Capacitors and RC Circuits
Introduction
A charging RC circuit consists of a battery of EMF ε, a resistor of resistance R, and a capacitor of capacitance C. The potential difference across the resistor and capacitor can be derived as follows:
1 / ε = ΔVc + ΔVr / Kirchoff’s voltage law applied to this specific circuit2 / ε = Q/C + IR / Definition of capacitance and Ohm’s law
3 / ε = Q/C + (dQ/dt)R / Definition of current
Everything in equation 3 is a constant except Q. The solution to this “differential equation” requires mathematics beyond the scope of this course, so it must simply be asserted.
4 / Q = Cε(1 – e-t/τ)τ = “time constant” = RC / Solution to equation 4, a “differential equation”
5 / ΔVc = Q/C / Definition of capacitance
6 / ΔVc = ε(1 – e-t/τ) / Equation 4 in equation 5
7 / ε = ε(1 – e-t/τ) + ΔVr / Equation 6 in equation 1
8 / ΔVr = εe-t/τ / Algebra
For a discharging capacitor consisting of only a capacitor and a resistor, a similar analysis can be done.
9 / 0 = ΔVc + ΔVr / Kirchoff’s voltage law10 / 0 = Q/C + IR / Definition of capacitance and Ohm’s law
11 / I = -1/(RC)*Q / Algebra
12 / dQ/dt = -1/(RC)*Q / Definition of current
13 / Q = Qoe-t/ τ
τ = RC / Solution to “differential equation”
14 / ΔVc = Q/C / Definition of capacitance
15 / ΔVc = (Q0/C)e-t/ τ / Equation 13 in equation 14
16 / ΔVc = Voe-t/τ / Definition of capacitance
The purpose of this lab is to test the equations for potential difference for charging and discharging RC circuits.
Physics is fun!
Experimental Procedures
RC Time Constant while Charging
1)Select a resistor and capacitor so that the time constant is approximately 100 seconds.
2)Measure the resistance of your resistor using a DMM.
3)Calculate the theoretical time constant for an RC circuit using your measured resistance and the listed value for capacitance. You may assume ±10% if the capacitor does not list an error.
4)Turn on the power supply and set it to about 5.5 V.
5)Assure that the capacitor has little or no charge by connecting its two prongs with a wire for about 10 seconds.
6)Connect the following items in a loop: resistor, capacitor, and power supply except leave one wire to the power supply disconnected.
7)Configure two DMMs as voltmeters and place them so that they measure the potential differences across the resistor and the capacitor
8)Start your stopwatch and plug in the remaining wire to the power supply
9)Measure the potential differences across the resistor and capacitor every 10 seconds or so for about 3 minutes.
10)Disconnect the circuit and turn off the power supply.
11)Graph the potential difference as a function of time for the capacitor. Test if the graph looks qualitatively like theory predicts. If so, then this is evidence in favor of equation 6.
12)Graph the potential difference as a function of time for the resistor.
13)For the graph of the resistor, add an exponential (not linear) trend line to obtain a curve and an equation for potential difference. Increase the number of digits displayed in the equation so that at least two significant digits appear in the exponent.Does this curve go through most of the error bars? If so, then this is evidence in favor of equation 8.
14)Take the magnitude of the constant in the exponent of the equation from the graph and raise it to the power of -1 to obtain an experimental value for the time constant. You will not be able to obtain an error in this value. Compare this value to the theoretical value calculated in step 3 as a test of equation 8.
15)Be sure to increase the number of digits displayed in the equation of the graph so that at least two significant digits appear in the exponent.
RC Time Constant while Discharging
16)Repeat steps 8 through 14 when you connect the charged capacitor directly to the resistor (no power supply). You will only need one voltmeter and one graph in this case as the resistor and capacitor will have the same potential difference. This curve and the comparison of time constants will be tests of equation 16. Begin measuring potential difference immediately after connecting the circuit as the capacitor will discharge through the resistor.
17)Be sure to increase the number of digits displayed in the equation of the graph so that at least two significant digits appear in the exponent.