connected in parallel n Capacitors connected in series ¤ RC
circuit
n Discharging
a capacitor n Charging a capacitor
Clicker question: 1 3
Capacitors in parallel 4
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When capacitors are connected in parallel, the potential difference across each one is the same. If we replace the capacitors with a single capacitor with an equivalent capacitance, the circuit is operationally equivalent. 𝑄 = 𝑄# + 𝑄% + 𝑄& The equivalent capacitance for n capacitors in parallel is n
Ceq = C1 + C2 + C3 + ⋅ ⋅ ⋅ = ∑ Ci i =1
Clicker question: 2 5
Capacitors in series 6
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The total charge in the sections between the two capacitors must be zero. Therefore the magnitude of the charge on the capacitors is the same. The total potential difference is the sum of the potential differences across each one: ℇ = 𝑉# + 𝑉% + 𝑉& . the equivalent capacitance for n capacitors in series can be calculated by n 1 1 1 1 1 = + + + ⋅⋅⋅ = ∑ Ceq C1 C2 C3 i =1 Ci
¤
Note that Ceq is smaller than any of C in series.
Equivalent circuits 7
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We can simplify complicated circuits by looking for capacitors in series or parallel and replacing it with their equivalent capacitor. We can then calculate the charge and potential difference for the equivalent capacitor.
Example: 1 8
a)
b) c)
What is the equivalent capacitance of three capacitors? What is the charge on C3? What is the charge on C1?
C1
C2
V C3
RC circuits 9
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A circuit containing a resistor and capacitor is called an RC circuit. In a circuit containing only batteries and capacitors, charge appears almost instantaneously on the capacitors when the circuit is connected. However, if the circuit contains resistors as well, this is not the case. Kirchhoff’s rules still apply; however, the current is not necessarily constant with time.
Discharging a capacitor 10
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The current reduces the charge on the capacitor. A capacitor initially has charge Q0. The initial potential difference across the capacitor is V0 =
¨
Right after the switch is closed, the initial current is I0 =
€
Q0 C
V0 Q0 = R RC
Discharging a capacitor: 2 11
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The charge and current as the capacitor discharges are: q ( t ) = Q0 e−t
( RC )
I ( t ) = I 0 e−t τ =
≡ Q0 e−t τ Q0 −t τ e RC
τ = RC is called the time constant. ¤ τ is the time it takes for the charge or current to decrease by a factor of e-1. ¤
q
Example: 2 12
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After the switch has been at contact b for a long time, the potential difference across the capacitor is 100 V. The switch is then rotated to contact a at t = 0. At t = 10.0 s, the potential difference across the capacitor is 1.00 V. a) What is the time constant of the circuit? b) What is the potential difference across the capacitor at t = 17.0 s? b
Charging a capacitor 13
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Charging an initially uncharged capacitor. When the switch is closed, a current starts, charging the capacitor. When the capacitor is fully charged, current stops. And the potential difference across the capacitor and the battery would be the same. E = Qf/C ¤
where Qf is the final charge on the capacitor.
Charging a capacitor: 2 14
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The charge on the capacitor and the current in the circuit as a function of time are: q
(
q ( t ) = Qf 1 − e−t
Qf = CE
( RC )
) I (t ) = I 0 e
−t / RC
I0 =
E R
Demo: 1 15
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RC circuit ¤ A
capacitor is charged by a battery through a resistor, all in series.
VC ( t ) = ¤ A
q ( t ) Qf ( ) = 1 − e−t RC C C
(
)
capacitor is discharged through a resistor.
VC ( t ) =
q ( t ) Q0 −t ( RC ) = e C C
Current after short time 16
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The switch was off for a long time, then turned on. Initially the capacitor can be treated as a short circuit.
Current after long time 17
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As charge builds up on the capacitor the potential builds up until no more current flows in that part of the circuit, and it can be treated as a open circuit.
Example: 3 18
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A resistor with a resistance of R = 145 Ω is connected to a switch, a capacitor, and a battery with an emf of E = 9.0 V, all in series. Assume that the capacitor is uncharged initially. a) What capacitance must be used in this circuit if the time constant is to be τ = 3.5 ms? b) Using the capacitance determined in part a), calculate the current in the circuit t = 7.0 ms after the switch is closed. c) What is the charge on the capacitor a long time after the switch is closed.
