RC Circuits (Charging)
!
!
_
+
Qo = 0
RC Circuits
a
b
R
C
a
R
+q -q
b
C
c
" # v ab # v bc = 0
" # iR # Cq = 0
" q
i= #
R
!
1
i
i
c
!
Current, Resistance, and emf
_
+
RC
Current, Resistance, and emf
2
!
RC Circuits (Charging)
"
i= #
R
RC Circuits (Charging)
q
RC
q
t
dq
dt
=$"
q
"
C
RC
0
0
$
#
$ q " C# '
t
ln&
)="
% "C# (
RC
q " C#
=e
"C#
"
!
!
dq
q
= #
dt R RC
!
dq
dt
="
q"C
RC
#
"t
!
RC
q
!
t
dq
dt
=$"
$
q
"
C
RC
0
0
(
#
Current, Resistance, and emf
q(t ) = Q f 1 " e
!
3
"t
RC
)
where Q f = C #
Current, Resistance, and emf
!
$
4
!
RC Circuits (Charging)
RC Circuits (Charging)
q
The charge on the capacitor varies according to:
Qf
(
q(t ) = Q f 1 " e
"t
RC
(
)
q(t ) = Q f 1 " e
RC
)
t
The current at any time is given by:
#t
"t
!
dq
i=
= e RC = Io e RC
dt
R
"
i
!
Io
RC is called the time constant (" ) and is the time it
takes the capacitor to !
become 63.2% charged.
!
!
Current, Resistance, and emf
"t
i = Io e
"t
RC
Io /e
t
RC
5
Current, Resistance, and emf
!
6
RC Circuits (Charging)
RC Circuits (Discharging)
q
i
Qf
i=0
a
R
i
+Qo -Qo
b
Increasing RC
C
c
a
iR +
t
i=
!
Current, Resistance, and emf
7
R
+q -q
b
C
c
q
=0
C
dq
q
="
dt
RC
Current, Resistance, and emf
8
!
RC Circuits (Discharging)
RC Circuits (Discharging)
dq
q
="
dt
RC
The charge on the capacitor varies according to:
q
q = Qoe
dq t dt
="#
Qo q
0 RC
"
!
q = Qoe
"t
RC
The current at any time is given by:
!
" q%
t
ln$ ' = (
RC
# Qo &
!
"t
i=
RC
"t
Q "t
dq
= " o e RC = Io e RC
RC
dt
!
Current, Resistance, and emf
9
Current, Resistance, and emf
!
!
!
RC Circuits (Discharging)
10
!
RC Circuits (Discharging)
q
q = Qoe
"t
q
Qo
RC
Qo
Qo /e
t
RC
i
!
Increasing RC
Io
i = Io e
"t
RC
t
Io /e
t
RC
Current, Resistance, and emf
!
11
Current, Resistance, and emf
12
Example 1
C
S
C
S
+
+
E#
E#
R
An ideal battery with an emf E = 100 V is connected to a resistor
with resistance R = 100 ! and an initially uncharged capacitor
with capacitance C = 1 µF. The circuit is completed when
switch S is closed at time t = 0.
a.) Find the current through and voltage across each device
immediately after the switch is closed.
b.) A long time after the switch is closed, find the charge on the
capacitor and the voltage across the resistor.
c.) What is the charge on the capacitor after 0.2 ms?
d.) Find the total energy dissipated in the resistor.
Current, Resistance, and emf
13
C
S
R1
R2
Example 2
An ideal battery with an emf E = 12 V, two resistors with
resistances R1 = 4 ! and R2 = 6 !, and an initially uncharged
capacitor with capacitance C = 6 µF. The circuit is completed
when switch S is closed at time t = 0.
a.) At time t = 2", what is the potential difference across the
capacitor?
b.) At time t = 2", what are the potential differences across
the two resistors? Do those potential differences increase,
decrease, or remain the same while the capacitor is being
Current, Resistance, and emf
14
charged?
Example 3
C
S
+
E#
+
E#
R1
R2
Example 2 (continued)
c.) A long time after the switch has been closed the switch is
opened.
i.) Write an equation for the charge on the capacitor after
the switch is opened.
ii.) Write an equation for the voltage on each resistor after
the switch is opened.
Current, Resistance, and emf
Current, Resistance, and emf
(continued)
E#
+
E#
C
C
R1
S2
16
S1
Example 4
+
R3
In the circuit above E = 1000 V, C = 10 µF, and R1 = R2 =
R3 = 1 M!. The capacitor is completely uncharged when
switch S is closed.
a.) Determine the current through each resistor at t = 0
and t = $.
b.) Draw qualitatively a graph of the potential difference
V2 across R2 for t = 0 to t = $.
c.) What are the numerical values of V2 at t = 0 and t = $?
15
S1
Example 4
R2
R1
R1
R2
S2
R2
At t = 0 the capacitor is uncharged and switch S1 is closed.
a.) Write a differential equation that can be solved to obtain the charge on the
capacitor as a function of time.
After switch S1 has been closed for a long time, switch S2 gets closed at a new
time t = 0.
b.) Solve the differential equation in part (a) to determine the charge on the
capacitor as a function of time t.
d.) Sketch graphs of the current I1 in R1 and of the current I2 in R2 versus time,
beginning when switch S2 is closed at new time t = 0. Clearly label which
graph is I1 and which is I2.
Given that E = 12 V, C = 0.06 F, and R1 = R2 = 5000 !
c.) Determine the time at which the capacitor has a voltage 4.0 V across it.
Current, Resistance, and emf
17
Current, Resistance, and emf
18