ELECTRICAL CIRCUITS IV
The RC Series Circuit
The charge Q on a capacitor’s plate is proportional to the potential difference V across the capacitor. We express this as
,
where Cis a proportionality constant known as the capacitance. C is measured in farads (1 farad = 1 coulomb/volt).
Figure 1 shows an RC series circuit connected to a battery with a potential Vo and a spdt (single pole, double throw) switch.
Figure 1
When the spdt switch is connected to position A at time t = 0, the battery will begin to “charge” the capacitor and the voltage across the capacitor will increase according to the expression
.
After a relatively long time, the capacitor of capacitance C will become fully charged to a potential Vo (volts).
When the switch is moved to position B, the battery will be disconnected from the RC circuit and the capacitor will “discharge” through the resistor. As the discharging current flows, the charge Q is depleted, reducing the potential (voltage) across the capacitor. This process creates an exponentially decreasing voltage across the capacitor, given by the expression
.
How rapidly the voltage increases or decreases across the capacitor depends on the product RC, known as the time constant of the circuit. A large time constantmeans that the capacitor will discharge slowly.
Objectives
- Measure an experimental time constant of an RC series circuit.
- Measure the potential across a capacitor as a function of time as it charges and as it discharges.
- Fit an exponential function to the data. One of the fit parameters corresponds to an experimental time constant.
- Compare the time constant to the value predicted from the component values of the resistance and capacitance.
Materials
Windows PC / 10-F non-polarized capacitorUniversal Lab Interface (ULI) / 100-k, 47-k resistors
Logger Pro / two D cells with battery holder
Voltage Probe / single-pole, double-throw switch (spdt)
5 connecting wires / 3 alligator clips
Preliminary Questions
1.Consider a candy jar, initially with 1000 candies. You walk past it once each hour. Since you don’t want to obvious about taking candy, each time you take 25% of the candies remaining in the jar. Sketch a graph of the number of candies for a few hours.
2.How would the graph change if instead of removing 25% of the candies, you removed 50%? Sketch your new graph.
3.These are exponential curves. Can you write the equations for them?
Procedure
1.Connect the circuit as shown in Figure 1 above with the 10-F capacitor and the 100-k resistor. Record the values of your resistor and capacitor in your data table, as well as any tolerance values marked on them.
2.Connect the Voltage Probe to the ULI and across the capacitor, with the red (positive lead) to the side of the capacitor connected to the resistor. Connect the black lead to the other side of the capacitor.
3.Prepare the computer for data collection by opening “Exp 27” from the Physics with Computers experiment files of Logger Pro. A graph will be displayed. The vertical axis of the graph has potential scaled from 0 to 4V. The horizontal axis has time scaled from 0 to 10s.
Procedure, Discharging
4.In order to observe the discharging, you must first charge the capacitor. Charge the capacitor for 30 s or so with the switch in position A 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. Wait until the potential is constant.
5.Click to begin data collection. As soon as graphing starts, throw the switch to position B to discharge the capacitor. Your data should initially show a constant voltage, then a decreasing voltage.
6.To compare your data to the model, select only the data after the voltage has started to decrease by dragging across the graph; that is, omit the constant portion. Click the curve fit tool , and from the function selection box, choose the Natural Exponential function, A*exp(–C*x )+B. Click , and inspect the fit. Click to return to the main graph window.
7.Record the value of the fit parameters in your data table. Notice that the C used in the curve fit is not the same as the C used to stand for capacitance. Compare the fit equation to the mathematical model for a capacitor discharge proposed in the introduction,
How is the fit constant C related to the time constant of the circuit, which was defined in the introduction? (The values of A and B may seem unreasonable, however, the fit constant C should give you a good result.)
8.Print the graph of potential vs. time.
Procedure, Charging
9.The capacitor is now discharged. To monitor the charging process, click . As soon as data collection begins, throw the switch to position A. Allow the data collection to run to completion.
10.This time you will compare your data to the mathematical model for a capacitor charging,
Select the data beginning after the voltage has started to increase by dragging across the graph. Click the curve fit tool, , and from the function selection box, choose the Inverse Exponential function, A*(1 – exp(–C*x))+B. Click and inspect the fit. Click to return to the main graph window. (Again the values of A and B may seem unreasonable.)
11.Record the value of the fit parameters in your data table. Compare the fit equation to the mathematical model for a charging capacitor.
12.Print the graph of potential vs. time.
Procedure, Smaller Resistance Value
13.Now repeat the experiment with a resistor of lower value. How do you think this change will affect the way the capacitor discharges and charges? Rebuild your circuit using the 47-k resistor and repeat Steps 4–12.
Data Table
Resistor / Capacitor / Fit ParametersTrial / R () / C (F) / A / B / C
Discharge 1
Charge 1
Discharge 2
Charge 2
Analysis and Calculations
1.Using the values of R and C from the data table, calculate the theoretical time constants for each of the four trials. (Note that 1 F= 1 s).
2.Calculate and enter in the results table the inverse of the fit constant (1/C) for each trial.
3.Find the time constant for each of the trials from your graphs. Explain or explicitly show on your graphs how you found the time constant.
4.Use a different part of the graphs for each trial and again find the time constant. Explain below or explicitly show on the graphs how you obtained your values.
5.What was the effect of reducing the resistance of the resistor on the way the capacitor discharged?
Results Table
Trial / Theoretical Time Constant, RC (sec) / Fit Parameter Time Constant, 1/C (sec) / Time Constant from Graph (sec) / Time Constant from another Part of Graph (sec) / Maximum Percentage DifferenceDischarge 1
Charge 1
Discharge 2
Charge 2
Questions
1.How would the discharge graph look if you plotted the natural logarithm of the potential across the capacitor vs. time? Sketch a prediction on the graph below. What is the significance of the slope of the plot of ln(V) vs. time for a capacitor discharge circuit? (Hint: Take the natural log of the discharging voltage, V = Voe-t/RC, and relate this equation to that of a straight line, y = mx + b.
2.When the capacitor is discharging, what percentage of the initial potential remains after one time constant has passed? After two time constants? What is the minimum number of time constants until less than one percent remains?
3.When the capacitor is charging, what percentage of the final potential is reached after one time constant? After two time constants? What is the minimum number of time constants until the voltage reaches over 99% of the final voltage?