Two Loop RC Circuit 2 1 2 3 4 5 6 A circuit is constructed with four resistors,

Question

Two Loop RC Circuit 2

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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 = 83 and R4 = 124 . The capacitance is C = 68 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(topen), the charge on the capacitor at a time topen = 593 s after the switch was opened?

C

4)

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.

A

5)

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.

A

Explanation / Answer

at t = 0 , Capacitor behaves as short circuit, hence

R2 and R3 being in series gives the combination as

R23 = R2 R3/(R2 + R3) = 39 x 83 /(39 + 83)= 26.5 ohm

Rt = total resistance of the circuit = R1 + R23 + R4 = 39 + 26.5 + 124 = 189.5 ohm

since the battery is in series with R4, i4(0) is same as the toital current coming from the battery

i4(0) = V/Rt = 12/189.5 = 0.063 A

2)

when switch is closed for long time , capacitor behaves as open circuit.

Rt' = total resistance of circuit after long time = R1 + R3 + R4 = 39 + 83 + 124 = 246 ohm

i' = current coming from the battery = V/Rt' = 12/246 = 0.049 A

V3 = Voltage across R3 = Voltage across Capacitor = i' R3 = 0.049 x 83 = 4.07 Volts

Q() = CV3 = 68 (4.07) = 276.8 uC

3)

T = time constant = (R2 + R3) C = (39 + 83) (68 x 10-6) = 0.0083 sec

t = 0.000593 sec

Qo = 276.8

using the equation

Q(t) = Qo e-t/T

Q(t) = (276.8) e-(0.000593/0.0083)

Q(t) = 257.7 uC

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