A circuit is constructed with four resistors, one capacitor, one battery and a s

Question

A circuit is constructed with four resistors, one capacitor, one battery and a switch as shown. The values for the resistors are: R1 = R2 = 39 , R3= 64 and R4 = 151 . The capacitance is C = 48 F and the battery voltage is V = 12 V. The positive terminal of the battery is indicated with a + sign.

1. What is Q(), the charge on the capacitor after the switch has been closed for a very long time?

2. After the switch has been closed for a very long time, it is then opened. What is Q(topen), the charge on the capacitor at a time topen = 697 s after the switch was opened?

3. What is IC,max(closed), the current that flows through the capacitor whose magnitude is maximum during the time when the switch is closed? A positive value for the current is defined to be in the direction of the arrow shown.

4. What is IC,max(open), the current that flows through the capacitor whose magnitude is maximum during the time when the switch is open? A positive value for the current is defined to be in the direction of the arrow shown.

Explanation / Answer

1. switch is closed for very longtime

then, Vc = V R3 / (R1 + R3 + R4)

Vc = 3.02 Volt

Q = C Vc = (48 uF) (3.02 v) = 145 uC

2. T = (R2 + R3)C = 4944 us

Q = Q0 [ e^(-t/T)]

Q = (145) (e^(-697/4944))

Q = 126 uC  

3. when closed,

I = Imax when Vc = 0

1/R = 1/R2 + 1/R3

R = 24.2 ohm

I = 12 / (24.2 + 39 + 151) = 0.056 A

Ic = 0.056 x 64 / (64 + 39) = 0.035 A ....Ans

4. Imax =3.02 / (39 + 64) = 0.0293 A

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