Problem 19.63: Charging and discharging a capacitor. A 1.57 F capacitor is charg
ID: 1622364 • Letter: P
Question
Problem 19.63: Charging and discharging a capacitor.
A 1.57 F capacitor is charged through a 123 resistor and then discharged through the same resistor by short-circuiting the battery.
A) While the capacitor is being charged, find the time for the charge on its plates to reach 1/e of its maximum value.
B) While the capacitor is being charged, find the current in the circuit at the time when the charge on its plates has reached 1/e of its maximum value.
C) During the discharge of the capacitor, find the time for the charge on its plates to decrease to 1/e of its initial value.
D) Find the time for the current in the circuit to decrease to 1/e of its initial value.
Explanation / Answer
a)
Capacitor charge equation .. Qt = Qo [1 - e^(-t/CR) ]
Qt/Qo = [1 - e^(-t/CR) ] = 1/e = 0.3679
1 - 0.3679 = e^(-t/CR)
Ln(0.6321) = -t/CR
t = -CR Ln(0.6321) .. - (1.56^-6F x 123 x -0.4587) .. .. t = 8.80^-5 s
b)
When charge on C has reached Q/e the pd across C and R = Vo/e (VQ)
Current .. i = pd/R = (Vo/e) / R .. .. i = Vo/(eR) .. (battery pd(Vo) required, e= 2.718)
c)
Discharge equation .. Qt = Qo e^(-t/CR)
Qt/Qo = e^(-t/CR) = 1/e = e¹ therefore ..
(-t/CR) = -1
t = CR .. .. 1.56^-6F x 123 .. .. t = 1.92^-4 s
d)
Current decay follows same decay curve as Q and V .. so the time for IoIo/e = time for QoQo/e which is calculated in part (c) .. .. t = 1.92^-4 s