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 = 56 Ω, R₃ = 45 Ω and R₄ = 91 Ω. The inductance is L = 255 mH and the battery voltage is V = 24 V.

1)

The switch has been open for a long time when at time t = 0, the switch is closed. What is I₁(0), the magnitude of the current through the resistor R₁ just after the switch is closed?

_______A

2)

What is I₁(∞), the magnitude of the current that flows through the resistor R₁ a very long time after the switch has been closed?

_______A

3)

What is V_L(0), the magnitude of the voltage across the inductor just after the switch is closed?

______V

4)

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

___________A

5)

What is I₂(0), the magnitude of the current through the resistor R₂ just after the switch is closed?

_______________A

Answer 1

1)

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

hence R_total = total resistance = R₁ + R₂ + R₃ + R₄ = 56 + 56 + 45 + 91 = 248 ohm

V = battery Voltage = 24 Volts

i₁ = current in R₁ = V/R_total = 24/248 = 0.097 A

2)

long time after the switch is closed, the inductor behaves as short circuit

hence R_total = total resistance = R₁ + R₄ = 56 + 91 = 147 ohm

V = battery Voltage = 24 Volts

i₁ = current in R₁ = V/R_total = 24/147 = 0.163 A

3)

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

hence R_total = total resistance = R₁ + R₂ + R₃ + R₄ = 56 + 56 + 45 + 91 = 248 ohm

V = battery Voltage = 24 Volts

i = current in circuit = V/R_total = 24/248 = 0.097 A

V_L = Voltage across indutor = Voltage across the series combination of R₂ and R₃ = i (R₂ + R₃) = (0.097) (56 + 45)

= 9.8 Volts

4)

long time after the switch is closed, the inductor behaves as short circuit

hence R_total = total resistance = R₁ + R₄ = 56 + 91 = 147 ohm

V = battery Voltage = 24 Volts

i₁ = current in R₁ = V/R_total = 24/147 = 0.163 A

i_L = current in inductor = i₁ = 0.163 A

5)

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

hence R_total = total resistance = R₁ + R₂ + R₃ + R₄ = 56 + 56 + 45 + 91 = 248 ohm

V = battery Voltage = 24 Volts

i₁ = current in R₁ = V/R_total = 24/248 = 0.097 A

i2= i1 = 0.097 A