A circuit is constructed with four resistors, one inductor, one battery and a sw

Question

A circuit is constructed with four resistors, one inductor, one battery and a switch as shown. The values for the resistors are: R1 = R2 = 24 , R3 = 114 and R4 = 134 . The inductance is L = 258 mH 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?

2)What is I4(), the magnitude of the current through the resistor R4 after the switch has been closed for a very long time?

3)What is IL(), the magnitude of the current through the inductor after the switch has been closed for a very long time?

4)What is VL,max(closed), the magnitude of the maximum voltage across the inductor during the time when the switch is closed?

Explanation / Answer

1)

just as the switch is closed , the inductor behaves as open circuit

so R1 , R3 and R4 are in series and their combination is given as

R134 = R1 + R3 + R4 = 24 + 114 + 134 = 272 ohm

current in R4 = Total current coming from battery = battery Voltage / R134 = 12 /272 = 0.044 A

2)

after long time , inductor behaves as short circuit , hence R2 and R3 are in parallel and their combination is given as

R23 = R2 R3 / (R2 + R3) = 24 x 114 / (24 + 114) = 19.83 ohm

Total resistance of the circuit is then the series cobination of R1 R23 and R4 which is given as

R1234 = R1 + R23 + R4 = 24 + 19.83 + 134 = 177.83 ohm

i = current in R4 = Total current coming from battery = battery Voltage / R1234 = 12 /177.83 = 0.068 A

3)

V3 = Voltage across R3 = V - V1 - V4 = 12 - 0.068 x 24 - 0.068 x 134 = 1.256 volts

Current in R3 = i3 = V3 /R3 = 1.256 / 114 = 0.011 A

current on R2 = current in inductor = i - i3 = 0.068 - 0.011 = 0.057 A

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