Please help me with this physics problem! A circuit is constructed with four res

Question

Please help me with this physics problem!

A circuit is constructed with four resistors, one capacitor, one battery and a switch as shown. The values for the resistors are: R1 - R2 - 40 Ohm, R3 - 107 Ohm and R4 - 100 Ohm. The capacitance is C - B9 mu F and the battery voltage is V - 12 V. The positive terminal of the battery is indicated with a + sign. The switch has been open for a long time when at time t - 0, the switch is closed. What is I4(0), the magnitude of the current through the resistor R4 just after the switch is closed? What is Q(-), the charge on the capacitor after the switch has been closed for a very long time? After the switch has been closed for a very long time, it is then opened. What is Q(t open), the charge on the capacitor at a time t open - 516 mu s after the switch was opened? 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. What is IC, max (open), 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

since the switch was open for long time capacitor is not charged capacitor acts like a wire (closed circuit)

so replacing capacitor with a wire

R3//R2= 107//40 = 29.11

I4= V/(R4+R1+R2//R3) = 0.070A

now general equatiob of rc circuit is Vc(t) = ( Vc(0+) - Vc(infinity)) e^-t/T + Vc(infinity)

Vc(0) =0v because capacitor was not charged for long time.

at t=infinity capacitor acts like a open circuit , now for finding the voltage between the open state that is Vc(infinity)

because capacitor is open at t=infinity no current passes through R2 junction,

so If = v/R3+R1+R4 = 0.048A = If = final current

Vc(infinty)=If * R3 = 5.19v

so Q(infinity)= Vc(infinity) * R3 = 462uc

so putting these values in GENERAL EQUATION we get

Vc(t) = (0 -5.19)e^-t/T +5.19

= 5.19(1-e^-t/T) where T is time constant = Rth * c = 100.6 * 89 = 0.00897

Vc(t) = 5.19( 1-e^-t/0.00897)

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