A circuit is constructed with four resistors one capacitor o

A circuit is constructed with four resistors, one capacitor, one battery and a switch as shown. The values for the resistors are: R₁ = R₂ = 52 , R₃ = 89 and R₄ = 88 . The capacitance is C = 55 F and the battery voltage is V = 12 V. The positive terminal of the battery is indicated with a + sign.

1)

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?

A

2)

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

C

3)

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 = 556 s after the switch was opened?

C

4)

What is I_C,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.

A

5)

What is I_C,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.

A

Solution

1)

I4(0) = V/Req

         = V/(R1 + R4 + 1/(1/R2 + 1/R3))

= 12/(52+88+1/(1/52+1/89))

        =  0.0694 A

2)

No current flow through R2: so, the current is,

i = V/Req = V/(R1 + R4 + R3) = 12/(52+88+89) = 0.0524 A

voltage: V3 = i R3 = 0.0524*89 = 4.66 V

charge is, Q = C V3 =55* 4.66 = 256.498 uC

3)

The charge is,

   Q = Qo e^-t/(R2+R3)C

       = [277.86X10^-6] e^{-(556X10-6)/(52+89)(55X10-6)}

          = 238.75X10^-6 C

= 238.75 uC

4)

the current is,

    I = V3/R3 = 4.66/89 = 0.05236 A

5) current: Im = V3 / (R2 + R3)