NE2 RC CIRCUIT – PARALLEL PLATE CAPACITOR DISCHARGING
Introduction
An RC circuit that consists of capacitor with capacitance C , resistor with resistance R connected in series
and the source of electromotive force (emf) U e can be seen in the circuit diagram in Fig. 1. The source U e
and ‘the voltmeter’ – measuring the potential difference on the capacitor – are included in the Coach Lab
panel.
Let’s assume that the key K is in the position 1, e.g. capacitor is charged and its initial voltage U0 is equal to
the emf of the source U e . If at the instant t = 0 the key K is switched from the position 1 to the position 2,
the discharging of capacitor begins via resistor R . If we want to describe the process of capacitor
discharging, we need to answer the following questions: how the charge on the capacitor plates depends on
time Q(t ) ; how the voltage (potential difference) on the capacitor depends on time U C (t ) and how the
current in the circuit changes with time I (t ) .
U0
1 K
2
CoachLab
C
+5V
3
+5V
4
R
Fig. 1: Electrical circuit of RC series.
When the key K is in the position 2, the source of emf U e is not connected to the circuit and according to the
II. Kirchhoff’s rule we can write
0 = I .R + U C
(1)
Q
The voltage on the capacitor U C is the ratio U C = , where Q is the charge on the capacitor plates and
C
dQ
the current is defined as I = −
. Substitution of the last two formulas into equation (1) gives the
dt
differential equation, which describes the time dependence of the charge Q during capacitor discharging
dQ Q
0 = R.
+
(2)
dt C
Using the initial conditions, if t = 0 then Q = Q0 , we can write the solution of the differential equation (2) as
function Q(t )
Q(t ) = Q0 .e
−
t
R .C
(3)
Taking into consideration that Q0 = C.U 0 , where U 0 is initial voltage on the capacitor at t = 0 , we obtain
the formula U C = Q(t ) /C , from which we obtain a similar time dependence for U C
−
Q(t )
U C (t ) =
= U 0 .e
C
t
R .C
(4)
Expressing the derivate of the equation (3) with respect to time t, results in the formula for current I (t )
during capacitor discharging:
I (t ) = −
where I 0 =
−
dQ(t )  Q0  − R.C U 0 − R.C
=
=
.e
= I0 .e R.C
.e
dt
R
 R.C 
t
t
t
(5)
U 0 Q0 1
=
.
is the current at time t = 0 , when the voltage at the capacitor is U 0 .
R
C R
Equations (3), (4), (5) describe exponential decay functions. The product R.C in these equations is the time
constant τ of RC circuit:
τ = RC
(6)
The total capacitance CP of two capacitors connected in parallel is
C P = C1 + C 2
(7)
and it is, of course, greater than capacitance C1 or capacitance C2 .
The total capacitance Cs of two capacitors connected in series is smaller than the capacitance C1 or the
capacitance C2 . The formula for Cs is
1
1 1
(8)
= +
Cs C1 C2
Experimental Procedure
1. After connecting elements of electrical circuit, see Fig. 1, connect the circuit to the emf source, which is in
the Coach Lab panel. The voltage at the capacitor Uc is measured by channel 3 on the Coach Lab panel.
The value of resistance in the circuit is R=50 kΩ. The position 1 of the key K means, that the capacitor is
charged to the initial voltage U0. The position 2 starts the process of capacitor discharging through resistor R.
Begin your work on computer (icon CoachLab) with Start, then choose Exploring Physics and open the file
RC-circuit. The graphs for measured quantities versus time are shown on the screen, also the table is
prepared. Before the measuring the key K it is to be in the position 1, and the initial voltage across capacitor is
U0=5V. To begin the measurement of capacitor discharging click on the green “measurement button” Start
and then switch the key K to the position 2. During discharging the graphs are drawn on the screen,
dependence Uc(t) is exponentially decreasing function. Uc is continuously measured by the Coach Lab
Panel.
U 
2. The linear function ln 0  = f (t ) is displayed in the second window of the screen. To analyze the this
 UC 
function click with right button on the option Analyse, Function-fit and choose the linear function f(x) = a.t.
To determine the parameter of linear regression a click on Auto Fit. After fitting the capacitance can be
calculated by means of parameter a :
C=
1
.
aR
Repeat the same procedure with the second capacitor.
3. To determine the total capacitance CP of the capacitors C1 and C2 connected in parallel, repeat the same
procedure with capacitors connected in parallel including calculation CP using parameter a. To determine the
total capacitance CS of the capacitors C1 and C2 connected in series, repeat the same procedure with
capacitors connected in series and calculate CS using parameter a.
4. The results of four experiments are the values C1,C2,Cp,Cs and corresponding measurement uncertainties.