1. RC Circuit: Charging CapacitorV 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 = CV(1 - 0); qmax = CV
The charge q rises from zero initially to its maximum value qmax = CV

2. The time t = RC is known as the time constant.
Example 1. What is the charge on a 4-mF capacitor charged by 12-V for a time t = RC?
Time, t
Qmax q
Rise in Charge
Capacitor t
0.63 Q
R = 1400 W V
4 mF + - a b i
The time t = RC is known as the time constant.
e = 2.718; e-1 = 0.63

3. Example 1 (Cont.) What is the time constant t?
Time, t
Qmax q
Rise in Charge
Capacitor t
0.63 Q
R = 1400 W V
4 mF + - a b i
The time t = RC is known as the time constant.
In one time constant (5.60 ms in this example), the charge rises to 63% of its maximum value (CV).
t = (1400 W)(4 mF)
t = 5.60 ms

4. RC Circuit: Decay of CurrentV C + - a b i
As charge q rises, the current i will decay.
Current decay as a capacitor is charged:

5. The current is a maximum of I = V/R when t = 0.
Current Decay R V C + - a b i
Time, t I i
Current Decay
Capacitor t
0.37 I
Consider i when t = 0 and t =  .
The current is a maximum of I = V/R when t = 0.
The current is zero when t =  (because the back emf from C is equal to V).

6. The time t = RC is known as the time constant.
Example 2. What is the current i after one time constant (t = RC)? Given R and C as before.
R = 1400 W V
4 mF + - a b i
Time, t I
Current Decay
Capacitor t
0.37 I
The time t = RC is known as the time constant.
e = 2.718; e-1 = 0.37

7. Charge and Current During the Charging of a Capacitor.
Time, t
Qmax q
Rise in Charge
Capacitor t
0.63 I
Time, t I i
Current Decay
Capacitor t
0.37 I
In a time t of one time constant, the charge q rises to 63% of its maximum, while the current i decays to 37% of its maximum value.

8. RC Circuit: Discharge C C
After C is fully charged, we turn switch to b, allowing it to discharge. R V C + - a b R V C + - a b i
Discharging capacitor. . . loop rule gives:
Negative because of decreasing I.

9. Discharging CapacitorV C + - a b i
Note qo = CV and the instantaneous current is: dq/dt.
Current i for a discharging capacitor.

10. From definition of logarithm:
Prob. 45. How many time constants are needed for a capacitor to reach 99% of final charge? R V C + - a b i
Let x = t/RC, then:
e-x = or e-x = 0.01
From definition of logarithm:
4.61 time constants
x = 4.61

11. Prob. 46. Find time constant, qmax, and time to reach a charge of 16 mC if V = 12 V and C = 4 mF.+ - a b i
1.4 MW C
12 V
t = RC = (1.4 MW)(1.8 mF)
t = 2.52 s
qmax = CV = (1.8 mF)(12 V);
qmax = 21.6 mC
Continued . . .

12. From definition of logarithm:
Prob. 46. Find time constant, qmax, and time to reach a charge of 16 mC if V = 12 V and C = 4 mF. R V
1.8 mF + - a b i
1.4 MW C
12 V
Let x = t/RC, then:
From definition of logarithm:
x = 1.35
Time to reach 16 mC:
t = 3.40 s