Physics 2112
Unit 11
Today’s Concept:
RC Circuits
Unit 11, Slide 1
RC Circuit (Charging)
Capacitor uncharged, Switch is moved to position “a”
aa
Kirchoff’s Voltage Rule
q
Vbattery   IR  0
C
Differential Equation
VVbattery
battery
C
C
bb
RR
q dq
Vbattery  
R0
C dt
Vbat
q
dq
1



(q  VC )
R RC dt
RC

q(t )  CV 1  e  t / RC

I (t )  I 0e t / RC
Unit 11, Slide 2
Example 11.1 (Charging Capacitor)
10V
C1 = 1uF
What is the charge on the
capacitor 1second after the
switch is closed?
R1 = 1.2MW
What is the current through the
resistor 1 second after the
switch is closed?
 Conceptual Idea:
Use charging equations for a capacitor.
 Plan:
• Find time constant, t
• Find Qf and Io
• Put in t=1sec
Example 11.2 (Charging Capacitor)
10V
C1 = 1uF
What is the charge on the
capacitor 10 seconds after the
switch is closed?
R1 = 1.2MW
What is the current through the
resistor 10 seconds after the
switch is closed?
 Conceptual Idea:
Use charging equations for a capactor.
 Plan:
• Find time constant, t
• Find Qf and Io
• Put in t=1sec
RC Circuit (Discharging)
Capacitor has q0  CV, Switch is moved to position “b”
Kirchoff’s Voltage Rule
q
+ + IR  0
C
Differential Equation
q dq
+ +
R0
C dt
dt dq


RC Q
V
q(t )  q0 e t / RC
aa
C
C
+ 
Vbattery
Vbattery
bb
I
RR
I
I (t )  I 0e t / RC
Unit 11, Slide 5
Example 11.3 (Discharging Capacitor)
10V
1
2
C1 = 1uF
R1 = 1.2MW
The switch is held in position 1 for
a long time and the capacitor
becomes fully charged. It is then
flipped to position 2.
What is the charge on the
capacitor 2 seconds after the
switch is flipped?
 Conceptual Idea:
Use discharging equations for a capacitor.
 Plan:
• Find time constant, t
• Find Qo
• Put in t=2sec
How do Exponentials Work?
Q(t )  Q0 e
t
RC
Q
(
t
)
“Fraction of initial
charge that remains” Q0
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
“How many time constants worth
of time that have elapsed”
4
5
6
7
8
9
10
t
RC
Unit 11, Slide 7
t
Q(t )  Q0e 
RC
Q (t )
Q0
1
0.9
0.8
0.7
0.6
0.5
RC  2
0.4
Time constant:
t  RC
0.3
0.2
0.1
The bigger t is,
the longer it takes to get
the same change…
RC  1
0
0
1
2
3
4
5
6
7
8
9
10
t
Unit 11, Slide 8