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A4 Charging and Discharging of a Capacitor(5).pdf

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Wagih Ghobriel

CHARGING AND DISCHARGING OF A CAPACITOR INTRODUCTION In this experiment, we will study charging a capacitor by connecting it to a voltage source through a resistor. The experiment also includes the study of discharging the capacitor through a resistor. OBJECTIVES: • To study the charging and discharging processes of capacitors. • To determine the time constant τ of an RC-circuit. EQUIPMENT TO BE USED: • ~200 kΩ resistors. 2 • ~330 µF capacitor. 1 • Xplorer GLX 1 • Multimeter. 1 • Voltage sensor 1 • Alligator clip leads 4 • 10 V supply 1 THEORY: Charging a Capacitor: Consider a circuit as shown in Figure 1. Capacitoris initially uncharged. By closing the switch , a current is set up in the loop and the capacitor begins to charge. Applying Kirchoff’s loop rule, we get (1) where is the supply voltage, is the resistance, is the charge on the capacitor and is the capacitance. Substituting for the current , Equation (1) becomes (2) Figure 1: Charging circuit Rearranging the terms, Equation (2) becomes (3) The solution of Equation (3) is given as ( ) (4) where is the maximum value of the voltage across the capacitor. This determines the charge on the capacitor as a function of time . The voltage across the capacitoisgiven as (5) Dividing Equation (4) by yields ( ) (6) At a specific value of time (called the time constant of the R-C circuit), ( ) (7) (8) Therefore, by plotting versus , the time constant may be determined, and hence, the value of can be calculated, provided is known. Equation (6) shows that the growth of the capacitor’s voltage is not linear, but rather grows exponentially reaching a saturation value which equals the voltage of the voltage source. The capacitor is considered to be fully charged after a period of about five time constants. The current in the circuit at a given time is given as (9) where represents the initial current in the circuit. Therefore we can write (10) At time , Discharging a Capacitor: For the discharging process, consider the circuit shown in Figure 2. After closing the switch for a long time (compared to the circuit’s time constant), the capacitor will be fully charged to a value of . As the switch is opened, the power supply is disconnected from the circuit and the capacitor starts to discharge through the resistor. Figure 2: Discharging circuit Following the same procedure as for the charging analysis, the differential equation for the discharging process is given by (11) The solution of the equation is (12) This equation determines charge on the capacitor as a function of time. The voltage across the capacitor is given by the following equation. (13) At time (the time constant of the circuit), the voltage across the capacitor becomes (14) The current expression becomes At time the current becomes (15) The minus sign indicates that the current is decreasing (i.e. charge is decreasing). Experimental Setup: INITIAL SETUP FOR GLX: To start a new experiment on the Xplorer GLX 1. Press the home button to go to the home screen. 2. Use the arrow keys to highlight the data files and press the button to open the data files screen. 3. Press to open the files menu and select new file. 4. When the GLX asks if you would like to save the previous file, presto save or not to save. PROCEDURE: 1. Create the circuit pictured on page 2, above. The voltage sensor is connected so that it will measure the voltage across the capacitor. 2. Record the values of capacitance and resistance (measure the resistance directly using a multimeter and read the capacitance value from capacitor). C = _______________ Farad (Convert microfarad to farad) R= _______________ Ohm (Kilo ohm to ohm) GLX SETUP: 1. Connect a voltage sensor to the GLX. 2. If there are other sensors connected, remove them. 3. Setup the Graph to plot Voltage versus Time. Press to return to the Home screen; press to open Graph. The display will be automatically setup to graph Voltage versus Time. Press to go to the home screen. Then press the Table option. Highlight the voltage and time tabs of the table by pressing and make sure that columns are Voltage and Time. Now go back to the graph by pressi
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