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					                                    Capacitors and RC Circuits
Introduction
An RC circuit typically has a battery of EMF ε, a resistor of resistance R, and a capacitor of capacitance
C. If we go around the single loop and set Σ|Vup| = Σ|Vdown|, then

                                  [1] ε = Vc + Vr
                                  ε = Q/C + IR
                                  ε = Q/C + (dQ/dt)R

The solution to this differential equation gives the charge on the capacitor as a function of time:

                                  [2] Q(t) = Cε(1 – e-t/τ)
                                  τ = Time Constant = RC

The definition of capacitance and equation [2] give a formula for the potential difference as a function
of time across a charging capacitor:

                                  [3] Vc(t) = Q/C = ε(1 – e-t/τ)

Placing equation [3] into equation [1] gives the potential difference as a function of time for the
resistor:

                                  [4] Vr = εe-t/τ

For a discharging capacitor, a similar analysis can be done.

                                  0 = Vc + Vr
                                  0 = Q/C + IR
                                  dQ/dt = -1/(RC)*Q
                                  Q = Qoe-t/ τ
                                  [5] Vc(t) = Voe-t/τ = Vr(t)

The purpose of this lab is to test these equations for charging and discharging RC circuits.

Experimental Procedures

RC Time Constant while Charging

    1) Select a 100 ohm resistor.
    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 of 1.0 F ± 10%.
    4) Turn on the power supply and set it to about 5.5 V.
    5) Connect the following items in a loop: resistor, capacitor, and power supply except leave one
       wire to the power supply disconnected.
    6) Configure two DMMs as voltmeters and place them so that they measure the potential
        difference across the resistor and the capacitor
    7) Start your stopwatch and plug in the remaining wire to the power supply
    8) Measure the potential differences across the resistor and capacitor every 5 seconds or so for
        about 3 minutes.
    9) Disconnect the circuit and turn off the power supply.
    10) Graph your results (including error bars) using Excel with the potential difference on the y axis
        and the time in seconds on the x axis.
    11) Use a least squares trend line (exponential, not linear!) to obtain a curve and an equation for
        potential difference across the resistor as a function of time. 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 preliminary evidence in
        favor of equation [4].
    12) 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. If they overlap,
        then the experiment supports the theory. If they do not overlap, but they are “close”, then you
        could explain the difference with the unknown error in the experimental value.
    13) 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

    14) Repeat steps 8 through 12 when you connect the charged capacitor directly to the resistor. You
        will only need one voltmeter in this case as the resistor and capacitor will have the same
        potential difference. This will be a test of equation [5].
    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.

				
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posted:8/1/2011
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