RC Circuit: Charging Capacitor R V C + + - - a b i q C / 1 t RC q CV e Instantaneous charge q on a charging capacitor: At time At time t t = 0: = 0: q = CV(1 - 1); q = 0 q = CV(1 - 1); q = 0 At time At time t t = = : : q = CV(1 - 0); q q = CV(1 - 0); q max max = = CV CV The charge q rises from zero initially to its maximum value q max = CV
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RC Circuit: Charging Capacitor R V C ++ -- a b i Instantaneous charge q on a charging capacitor: At time t = 0: q = CV(1 - 1); q = 0 At time t = : q.
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RC Circuit: Charging Capacitor
R
V C
++
--
a
bi
q
C
/1 t RCq CV e
Instantaneous charge q on a charging
capacitor:
At time At time t t = 0: = 0: q = CV(1 - 1); q = q = CV(1 - 1); q = 00At time At time t t = = : : q = CV(1 - 0); qq = CV(1 - 0); qmaxmax = = CVCV
The charge q rises from zero initially to its maximum value qmax = CV
The charge q rises from zero initially to its maximum value qmax = CV
Example 1. What is the charge on a 4-F capacitor charged by 12-V for a time t =
RC?
Time, t
Qmaxq
Rise in Rise in ChargeCharge
Capacitor
0.63 Q
The time The time = RC = RC is is known as the time known as the time
constant.constant. /1 t RCq CV e
11q CV e
R = 1400
V 4 F
++
--
a
bi
e e = 2.718= 2.718; e; e-1-1 = 0.63 = 0.63
1 0.37q CV
0.63q CV0.63q CV
Example 1 (Cont.) What is the time constant ?
Time, t
Qmaxq
Rise in Rise in ChargeCharge
Capacitor
0.63 Q
The time The time = RC = RC is is known as the time known as the time
constant.constant.
R = 1400
V 4 F
++
--
a
bi
In one time In one time constant (5.60 ms constant (5.60 ms in this example), in this example), the charge rises to the charge rises to 63% of its 63% of its maximum value maximum value (CV).(CV).
= (1400 = (1400 )(4 )(4 F)F)
= 5.60 ms = 5.60 ms
RC Circuit: Decay of CurrentR
V C
++
--
a
bi
q
C
/1 t RCq CV e
As charge q rises, the current i will
decay.
/ /t RC t RCdq d CVi CV CVe edt dt RC
Current decay as Current decay as a capacitor is a capacitor is
charged:charged:
/t RCVi eR
/t RCVi eR
Current Decay
Time, t
I i
Current Current DecayDecay
Capacitor
0.37 I
R
V C
++
--
a
bi
q
C
The current is a The current is a maximum of I = V/R maximum of I = V/R
when t = 0.when t = 0.The current is zero The current is zero when t = when t = (because (because the back emf from C is the back emf from C is equal to V).equal to V).
/t RCVi eR
/t RCVi eR
Consider Consider ii when t = 0 when t = 0 and t = and t = . .
Example 2. What is the current i after one time constant (RC)? Given R and C as
before.
The time The time = RC = RC is is known as the time known as the time
constant.constant.
e e = 2.718= 2.718; e; e-1-1 = 0.37 = 0.37
max0.37 0.37V
i iR
max0.37 0.37V
i iR
R = 1400
V 4 F
++
--
a
bi
Time, t
I i
Current Current DecayDecay
Capacitor
0.37 I
/ 1t RCV Vi e eR C
Charge and Current During the Charging of a Capacitor.
Time, t
Qmaxq
Rise in Rise in ChargeCharge
Capacitor
0.63 I
Time, t
I i
Current Current DecayDecay
Capacitor
0.37 I
In a time In a time of one time constant, the of one time constant, the charge charge q q rises to 63% of its maximum, rises to 63% of its maximum, while the current while the current ii decays to 37% of its decays to 37% of its maximum value.maximum value.
RC Circuit: Discharge
R
V C
++
--
a
b
After C is fully charged, we turn switch to b, allowing it to discharge.