dX RC dt - physics.purdue.edu
Transcript of dX RC dt - physics.purdue.edu
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RC Circuits RC Circuits
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• 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: ε − qC− IR = 0
• Short term:
• Long term:
ε − 0 − I0R = 0
I0 =εR
(q = q0 = 0)
ε − q∞C
− 0 ⋅ R = 0 q∞ = Cε
(Ic = 0)
Intermediate term:
ε − qC−dqdtR = 0
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Solution dqdt
=εR−
qRC
dqε / R − q / RC0
Q
∫ = dt0
t
∫
X = ε / R − q / RC dX =−1RC
dq
−RC dXXε
R
εR−QRC
∫ = dt0
t
∫ ln x εR
εR−QRC = ln
εR− QRC
εR
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥=
−tRC
e−
tRC = 1− Q
εC⎛⎝⎜
⎞⎠⎟
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Continued
τ = RCQ = Cε(1− e− t /τ )
Capacitive Time Constant: "
The greater the , the greater the charging time.
Vc =QC
= ε(1− e− t /τ )
I = dQdt
=εRe− t /τ
Units of :
ΩF =VACV
=CC/s
= s
2
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Charging a Capacitor Q = Cε(1− e− t /τ )
I = εRe− t /τ
t = 0t = ∞t = τ
t = 0t = ∞t = τ
at
at
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Charging a Capacitor DEMO
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RC Circuits • Discharging a capacitor: • C initially charged with Q=C"• Connect switch S2 at t=0.
• Apply Kirchhoff’s Voltage Law: qC+ IR = 0
• Short term:
• Long term:
(q = q0 = 0)
(Ic = 0)
Intermediate term: ε + IR = 0
I0 =−εR
q∞ = 0
qC+dqdtR = 0
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Solution
R dQdt
+QC
= 0 dqq
= −dtRC0
t
∫Cε
Q
∫
−tRC
= lnQ CεQ = ln Q
Cε
Q = Cεe−
tRC
I = dQdt
=−εRe−
tRC
3
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Discharging a Capacitor
DEMO
t = 0t = ∞t = τ
t = 0t = ∞t = τ
at
at
Q = Cεe−
tRC
I = −εRe−
tRC
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Behavior of Capacitors
• Charging – Initially, the capacitor behaves like a wire. – After a long time, the capacitor behaves like an open
switch in terms of current flow.
• Discharging – Initially, the capacitor behaves like a variable battery. – After a long time, the capacitor behaves like an open
switch
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Question 3 "
R
R
C
C
"
RRC
C
CIRCUIT 2 CIRCUIT 1
. 1 < 2
. 1 = 2
C. 1 > 2
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Magnetic Field
• 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
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Bar Magnets
From North to South
N
S
N
S
Attraction
S
N
N
S
Repulsion
• 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)
NS
DEMO
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DEMO of Magnetic Field Lines
Magnetic Field Lines of a bar magnet
Electric Field Lines of an Electric Dipole
NS
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Magnetic Monopoles
• How can you isolate this magnetic charge?
Try cutting a bar magnet in half:
N S N N S S
• In fact no attempt yet has been successful in finding magnetic monopoles in nature but scientists are looking for them.
• 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).
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Earth’s Magnetic Field
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Earth’s Magnetic Field
Magnetic North Pole 1999
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Earth’s Magnetic Field
Magnetically Quiet Day
Magnetically DisturbedDay
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Earth’s Magnetic Field
Since 1904: 750 km, an average of 9.4 km per year.
From 1973 to late 1983: 120 km, an average of 11.6 km per year
Earth’s Magnetic Field