E X P E R I M E N T 7


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1 E X P E R I M E N T 7 The RC Circuit Produced by the Physics Staff at Collin College Copyright Collin College Physics Department. All Rights Reserved. University Physics II, Exp 7: The RC Circuit Page 1
2 Purpose In this experiment, you will both charge and discharge capacitors in an RC circuit. We will explore how the charging and discharging times depend on the resistance and capacitance in the circuit. Equipment Power amplifier RLC circuit board Voltage sensor Digital multimeter Resistor, 10 kw Stop Watch Doublethrow switch Theory When you connect a battery or power supply to an uncharged capacitor, the charge does not instantaneously appear on the capacitor plates. The charge on (and voltage across) the capacitor increases exponentially from zero. Similarly, the charge on (and voltage across) the capacitor decreases exponentially toward zero when you discharge it. As with all properties that change at an exponential rate, the rate of change of the charge (or voltage) on a capacitor at any point in time is proportional to the amount of charge (or voltage) already present at that time. And like all exponential curves, both the halflife of the measured property (the time required for the property to change to half or double its value) and its time constant (the time required for the property to change to 1/e of its initial or final value) are constant. The values of the halflife and of the time constant depend on the values of C and R in the circuit. Figure 7.1 shows a capacitor, a resistor, a switch, and a battery (power source) connected in series. Assume initially that the switch is open and there is no charge on the capacitor. When the switch is closed at time t = 0, current begins to flow in the circuit. Figure 7.1 University Physics II, Exp 7: The RC Circuit Page 2
3 Kirchhoff s second rule states that the sum of all the voltage drops across the components of a loop (or series circuit) is zero. The voltage across the capacitor is q/c; the voltage across the resistor is IR; and the battery has a voltage V 0. Assume that the voltages across the other circuit components (wires and switch) are negligible. Setting the sum of all the voltage drops to zero, you get V 0 q/c IR = 0 Equation 7.1 Since the current cannot flow across the capacitor, charge must accumulate on the two capacitor plates. The time rate of change or accumulation of charge is, of course, the current I, so dq/dt = I Equation 7.2 Eliminating I from equation (7.1) by using its value in equation (7.2), you get V 0 q/c Rdq/dt = 0 Equation 7.3 Equation (7.3) is a differential equation for the charge on the capacitor as a function of time. Solving this equation, you get q(t) = Q(1 e t/rc ), Equation 7.4 where Q is the charge on the capacitor after an infinite time. To find the current, differentiate Equation 7.4 with respect to time to get I(t) = (V 0 /R)e t/rc Equation 7.5 where C = Q/V 0 Note from Equations 7.4 and 7.5 that the initial charge on the capacitor is zero. This is consistent with the requirement that the capacitor be uncharged initially. The initial current (at t = 0) is V 0 /R. This is the current that would flow if you replaced the capacitor with a short piece of wire. After an infinite time, the charge on the capacitor is Q, and the current in the circuit is zero. The denominator in the exponent of Equation 7.5 is RC. This factor is called the time constant τ c of the series RC circuit. After a time equal to τ c, the current in the circuit is reduced to 1/e (36.8%) of its initial value. This is true for all values of time: if you measure the current at any time while the capacitor is charging, the current at a time τ c later will be 36.8% of the value you measured. Another way of expressing the rate of current decrease is the half life of the circuit. The half life, τ h, is the time for the current to be reduced to 50% of its value. The half life is related to the time constant by τ h = τ c ln 2 Equation 7.6 University Physics II, Exp 7: The RC Circuit Page 3
4 The half life is the time for the capacitor to discharge from any given charge to half that charge, or the time required to charge a capacitor to one half the voltage of the power source. Procedure The basic procedure is to connect an uncharged capacitor, a resistor and a voltage source in series and measure the rate at which the voltage across the capacitor increases. You will use two separate techniques to measure the change in V C, and you will measure the time both manually and automatically. In the first technique you will observe how the voltage across the capacitor changes by using the Data Studio Scope display. The voltage source will be the Power Amplifier controlled by the computer. When the Power Amplifier applies a squarewave voltage to the circuit, the voltage across the capacitor shown on the Scope display will be a modified square wave as shown in Figure 7.2. Its leading edge is the capacitor s charging curve and its trailing edge is its discharge curve. You will measure the half life of the charging curve on a display of the voltage across the capacitor. Figure 7.2 In the second technique you will observe the