Activity 25 PS-2826 Time Constant of a Resistor-Capacitor Circuit Electricity: resistor-capacitor circuit, time constant GLX setup file: time constant Qty Equipment and Materials Part Number 1 PASPORT Xplorer GLX PS-2002 1 Voltage Probe (included with GLX) 1 CASTLE Kit EM-8624A 2 D Cell PI-6602 2 Alligator Clip Adapters SE-9756 Purpose The purpose of this activity is to measure the voltage across a capacitor as it is charged and then discharged through a resistor that is in series in a circuit with the capacitor. Background When a DC voltage source is connected across an uncharged capacitor, the rate at which the capacitor charges up decreases as time passes. At first, the capacitor is easy to charge because there is very little charge on the plates. But as charge accumulates on the plates, the voltage source must do more work to move additional charges onto the plates because the plates already have charge of the same sign on them. As a result, the capacitor charges exponentially, quickly at the beginning and more slowly as the capacitor becomes fully charged. If the capacitor is in a circuit with a resistor, the time it takes to charge the capacitor to a maximum value depends on both the capacitance, C, of the capacitor and the resistance, R, of the resistor. The product of R Fig. 1: RC circuit and C is called the capacitive time constant and it is symbolized by where = RC. NOTE: The stated value of a capacitor may vary by as much as ±20% from the actual value. The capacitive time constant is the amount of time it takes to charge the capacitor to 63.2% of its maximum charge, or the amount of time it takes to discharge a capacitor to 36.8% of its maximum charge. Can the charge on a capacitor be determined by measuring the voltage across the capacitor? The charge, q, is the product of the capacitance, C, and the voltage, V. Therefore, the charge is directly proportional to the voltage, or q = CV. Preview Use a Voltage Probe to measure the voltage across a capacitor as it is charged and then discharged through a resistor. Use the Xplorer GLX to record and display the voltage. Determine the time constant of the circuit from the graph of voltage versus time. Compare the time constant to the product of the capacitance, C, of the capacitor and the resistance, R, of the resistor. Physics with the Xplorer GLX 2006 PASCO p. 187
Safety Precaution Follow all directions for using the equipment. Procedure GLX Setup 1. Turn on the GLX ( ) and open the GLX setup file labeled time constant. (Check the Appendix at the end of this activity.) 2. The GLX displays a Graph screen of Voltage (V) versus Time (s). The file is set to measure voltage 100 times per second (100 Hz). 3. Plug a Voltage Probe into the voltage input port on the left side of the Xplorer GLX. Equipment Setup 1. Put two D cells into the battery holder (from the CASTLE Kit). 2. Set up a circuit with one 10 ohm (10 ) resistor, the 25,000 microfarad (25,000 F or 0.025 F) capacitor, and the voltage source as shown. Be careful to connect an alligator clip to the spring in the battery holder as shown. 4. Leave one clip disconnected until you are ready to collect data. 5. Put alligator clip adapters on the ends of the Voltage Probe. 6. Connect the Voltage Probe to the posts on the top of the capacitor. Record Data NOTE: The procedure is easier if one person handles the equipment and a second person handles the Xplorer GLX. Fig. 2: GLX Graph Fig. 3: Equipment setup Physics with the Xplorer GLX 2006 PASCO p. 188
Charge the Capacitor 1. Press Start ( ) on the GLX to start recording data. 2. Wait about 2 seconds and then connect the clip to the voltage source to complete the circuit. Watch the Voltage-Time Graph on the GLX. 3. When the voltage reaches its maximum value and does not change, disconnect the resistor from the voltage source to open the circuit. DO NOT STOP RECORDING DATA YET. Discharge the Capacitor 1. Move the clip that had been connected to the voltage source to the end of the resistor that was previously connected to the battery holder as shown. 2. Watch the Voltage-Time Graph on the GLX. 3. When the voltage reaches its minimum value and does not change anymore, press to stop data recording. Analysis Fig. 4: Discharge capacitor Examine your graph of voltage versus time to find the maximum voltage. 1. In the Graph screen, move the cursor to the maximum value of voltage and record the value in the Data Table. Use the data during the charging of the capacitor to find the time to charge to 63.2% of the maximum voltage. 2. Calculate the voltage that is 63.2% of the maximum voltage and record the value. 3. Move the cursor to the point on the Graph screen where the capacitor begins to charge. Press to open the Tools menu and select Delta Tool. 4. Carefully move the cursor to the point closest to the voltage that is 63.2% of maximum and record the time displayed by the Delta Tool. Fig. 5: Select Delta Tool Use the data for the discharging of the capacitor to find the time to discharge to 36.8% of maximum voltage. 5. Calculate the voltage that is 36.8% of the maximum voltage and record the value. Subtract that voltage from the maximum voltage to determine the change in voltage from the maximum to the value that is 36.8% of maximum. (For example, if the maximum voltage is 3 V, and the value that is 36.8% of maximum is 1 V, the change in voltage is 2 V.) Physics with the Xplorer GLX 2006 PASCO p. 189
6. Turn off Delta Tool for the moment. 7. Move the cursor to the point on the Graph screen where the capacitor begins to discharge, and turn on the Delta Tool again. 8. Carefully move the cursor until the Delta Tool shows the change in voltage you calculated previously. Record the time displayed by the Delta Tool. 9. Calculate the average of the two times. Calculate the theoretical value for the capacitive time constant based on the resistance and capacitance values on the resistor and capacitor, respectively. Compare the average of your experimental results to the theoretical value. Extension Repeat the procedure with a light bulb instead of a resistor. Record your results and answer the questions in the Lab Report section. Appendix: Opening a GLX File To open a specific GLX file, go to the Home Screen ( ). In the Home Screen, select Data Files and press to activate your choice. In the Data Files screen, use the arrow keys to navigate to the file you want. Press to open the file. Press the Home button to return to the Home Screen. Press to open the Graph. Physics with the Xplorer GLX 2006 PASCO p. 190
Lab Report Activity 25: Time Constant of a Resistor-Capacitor Circuit Name Date Data Sketch a graph of voltage versus time. Include units and labels for your axes. Data Table Item Resistance of resistor Capacitance of capacitor Maximum voltage 63.2% of maximum voltage Time to 63.2% of max. voltage 36.8% of maximum voltage Time to 36.8% of max. voltage Average time constant Theoretical time constant Percent difference Value %diff = theoretical average theoretical 100% Physics with the Xplorer GLX 2006 PASCO p. 191
Calculations Based on the maximum voltage, calculate and record the value that is 63.2% of maximum, and the value that is 36.8% of maximum. Calculate the average time constant based on the two experimental values. Calculate the theoretical capacitive time constant,, based on R and C, where = RC. Calculate the percent difference between the theoretical and average values of the time constant. Questions 1. Is the Voltage Probe placed in parallel or in series in this circuit? Explain. 2. What component of the circuit is the Voltage Probe measuring? 3. How does your average experimental value for the time constant compare to the theoretical value? 4. What are some reasons for the difference, if any, between the average experimental value and the theoretical value of the time constant? 5. The time constant is the produce of resistance, measured in ohms, and capacitance, measured in farads. Use the following definitions of units to show algebraically that the unit for the time constant is seconds. ohm = volt amp, coulomb farad = volt, amp = coulomb second Physics with the Xplorer GLX 2006 PASCO p. 192