apacitors THEORY The charge Q on a capacitor s plate is proportional to the potential difference V across the capacitor. We express this with Q = V (1) where is a proportionality constant known as the capacitance. is measured in units of Farads (F), (1 Farad = 1 oulomb/volt). If a capacitor of capacitance (in farads), initially charged to a potential V0 (volts) is connected across a resistor R (in Ohms), a timedependent current will flow according to Ohm s law. This situation is shown by the (resistorcapacitor) circuit below when the switch is changed. Figure 1 As current flows, the charge q on the capacitor plates is depleted, reducing the potential difference across the capacitor, which in turn reduces the current. This process creates an exponentially decreasing voltage, modeled by e! t (2) The rate of the decrease is determined by the product, known as the time constant of the circuit. A large time constant means that the capacitor will discharge slowly.
When the capacitor is charged, the potential across it approaches its final value exponentially, modeled by (1! e! t ) (3) The same time constant describes both the rate of charging and discharging. OBJETIVES _ Measure an experimental time constant of a resistorcapacitor circuit. _ ompare the time constant to the value predicted from the component values of the resistance and capacitance. _ Measure the potential across a capacitor as a function of time as it discharges and as it charges. _ Fit an exponential function to the data. One of the fit parameters corresponds to an experimental time constant. MATERIALS omputer Logger Pro software Vernier Voltage Probe onnecting wires 2Low leakage capacitors (1 to 10 µf) 2kΩ resistors (50 to 150 kω) Power supply Singlepole, doublethrow switch PROEDURE 1. Measure the values of the capacitors and resistors and record the values in the data table. The lab instructor will show you how to measure the values. 2. onnect the circuit as shown in Figure 1 above with the first capacitor and the first resistor. 3. onnect the Voltage Probe to the Logger Pro and across the capacitor, with the red (positive lead) to the side of the capacitor connected to the resistor. onnect the black lead to the other side of the capacitor. 4. Prepare the computer for data collection by opening Exp 27 from the Physics with omputers experiment files of Logger Pro 2.1. The graph displayed will have a vertical axis corresponding to potential (scaled 0 to 4 V) while the horizontal axis corresponds to time (scaled 0 to 10 s). 5. harge the capacitor for 30 s or so with the switch in the position as illustrated in Figure 1.
You can watch the voltage reading at the bottom of the screen to see if the potential is still increasing. Adjust the voltage from the power supply so that the voltage lies close to 3 volts. Wait until the potential is constant. 6. lick to begin data collection. As soon as graphing starts, throw the switch to its other position to discharge the capacitor. Your data should show a constant value initially, then an exponentially decreasing function. 7. To compare your data to the model, select only the data after the potential has started to decrease by dragging a rectangle across the graph; that is, omit the initial constant portion. lick the curve fit tool, and from the function selection box, choose the Natural Exponential function, A*exp ( *x ) B. lick Try Fit and inspect the fit. lick Done to return to the main graph window. 8. Record the value of the fit parameters in your data table. Notice that the used in the curve fit is not the same as the used to stand for capacitance. ompare the fit equation to the mathematical model for a capacitor discharge proposed in the introduction, e! t (2) Notice that the curve fit is equal to one over the product of resistor and the capacitor. Find the percent error between 1/ (curve fit) and. 9. Print the graph of potential vs. time. 10. The capacitor is now discharged. To monitor the charging process, click. As soon as data collection begins, throw the switch the other way. Allow the data collection to run to completion. 11. This time you will compare your data to the mathematical model for a capacitor charging, (1! e! t ) (3) Select the data beginning after the potential has started to increase by dragging a rectangle across the graph. lick the curve fit tool, and from the function selection box, choose the Inverse Exponential function, A*(1 exp ( *x )) B. lick Try Fit and inspect the
fit. lick Done to return to the main graph window. 12. Record the value of the fit parameters in your data table. ompare the fit equation to the mathematical model for a charging capacitor. 13. Now you will repeat the experiment with a different resistor and capacitor. 14. Repeat the experiment with two capacitors hooked in series. apacitors in series add as the reciprocal. That is, 1 & 1 = $ % 1 # & 1! $ " % 2 #! " (4) 15. Repeat the experiment with two capacitors hooked in parallel. apacitors in parallel add directly. That is, = 1 2 (5)
DATA TABLE R 1 1 R 2 2 Fit Parameters Resistor apacitor Time onstant Percent Error Trial A B 1 R (Ω) (F) (s) % Discharge 1 harge 1 Discharge 2 harge 2 apacitors in Series Equivalent apacitor _ (Equation 4) Fit Parameters Resistor apacitor Time onstant Percent Error Trial A B 1 R (Ω) (F) (s) % Discharge 1 harge 1
apacitors in Parallel Equivalent apacitor (Equation 5) Fit Parameters Resistor apacitor Time onstant Percent Error Trial A B 1 R (Ω) (F) (s) % Discharge 1 harge 1 QUESTIONS 1. A capacitor is a device that can store energy as it stores charge. The amount of energy stored is just equal to work done in moving the charge into the capacitor; the work done increases as more charge is added to the plate. Therefore, the energy stored on the capacitor is just 1 1 W= ( ) Q V = ( ) V 2. 2 2 Using the data from your experiment, calculate the energy stored in the capacitor for the first trial. What became of that energy when the capacitor is discharged? 2. Below shows a diagram of a charged parallel plate capacitor without a dielectric between the plates and the other diagram shows the same parallel plate capacitor that has a dielectric placed between the plates. (a) Sketch the electric field lines between the plates on each diagram. (b) If the charge on the plates remains constant while the dielectric is slipped between the plates, will the voltage on the capacitor with the dielectric increase, decrease, or remain constant when compared to the capacitor without the dielectric.
3. (a) Find the equivalent capacitance of the group of capacitors shown in the figure below. (b) Find the charge stored on the 8 µf capacitor. 24 V 6µF 8µF 4µF 2µF 2µF