Assume the capacitor below starts out with no charge and the

Assume the capacitor below starts out with no charge, and the resistors all have resistance, R = 2.0 Ohms. the capacitor has C = 12?F. The power supply provides 12 Volts.
a.   What is the current through the resistors immediately after the switch is thrown.
b.   What is the current through each resistor after the switch has been closed for a long time.
c. What is the charge on the capacitor after the switch has been closed for a long time.

Please show the steps on how to solve this problem.

Solution

a)

Immediately after switch is closed ,capacitor will act as short circuit .so equivalent resistance is

R_eq=R₁+(R₂||R₃) =R₁+R₂*R₃/(R₂+R₃)

R_eq = 2+2*2/(2+2) =3 ohms

Total current flowing in the circuit

I=E/R_eq=12/3 =4 A

So current through each resistor is

I₁=I=4 A

I₂=I/2 =2 A

I₃ =I/2 =2 A

b)

After long time switch is closed inductor will act as open circuit ,so equivlaent resistance is

R_eq = R₁+R₂ =2+2 =4 A

Total current

I=V/R_eq =12/4 =3 A

So current flowing in each resistor is

I₁=3 A

I₂=3 A

I₃=0 A

c)

Voltage across capacitor

V_C=V₂=IR₂=3*2 =6 Volts

Charge on the capacitor is

Q=CV_c =12u*6

Q=72 uC or 7.2*10^-5 C