 3, plot VV CC -versus-tt. Use Equation (5), VV CC (tt) = VV 0 1 ee tt/(rrrr), and find the best parameters VV 0 and RRRR-product to fit your experimental data. Record the fitting parameters: VV 0 = ττ 02 = RRRR = Questions 1. Using the fitted parameters from analysis 1, calculate RR and CC? 2. Should the fitted ττ constants be the same from analysis 1 and analysis 2? 10
3. Calculate ττ 02 = RRRR with RR = 50 Ω and CC = 1.0 F. Calculate the percentage error using the theoretical and the measured τ02 values from analysis 3. 11