Consider the circuit shown with R 1 = 23 , R 2 = 31 , C 1 = 9 µF, C 2 = 28 µF, C
ID: 1531861 • Letter: C
Question
Consider the circuit shown with R1 = 23 , R2 = 31 , C1 = 9 µF, C2 = 28 µF, C3 = 35 µF, and V = 8.0 V.
(a) Draw an equivalent circuit with one resistor and one capacitor and label it with the values of the equivalent resistor and capacitor.
(b) A long time after switch S is closed, what is the charge on capacitor C3?
What is the current in resistor R1?
(c) What is the time constant of the circuit?
(d) At what time after switch S is closed is the voltage across the combination of three capacitors 50% of its final value?
Explanation / Answer
a)Req = R1 + R2 = 23 + 31 = 54 Ohm
C = C1 + C2 = 9 + 28 = 37 uF
Ceq = 37 x 35/(37+ 35) = 18 uF
the circuit can be replaced with a single resistir of R = 54 Ohm in series with a single capacitor of C = 18 uF.
b)After a very long time, the charge on capacitor will be:
Q = CV = 18 x 10^-6 x 8 = 144 uC
Hence, Q = 144 uC
The current through the resistor is zero after a very long time.
I(R) = 0
c)Time constant will be:
Tau = RC = 54 x 18 x 10^-6 = 972 x 10^-6 s = 9.72 x 10^-4 = 0.972 ms
Hence, tau = 0.972 ms
d)V' = V ( e^-t/RC)
0.5 = 8 e^-t/RC)
0.0625 = e^-t/RC
e^-t/RC = 0.0625
taking natural log both sides:
-t/RC = -2.77
t = 2.77 x Rc = 0.065 x 0.972 x 10^-3 = 2.69 x 10^-3 s = 2.69 ms
Hence, t = 2.69 ms