discharge of a capacitor through a resistor. You will first charge the capacitor with a DC voltage supplied by the Power Amplifier, then use a doublethrow switch to disconnect the voltage source and connect the resistor across the capacitor. You will use a digital multimeter to measure V C during both the charging and discharging of the capacitor, but you will record your measurements only for the discharge. You will start a stop watch when V C has an arbitrary value; then stop the stop watch when V C drops to half that value. A. Automated Time Measurement Experimental Setup 1. Connect the Power Amplifier to analog input channel A of the Interface. The RLC circuit board contains various resistors, capacitors, and other components. Each component on the circuit board is labeled with its type and value. For each component, there are banana jacks for wiring the components together and to external equipment. You will use only one resistor and one capacitor. 2. Select the 100 Ω resistor and the 330 μf capacitor, and use the patch cords to wire them in series with the output of the Power Amplifier. Record the values of the components in Table Connect the Voltage Sensor across the capacitor and to analog input channel B of the Interface. Switch on the computer and the Interface. University Physics II, Exp 7: The RC Circuit Page 4
5 4. Open the Data Studio program and select Create Experiment. Doubleclick the Power Amplifier icon in the Sensors list. Change Sine Wave to Square Wave, set the Amplitude to V, and set the Frequency to Hz. 5. Doubleclick the Scope display to view the voltage signal from the sensor connected to Channel B. Data Collection 1. Switch on the Power Amplifier. 2. Click the Start button to start collecting data. Continue data collection for about 4 seconds and then press Stop. The Data List will show the data as Run #1. 3. Switch off the Power Amplifier. 4. Adjust the V/div on the SCOPE display so the wave pattern fills the graph. 5. Drag a rectangle around the rising portion of one cycle of the displayed voltage. Start the rectangle before the exponentially falling voltage reaches 4 V and end it before the rising voltage reaches 4 V. 6. Using the Smart Tool, measure and record (a) the time t a when the voltage just starts to raise, and (b) the time t b when the rising voltage crosses 0 volts. 7. Calculate and record both the half life τ h and the time constant τ c of the circuit. 8. Remove the resistor from the circuit. Measure and record its resistance with the multimeter using the appropriate resistance scale. 9. Using the measured value of R, calculate and record both the theoretical time constant and halflife of the circuit. 10. Calculate and record the percent error in your experimental value of the time constant. B. Manual Time Measurement Equipment Setup 1. Connect the Power Amplifier to analog input channel A. 2. Select the 330 μf capacitor on the RLC circuit board and use the patch cords to connect the capacitor, a 10 kω resistor, the Power amplifier, a SPDT switch and a voltmeter as shown in Figure 7.3. Figure 7.3 University Physics II, Exp 7: The RC Circuit Page 5
6 Note. The singlepole doublethrow (SPDT) switch has three positions: When you close it upward, the Power Amplifier, the capacitor, and the resistor are connected in series so the capacitor is charged through the resistor by the Power Amplifier. The voltmeter is connected across the capacitor. When the switch is open, the capacitor and voltmeter are disconnected from the circuit. When you close it downward, the power amplifier is disconnected from the capacitor and voltmeter so the capacitor discharges through the resistor. 3. In the Data Studio SETUP window, doubleclick Power Amplifier in the Sensors list. In the Signal Generator window, select Square Wave, set the Amplitude to V, and set the Frequency to Hz. Data Collection 1. Switch on the Power Amplifier. Select ON in the Signal Generator window. 2. Put the SPDT switch in the open position. 3. Discharge the capacitor by shorting it for a moment with a clip lead. 4. Charge the capacitor by closing the switch to the power amplifier position. Let the capacitor charge until the voltage across it is nearly 4 volts. 5. Open the SPDT switch. The capacitor will retain its charge (except for a small amount which leaks off through the voltmeter). 6. Select OFF in the Signal Generator window and switch off the Power Amplifier. 7. Start discharging the capacitor by closing the SPDT switch to the resistor position. 8. When the voltage across the capacitor passes 3 volts, start the stop watch. When the voltage reaches 1.5 volts, stop the watch and record the time interval t. 9. Calculate and record both the half life and the time constant of the RC circuit. 10. Remove the resistor from the circuit and measure and record the value of its resistance with the multimeter using the appropriate resistance scale. 11. Using the measured value of R, calculate and record both the theoretical time constant and halflife of the circuit. 12. Calculate and record the percent error in your experimental value of the time constant. University Physics II, Exp 7: The RC Circuit Page 6
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