SERIES RC CIRCUIT



PURPOSE:	The dynamic behavior of a series RC circuit will be studied by applying a 
constant emf to a resistor and an initially uncharged capacitor.  Measurements of the circuit 
current as a function of time will provide the data for a graphical demonstration of the 
exponential nature of this behavior.

THEORY:	Examine the circuit diagram shown in Figure 1.  When the switch S1 is closed 
to complete the series circuit composed of an emf, the resistance R, and capacitance C (switch S2 
open), a current will pass through the resistance and charge will be deposited on the plates of the 
capacitor.  By conservation of energy, the potential drop across the resistance, IR, plus the 
potential drop across capacitance, q/C, must equal the emf ???that is,

? = IR + q/C

Since the current is the rate at which charges flow though the circuit, this 
expression represents a differential equation in q.  The solution yields the charge 
on the capacitor as a function of time; specifically,

q  = Qmax {1 - e- t/?},  Qmax = C?,  ? = RC

Using this result, one can determine the expression for the current flowing 
through the circuit as a function of time; specifically,

I = Ioe- t/?,  Io = ?/R,  ? = RC

This relation indicates that the current will decrease exponentially as time 
increases.  For a time interval of magnitude RC, the current value becomes 1/e 
times the initial value at the start of the interval.  This specific time interval is 
called the time constant (?) of the circuit and can be used to determine either R 
or C if the other is known.

				 



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PROCEDURE:	In this experiment you will demonstrate the exponential time dependence of the 
current.

Figure 1 shows the circuit to be studied.  S2 represents a switch for by-passing 
the capacitance.  Physically it is a wire with one end fixed to one plate of the 
capacitance and a free end which can be attached to the other plate.

With the circuit set up as shown, close both switches and adjust the power 
supply to provide as large an initial current as possible.  After this current value 
is read from the microammeter and recorded, open switch S2 and roughly 
determine the time constant of the circuit.  This is done by measuring the time 
for the current to decrease to 0.4 (1/e = .368) of the initial value.  If the value 
found is not in the range of 10 - 20 seconds, see your instructor because you 
have a defective circuit.

In order to calculate an accurate value of R you must measure and record the 
magnitude of the emf ??with the voltmeter.  It is important to keep this value 
constant for the duration of the experiment.

Take and record data for current versus time for at least 5 reasonably spaced 
values of current (for example, 70%, 55%, 40%, 25%, and 10% of initial 
current).  You should repeat and record each point at least 3 times.  In the 
graphing to be done, the average value of the time for each current value is to be 
used.

Plot the current I versus the time t on Cartesian paper.  Does your graph show a 
linear relationship between current and time?  Why or why not?  Locate on the 
current axis the current corresponding to 0.368Io (Ioe- 1).  Draw a dotted line 
horizontally to intersect your curve and draw a dotted line vertically down until 
it intersects the time axis.  Record this time.  It is the value of the time constant 
?.

Plot the current I versus time t on two cycle semi-log graph paper.  Does your 
graph show the expected decreasing exponential dependence of current with 
increasing time?  Calculate the slope of the line;

	slope = ln(I2/I1)
						      (t2 - t1)

where (t1, I1) and (t2, I2) are the coordinates of any two points on the ObestO 
straight line.  Using the slope, calculate the time constant ?.

	Using the measured value of R and the value of C stamped on the 
capacitor, calculate the time constant ??	

	Finally, compare the three values of ??and discuss any differences which 
occur.
                                                                              









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