Charge and Discharge of a Capacitor - Notes | PY 212, Study notes of Physics

Download Charge and Discharge of a Capacitor - Notes | PY 212 and more Study notes Physics in PDF only on Docsity! Charge and Discharge of a Capacitor PY 212 1 Introduction and Purpose Capacitors are devices that can store electric charge and energy. A capacitor can be slowly charged to the necessary voltage and then discharged quickly to provide the energy needed. It is even pos- sible to charge several capacitors to a certain voltage and then discharge them in such a way as to get more voltage (but not more energy) out of the system than was put in. In this laboratory exercise you will observe the charging and discharging of a capacitor and you will measure the characteristic response time. You will use the response time to determine the value of the element’s capacitance. This experiment features an RC circuit, which is the simplest circuit that uses a capacitor. You will investigate ways to change the effective capacitance by combining capacitors in series and parallel arrangements. 2 Discussion of Principles Review of Capacitor Basics A capacitor consists of two conductors separated by a small distance. When the conductors are connected to a charging device (for example, a battery), charge is transferred from one conductor to the other until the difference in potential between the conductors (due to their equal, but opposite, charge) becomes equal to the potential difference between the terminals of the charging device. The amount of charge stored on either conductor is directly proportional to the voltage, and the constant of proportionality is known as the capacitance. This is written algebraically as Q = C∆V . The charge is measured in units of coulombs (C), the potential difference in volts (V), and the capacitance in units of farads (F). Capacitors are physical devices; capacitance is a property of those devices. Charging and Discharging In a simple RC circuit a resistor and a capacitor are connected in series with a battery and a switch. When the switch is closed, charge on the conductors builds to a maximum value after some time. If we wish to examine the charging and discharging of the capacitor, we are interested in what happens immediately after the switch is closed or opened, not the later behavior of the circuit in its steady state. The charge on a capacitor in a simple RC circuit during the charging process at some time is given by Q = Qf [1− e−t/RC ] (1) 1 where Qf represents the final charge on the capacitor that accumulates after an infinite length of time, R is the circuit resistance and C is the capacitance of the capacitor. From this expression you can see that charge builds up exponentially after the switch is closed. On the other hand, as an initially charged capacitor discharges - for example after its conductors are connected through a resistor - the charge is given by Q = Q0e −t/RC (2) where Q0 represents the initial charge on the capacitor at the beginning of the discharge, i.e., at t = 0 . You can see from this expression that the charge decays exponentially when the capacitor discharges, and that it takes an infinite amount of time to fully discharge. Time Constant The product RC (having units of time) has a special significance; it is called the time constant of the circuit. The time constant is the amount of time required for the charge on a charging capacitor to rise to 63% of its final value. In other words, when t = RC, Q = Qf (1− e−1) = 0.632 ·Qf . (3) Another way to describe the time constant is to say that it is the number of seconds required for the charge on a discharging capacitor to fall to 36.8% (e−1 = 0.368) of its initial value. It is useful to describe charging and discharging in terms of the potential difference between the conductors (i.e., the voltage across the capacitor), since the voltage across a capacitor can be measured directly in the lab. By using the relationship Q = C∆V , Eqs. (1) and (2) that describe the charging and discharging of a capacitor can be rewritten in terms of the voltage. Merely divide both equations by C, and the relationships become: For charging: V = Vf [1− e−t/RC ] (4) and for discharging: V = V0e −t/RC (5) In this experiment you will monitor the potential difference, and thus, indirectly, the charge on a capacitor. The voltage measurements will be used in two different ways to compute the time constant of the circuit. Finally, two capacitors will be connected in parallel to examine their equivalent circuit capacitance. 3 Procedure Charging a Capacitor 1. Set up a circuit like the one shown in figure 1using the 33 Ω resistor and the 100 µF capacitor. Bypass the inductor coil by simply inserting a single wire into each socket on either side of the coil. 2 By using the voltage relationship for charging a capacitor, the capacitance can be determined by graphical means. If we rearrange Equation (3) we can see: Vf − V Vf = e−t/RC (6) Taking the natural log of both sides an multiplying by -1 gives −ln ( Vf − V Vf ) = t RC (7) In your Data Summary table on the last page of this handout, fill in the last two columns to com- plete these calculations. Plot a graph of −ln[(Vf − V )/Vf ] as a function of time using the values from the fourth and second columns of your Data Summary table. Don’t forget to show any relevant sample calculations in your report. Calculate the slope of the line and determine the time constant and capacitance. Record these results on the last page. Be sure to include these results and a discussion of the results in the analysis section of your report. Measuring Effective Capacitance Recall that capacitance adds directly when capacitors are connected in parallel and inversely when connected in series Ceffective,parallel = C1 + C2 + C3 + ... (8) 1 Ceffective,series = 1 C1 + 1 C2 + 1 C3 + ... (9) 1. Connect the 330 µF capacitor in parallel with the capacitor that you used in the previous procedure by connecting a jumper wire across the two lower terminals of the capacitors. 2. Switch the resistor to the 10 Ω resistor. 3. After you have completed modifying the circuit, record another data set on your graph by pressing the “Start” button. It is easiest to analyze the new data set if it is the only thing in the window. Remove the first data set so only the newest graph is showing. You may also have to adjust the axes as you did before in order to get one full wavelength in the window for highest precision. 4. Follow the same procedure that you first used to determine the time constant and the capac- itance of the parallel combination from the graph. You can record any useful information on the last page of this handout. Be sure to include this information in your report. 5. Calculate the “theoretical” effective capacitance using the values printed on the circuit board and Eqn. 8. 6. Calculate the percent difference between the two values for effective capacitance. Comment on this percent difference in your report. 5 Some things to consider while writing your report... • In this lab we used two different methods for determining the time constant for the 100µF capacitor. Which method was the most accurate? Provide justification for your answer. • How do the charging times compare between the single capacitor and the parallel arrangement? Can you justify this with a theoretical argument? • What was the relationship between between voltage across the capacitor and time? Was the relationship proposed in the theory section of the procedure verified with any observations you made. • Why did we plot −ln ( Vf−V Vf ) v. time? How did graphing our data in this way makes things easier to interpret? 6 Charging a capacitor t = s (when V = 0) t = s (when V = 0.632Vf ) RC = s (experimental) Calculation of capacitance R = Ω C = F (accepted) C = F (experimental) Percent Error = % The final value of V was Vf = V Table 1: Data Summary Table Voltage (V) Time (s) Vf−V Vf -ln ( Vf−V Vf ) 1 2 3 4 1/RC = s−1 Time Constant (RC) = s Capacitance C = F Measuring effective capacitance t = s (when V = 0) t = s (when V = 0.632Vf ) RCeff = s (experimental) Ceff = F (experimental) Ceff = F (accepted) Percent Error = % Show relevant sample calculations in your final report 7