Physics 260 Calculus Physics II: E&M. RC Circuits


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1 RC Circuits Object In this experiment you will study the exponential charging and discharging of a capacitor through a resistor. As a byproduct you will confirm the formulas for equivalent capacitance of capacitors connected in series and parallel. Equipment Low voltage power supply Two capacitors (nominally between 0.5 µf and 1 µf) Resistance Box Two connection boxes to support the capacitors Digital multimeter Stopwatch Switch Patch cords Theory Consider a capacitor of capacitance C with an initial charge Q 0 allowed to discharge through a resistor of resistance R. t = 0 C R Q 0 Figure 1 The voltage across the plates of the capacitor decays exponentially with time according to V(t) = V 0 e t / RC (Eq. 1) where V 0 is the initial voltage. (The charge on the plates and current to the plates varies in a similar exponential fashion.) Figure 2 shows a graph of V vs. t for this exponential decay.
2 V V 0 V 0 /e τ (Time constant) t Figure 2 The time constant for the circuit, τ, is the time for the voltage (or current, or charge) to decay to 1/e ( 0.368) of its initial value. It is a measure of the response time for the circuit. It is easy to show (see questions) that τ = RC (Eq. 2) It is instructive to take the logarithm of both sides of Eq. 1 to obtain lnv = lnv 0 t RC (Eq. 3) Hence a graph of ln V vs. t will yield a straight line with slope equal to 1/RC = 1/τ, as illustrated in Figure 3. ln V ln V 0 slope = Š1/RC Figure 3 t
3 Procedure I: Using the multimeter to measure the resistance and capacitance of the circuit 1. I will explain how to use the multimeter to measure resistance and capacitance at the beginning of the lab. Pay attention! If you blow a fuse, you pay for it. 2. Set your resistance box to a nominal resistance of 10 MΩ. (Make sure all the other keys are down; sometimes one will sneak up on you.) Use the multimeter to measure the resistance more accurately and record in Table 1. The internal resistance of the voltmeter is very nearly 10 MΩ and you do not need to measure it. 3. Use the multimeter to measure the capacitance of each of the two capacitors that were supplied to you. Procedure II: Observing the exponential decay 1. Set up the circuit shown below. (The figure shows approximate nominal values; use the circuit components supplied to you and the measured values from Procedure I.)The internal resistance of the voltmeter is 10 MΩ so the total resistance of the right loop is about 20 MΩ. Have the instructor check your circuit before you turn on the power. S R box  10 Mž R int Å 10 k½ Low Voltage Power Supply 6.0 V C µf V R meter  10 Mž Figure 4 Note the unusual circumstance that the voltmeter is not connected in parallel with any circuit element. This is usually considered a sacrilege, but there is a good reason for it here. We want as much resistance as possible in the circuit so that you will be able to measure the exponential decay with a stopwatch. It is also possible to show (see Questions) that although the reading of the voltmeter in this circuit is not equal to the voltage across the capacitor it is proportional to it, and that will be sufficient for our purposes.
4 2. Close switch S. The internal resistance of the power supply is only about 10 kω, so the voltmeter should read half the power supply voltage almost immediately. (See Questions.) 3. Open switch S and observe the voltmeter reading begin to decrease rapidly at first, and then more slowly. This time the capacitor is discharging through about 20 MΩ and has a much larger time constant. The response of the circuit is slow enough that you can time the decay with a stopwatch. Record the voltage at 3.0s intervals for a total of about 30 s. Record in Data Table Repeat steps 2 3 twice more and calculate the average voltage for each time. Record in Data Table Plot a graph of V vs. t to see the exponential decay explicitly. 6. Plot a second graph, this time of ln V vs. t, and do a linear regression analysis. What is the time constant for this circuit, and how does it compare to what you would expect from the measured circuit values? Remember, the total resistance is now R = R box + R meter. Procedure III: Verify formula for equivalent capacitance of series combination. 1. Replace the 1.0µF capacitor with a series combination of the 1.0µF capacitor and the 0.68µF capacitor. (The figure shows approximate nominal values; use the circuit components supplied to you and the measured values from Procedure I.) S R box  10 Mž R int Å 10 k½ Low Voltage Power Supply 6.0 V C µf C 2  o.68 µf V R meter  10 Mž Figure 5 2. Close the switch so the capacitors charge up almost immediately. 3. Open the switch and begin timing. Measure the voltage decay every 3.0 s for about 30 s. Record in Data Table 3.
5 4. Repeat steps 2 3 twice more and calculate the average voltage for each time. Record in Data Table Plot a graph of ln V vs. t (you do not need a graph of V vs. t) and do a linear regression analysis. What is the time constant for this circuit, and how does it compare to what you would expect from the measured circuit values? Procedure IV: Verify formula for equivalent capacitance of parallel combination. 1. Replace the series combination with a parallel combination. (The figure shows approximate nominal values; use the circuit components supplied to you and the measured values from Procedure I.) S R box  10 Mž R int Å 10 k½ Low Voltage Power Supply 6.0 V V R meter  10 Mž C µf C 2  o.68 µf 2. Close the switch so the capacitors charge up almost immediately. 3. Open the switch and begin timing. Measure the voltage decay every 3.0 s for about 30 s. Record in Data Table Repeat steps 2 3 twice more and calculate the average voltage for each time. Record in Data Table Plot a graph of ln V vs. t (you do not need a graph of V vs. t) and do a linear regression analysis. What is the time constant for this circuit, and how does it compare to what you would expect from the measured circuit values?
6 Name RC Circuits Data Table 1 Circuit component Nominal Value Measured Value R box MΩ MΩ R meter 10 MΩ 10 MΩ C 1 µf µf C 2 µf µf Data Table 2: Single capacitor Time (s) V 1 (V) V 2 (V) V 3 (V) V avg (V) ln V avg Time constant from graph Time constant from circuit values Percent Difference Attach graphs of V vs. t and of ln V vs. t.
7 Data Table 3: Series capacitors Time (s) V 1 (V) V 2 (V) V 3 (V) V avg (V) ln V avg Time constant from graph Time constant from circuit values Percent Difference Attach graph of ln V vs. t. Data Table 4: Parallel capacitors Time (s) V 1 (V) V 2 (V) V 3 (V) V avg (V) ln V avg Time constant from graph Time constant from circuit values Percent Difference Attach graph of ln V vs. t.
8 Questions 1. Show that for an exponential decay like Eq. 1 the time for the voltage to decay to V 0 /e is given by the time constant shown in Eq When the switch is closed in Fig. 4 and the capacitor is charging, what is the equivalent resistance of the circuit? What is the time constant for the circuit? (Do you see why the capacitor charged almost immediately?) 3. The voltmeter reads the voltage across its terminals, and in Fig. 4 that is not the same as the voltage across the capacitor. Show, however, that the voltmeter reading is proportional to the capacitor voltage. What is the constant of proportionality? 3. Each time you closed the switch and charged the capacitor (or capacitor combination) the voltmeter read only about half the power supply voltage. Explain this in view of your answer to Question 2.
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