The RC Circuit

General physics lab

1. 1
The RC Circuit

Objectives:
1. To study the change of current and charge of a capacitor
through the discharge process.
2. To measure the time constant of RC circuit.

Apparatus:
DC power supply, voltmeter, capacitor, resistor and stop watch.

Theory:
Charge the capacitor
To charge the capacitor C we close switch s as in position 1 shown in
Fig. 1. This position completes an RC series circuits consisting of the
capacitor C, the resistance R and an ideal power supply of emf ε.
Current begins to flow in the circuit and increase the charge stored
on the plates of C and also the potential difference V across the
capacitor according to the following equation:
)1.....(
)(
)(
C
tq
tV
C
=
The current will decrease exponentially with time and will stop (i=0)
when the capacitor is fully charged or when the final charge Q is
given by the equation:
)2.....(εCQ=
According to Kirchhoff's loop rule, we may write:
( )3.....00 =−−⇒=−−
C
q
iRVV
CR
εε
And since the current is defined as dt
dq
i=
we have the equation:
( )4.....
C
q
dt
dq
R+=
ε
The solution of eq. (4) which satisfies the initial conditions, that is
q=0 when t=0 is:
)5).....(1(
RC
t
eQq
−
−=
1
s
2
R
V
C
C ε
+

-
Fig. (1)
0
1
2
3
4
5
6
7
8
9
0 t 2t 3t 4t 4.05t
Q(t)

t

Qo
0
1
2
3
4
5
6
7
8
9
0 t 2t 3t 4t 4.05t
I(t)

t

Io
Fig. (2)
Fig. (3)
RC
slope
1
=

t






V
n
ε
l
Fig (4)

2. 2
When the total charge Q is given by eq. (2), the charging current is:
( )6.....
RC
t
eI
dt
dq
i
−
==
o

Where R
I
ε
=
o
is the initial or maximum current.

Discharge the capacitor
To discharge the capacitor, connect switch s as in position 2 so the emf is out of the circuit or
0
=ε and eq. (4)
becomes:
( )7.....0=+
C
q
dt
dq
R

The solution of this equation is:
( )8.....
RC
t
Qeq
−
=
And the current is:
( )9.....
RC
t
eIi
−
−=
o

The two equations (8,9) shows that charge q and current i decreases exponentially with time in the discharging
process as shown in Fig. 2 and Fig. 3.
The product RC has the dimension of time and is called the time constant (
τ) of the circuit, which is equal:
()10.....RC=τ
From eq.(5) and at time t=RC, the charge on the capacitor has increase from zero to
,63.0)1(
1
QeQq =−=
−
i.e.
to 63% of it's final value.
Since q=CV, eq.(8) may be written in terms of V:
)11.....(
RC
t
eV
−
=ε
To obtain linear relationship, we take the (ln) of the both sides:
( )12.....ln
RC
t
V
=
ε

If
V
ε
ln is plotted versus t, the time constant of the circuit can be found, hence:
)13.....(
11
τ
==⇒
RC
slope

3. 3
Procedure and data Analysis
1. Connect the digital voltmeter across the power supply terminals and turn on to adjust the output voltage to
about 6 volts then turn off the power supply.
2. Use the multimeter to measure the accepted values of both R and C, then record them below:
R=_______________________ Ω, C=_______________________F.
3. Calculate the accepted value of (τ), where τ=RC [in seconds]:
τ = _______________________seconds.
4. Connect the circuit as shown in Fig. 1.
5. Charge the capacitor by connecting switch s as in position 1 for about 12 minutes until the potential across
the capacitor become constant around 6 volts.
6. The capacitor now is fully charged. Turn switch s to position 2 to start discharging process and switch on
the stop watch at the same time.
7. Note the reading of the voltmeter every 30 seconds and record the data in Table (1).
8. Plot V
C versus the time (t). At .37.0
1
εετ==⇒==
−
eVRCt
Caccepted
mark it on your graph.
9. Complete Table (1), then plot
()
CV
n
ε
l versus time. Find the slope in order to calculate the experimental
value of (τ) using eq. 13.
10. Compare between the theoretical and measured values of (τ)
Table (1)
t(sec) V(volt)
V
ε
()
V
n
ε
l
0
30
60
90
120
150
180
210
240
270
300
330
360
390
420
450

4. 4
480
510
540
570
600

5. 5
PRELAB
LAB REPORT FOR EXPERIMENT 6
THE RC CIRCUIT
Grade: [………..…………..]
ﻲﻌﻤﺎﺠﻝا مﻗرﻝا:
................................................... مﺴﻻا: ................................................................................................................
تﻗوﻝا : 8 – 11□ 11 - 2□ 2 - 5□ موﻴﻝا :دﺤأ □ نﻴﻨﺜإ □ ءﺎﺜﻼﺜ □ ءﺎﻌﺒرأ □ سﻴﻤﺨ □
ﺦﻴرﺎﺘﻝا:............................................................... سردﻤقﺎﺴﻤﻝا : ................................................................................................

Objectives:
Objectives: what you are trying to do
__________________________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________________________

Theory:
Theory: a brief statement of the important equations and what you have to measure.

__________________________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________________________

6. 6
REPORT

Data Analysis
Data Analysis include:

1. All data taken from the experiment (raw data). not something you calculated later. Data should be taken in tables with all the columns
labeled, units given, and estimated errors between experimental and accepted (theoretical) results.
2. Each graph must be titled, scaled, and the axes labeled.
3. The calculations take you from the data to the final result.
4. If many similar calculations are involved, just include one sample calculation in detail.

Raccepted=_______________________ Ω, C accepted=_______________________F.
τ
accepted = _________________________________________________________seconds.

Table (1)
t(sec) V(volt)
V
ε
()
V
n
ε
l
0
30
60
90
120
150
180
210
240
270
300
330
360
390
420
450
480
510
540
570
600

7. 7
Graph:
a. From table (1), plot VC versus time. At .37.0
1
εετ==⇒==
−
eVRCt
Caccepted
determine VC(τ) on the
graph

b.
Plot ln(
C
V
ε
) versus t. Determine the measured time constant where
slope
measured
1
=τ .
_____________________________________________________________________________
c. compare between accepted and measured τ
_____________________________________________________________________________

Conclusion
Conclusions: All the results should be stated and summarized in a table or a graph, including the errors. Summarize the results briefly
in words and try to make some perceptive comments on what you learned and how to improve the experiment.
__________________________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________________________