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 = 43 , R3 = 79 and R4 = 84 . The inductance is L = 238 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?

A

2)

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

A

3)

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

A

4)

After the switch has been closed for a very long time, it is then opened. What is I3(topen), the current through the resistor R3 at a time topen = 4.1 ms after the switch was opened? The positive direction for the current is indicated in the figure.

A

5)

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

V

6)

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

V

Explanation / Answer

here,

resistor in circuit :
r1 = r2 = 43 ohms
r3 = 79 ohms
r4 = 84 ohms

inductor, L = 0.238 H

voltage, v = 12 V

Part a:
Since Inductor will resist change in current so I through L and r2 will be zero
I4 = V/(r1 + r2 + r3)
i4 = 12/(43 + 43 + 49)
i4 = 0.073 A

Part b:
now L act as a wire ( which allows current)
r23 = 1/(1/r2 + r3)
r23 = 1/(1/43 + 1/79)
r23 = 27.844 ohms

therefore
equivalent resistance, req = r1 + r23 + r4
equivalent resistance, req = 43 + 27.844 + 84
equivalent resistance, req = 154.844 ohms

Current, I = 12/154.844
Current, I = 0.0775 A

Part C:
Current through L is same through r2, so
Il = Ir2 = I2

current, I2 = v2/r2
current, I2 = I * R23 /r2
current, I2 = 0.0775 * 27.844 /43
current, I2 = 0.05018 A

Part D:
time, t = 4.1 ms = 4.1*10^-3 s

Current, I through L remain same due to inductor, so Imax = Io = 0.05018 A

for RLC circuit, current at any give time
I(t) = Imax * e^-(t*(r2+r3)/L)
I(t) = 0.05018 * e^-((4.1*10^-3)*(43 + 79)/(0.238))
I(t) = - 0.00613 A

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