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

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

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

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

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).

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

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.

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.

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

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 = 1-0.99 or e-x = 0.01 From definition of logarithm: 4.61 time constants x = 4.61

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 . . .

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