RC Circuits
RC Circuits
•  Charging a capacitor:
C initially uncharged; connect
switch to a at t=0
Calculate current and charge
as function of time.
•  Apply Kirchhoff’s Voltage Law:
•  Short term:
Intermediate term:
•  Long term:
9/26/12
1
Solution
dq ε
dt
=
R
∫
ε
R
−
q
RC
dq
t
Capacitive Time Constant: "
∫ ε / R − q / RC = ∫ dt
0
dX
= dt
X ∫0
−
ln x
t
RC
0
dX =
t
e
9/26/12
2
Continued
Q
X = ε / R − q / RC
ε Q
−
R RC
9/26/12
ε Q
−
R RC
ε
R
τ = RC
−1
dq
RC
The greater the , the
greater the charging time.
Q = Cε (1 − e−t /τ )
Q ⎤
⎡ε
⎢ R − RC ⎥ −t
= ln ⎢
⎥=
ε
⎢
⎥ RC
⎣ R ⎦
I=
dQ ε −t /τ
= e
dt
R
Vc =
Q⎞
⎛
= ⎜1 −
⎟
⎝
εC ⎠
3
9/26/12
Q
= ε (1 − e−t /τ )
C
Units of  :
ΩF =
VC
C
=
=s
A V C/s
4
1
Charging a Capacitor
Q = Cε (1 − e
at
I=
−t / τ
DEMO
Charging a Capacitor
)
t=0
t=∞
t =τ
ε −t /τ
e
R
at t = 0
t=∞
t =τ
9/26/12
5
RC Circuits
9/26/12
6
Solution
•  Discharging a capacitor:
•  C initially charged with Q=C"
R
dQ Q
+ =0
dt C
−
t
= lnQ
RC
•  Connect switch S2 at t=0.
•  Apply Kirchhoff’s Voltage Law:
•  Short term:
Intermediate term:
Q = Cε e
•  Long term:
I=
9/26/12
7
9/26/12
−
Q
t
dq
dt
∫Cε q = ∫0 − RC
Q
Cε
= ln
Q
Cε
t
RC
dQ −ε − RCt
=
e
dt
R
8
2
Discharging a Capacitor
Q = Cε e
−
Behavior of Capacitors
t
RC
•  Charging
–  Initially, the capacitor behaves like a wire.
–  After a long time, the capacitor behaves like an open
switch in terms of current flow.
at t = 0
t=∞
t =τ
I=
•  Discharging
–  Initially, the capacitor behaves like a variable battery.
–  After a long time, the capacitor behaves like an open
switch
−ε − RCt
e
R
at t = 0
t=∞
t =τ
9/26/12
DEMO
9
Magnetic Field
9/26/12
10
Bar Magnets
•  Bar magnet ... two poles: N and S
Like poles repel; Unlike poles
attract.
•  Magnetic Field lines: (defined in
same way as electric field lines,
direction and density)
N
S
S
N
N
N
S
S
Attraction
•  Large Magnetic fields are used
in MRI (Nobel prize for
medicine in 2003)
•  Extremely Large magnetic field
are found in some stars
•  Earth has a Magnetic Field
S
N
Repulsion
From North to South
DEMO
9/26/12
11
9/26/12
12
3
DEMO of Magnetic Field Lines
Magnetic Monopoles
•  One
explanation: there exists magnetic charge, just like
electric charge. An entity which carried this magnetic
charge would be called a magnetic monopole (having +
or - magnetic charge).
Electric Field Lines
of an Electric Dipole
•  How can you isolate this magnetic charge?
Try cutting a bar magnet in half:
S
Magnetic Field Lines of
a bar magnet
9/26/12
S
N
S
N
S
N
•  In fact no attempt yet has been successful in finding
magnetic monopoles in nature but scientists are
looking for them.
N
13
Earth’s Magnetic Field
9/26/12
14
Earth’s Magnetic Field
Magnetic
North Pole
1999
9/26/12
15
9/26/12
16
4
Earth’s Magnetic Field
Earth’s Magnetic Field
Since 1904:
750 km, an average of 9.4 km per year.
Magnetically
Quiet
Day
From 1973 to late 1983:
120 km,
an average of 11.6 km per year
Magnetically
Disturbed
Day
9/26/12
17
9/26/12
18
5