Capacitors in Circuits

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Capacitors in Circuits
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Capacitance
Two parallel plates charged Q and Q
respectively constitute a capacitor
C Q / V
The relationship C Q / V is valid for any
charge configuration (Indeed this is the
definition of capacitance or electric capacity)
In the particular case of a parallel
plate capacitor C ?0 A / d vacuum or C ?
?0 A / d dielectric
The capacitance is directly proportional to the
area of the plates and inversely proportional to
the separation between the plates
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Capacitors in Circuits
A piece of metal in equilibrium has a constant
value of potential. Thus, the potential of a
plate and attached wire is the same. The
potential difference between the ends of the
wires is V, the same as the potential difference
between the plates.
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Parallel and Series

Series
Parallel
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Capacitors in Parallel
C1 - q1

• Suppose there is a potential
• difference V between a and b.
• Then q1 V C1 q2 V C2

a
b
C2 - q2
V

• We want to replace C1 and C2 with an
• equivalent capacitance C q V
• The charge on C is q q1 q2
• Then C q V (q1 q2 ) V q1 V q2 V C1
  C2

b
a
C - q
C C1 C2

• This is the equation for capacitors in parallel.
• Increasing the number of capacitors increases the
  capacitance.

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Capacitors in Series
C2
C1
C
a
b
q
-q
-q
q
a
b
-q
q
V1
V2
V

• Here the total potential difference between a and
  b is V V1 V2
• Also V1 (1/C1) q and V2 (1/C2) q
• The charge on every plate (C1 and C2) must be the
  same (in magnitude)
• Then V V1 V2 q / C1 q / C2 (1/C1)
  (1/C2) q
• or, V (1/C) q ?

1 / C 1 / C1 1 / C2

• This is the equation for capacitors in series.
• Increasing the number of capacitors decreases
  the capacitance.

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Parallel and Series
Capacitors
Parallel C C1 C2
Series 1/C 1/C11/C2
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Parallel and Series
Resistors Capacitors
Parallel 1/R1/R11/R2 CC1C2
Series RR1R2 1/C1/C11/C2
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How does the capacitor acquire the charge ?
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RC Circuits Charging
V I(t)R q(t)/C
When the switch closes, at first a high current
flows VR is big and VC is small. As q is
stored in C, VC increases. This fights against
the battery, so I gradually decreases. Finally,
I stops (I 0), C is fully charged (VC Q/C
V), and Q C V
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RC Circuits Charging
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RC Circuits Charging
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Discharging an RC Circuit
Current will flow through the resistor for a
while. Eventually, the capacitor will lose all
its charge, and the current will go to zero.
During the transient q(t) / C I(t) R 0
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Discharging an RC Circuit
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Charging and Discharging a Capacitor
Charging and discharging of a capacitor occurs
gradually with a characteristic time ? RC ?
time constant At t 0, (switch closed or open)
a large current flows, the capacitor behaves
like a short circuit. At t ? ?, the current is
essentially zero, the capacitor behaves like an
open switch. The current decreases exponentially.
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Measuring Current and Voltage in a circuit
The ammeter measures current, and is connected in
series.
The voltmeter measures voltage, and is connected
in parallel.
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A modern digital multimeter combines the
functions of ammeter, voltmeter, and
ohmmeter. (i.e. can measure current, voltage, and
resistance)
In addition, modern multimeters can
measure capacitance, temperature, and more, and
can be connected to computers too