A circuit is constructed with a resistor, two inductors, one capacitor, one batt

Question

A circuit is constructed with a resistor, two inductors, one capacitor, one battery and a switch as shown. The value of the resistance is R1 = 353 . The values for the inductances are: L1 = 290 mH and L2 = 136 mH. The capacitance is C = 136 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 closed for a long time when at time t = 0, the switch is opened. What is UL1(0), the magnitude of the energy stored in inductor L1 just after the switch is opened?

J

2)

What is o, the resonant frequency of the circuit just after the switch is opened?

radians/s

3)

What is Qmax, the magnitude of the maximum charge on the capacitor after the switch is opened?

C

4)

What is Q(t1), the charge on the capacitor at time t = t1 = 3.43 ms. Q(t1) is defined to be positive if V(a) – V(b) is positive.

C

5)

What is t2, the first time after the switch is opened that the energy stored in the capacitor is a maximum?

ms

6)

What is the total energy stored in the inductors plus the capacitor at time t = t2?

J

Explanation / Answer

part a )

at t = 0 capacitor is uncharged , so current will flow in outer circuit

I = V/R = 12/353 = 0.034 A

U = LI^2/2 = 290 x 10^-3 * (0.034)^2/2 = 1.68 x 10^-4 J

part b )

w = 1/sqrt(LC)

since L1 and L2 in series , so L = L1+L2

L = 290+136 = 426 mH

wo = 131.38 rad/s

part c )  

I = dQ/dt

Qmax = Imax/w

Qmax = 0.034/131.38 = 2.59 x 10^-4 C = 259 uC

part d )

Q(t) = -Qmax*sin(wt)

t = 3.43 ms

Q(t) = -112.66 uC

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