A circuit is constructed with a resistor two inductors one c

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 = 162 . The values for the inductances are: L₁ = 203 mH and L₂ = 144 mH. The capacitance is C = 142 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 U_L1(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 Q_max, 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 = 4 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

I need right answers !

Thank you!

Solution

1) CURRENT: i = v / R = 12 / 162 = 0.074 A

the energy:

u1 = 1/2 * L1 * i^{2 =} 1/2 * 203 * 10-3 * 0.074² = 5.57 * 10^-3 J

2)

the resonance frequency:

o = 1 / sqrt[LC]

= 1 / sqrt[(L1+L2)C]

   = 1 / sqrt[(203 mH + 144 mH)(142 * 10-6)]

= 142.46 rad/s

3) we know that, u_L= u_C

so,

1/2 L i² = 1/2 Q² / C

thus, the max charge:

Qmax = sqrt [L C i²]

= sqrt [L C i²]

=  sqrt (203 mH + 144 mH)(142 * 10-6)0.074²]

= 5.19 * 10-4 C

= 519 C

= 520  C

4)

Charge: Q = Q_maxsin wt =5.19 * 10-4 C * sin (142.46*4 * 10-3)

   =2.8e-4 C

= 280 C

5)

Charge:

Q = Q_max, sin wt = 1, wt = pi / 2, t = pi / 2 w

= pi / (2 * 142.46)

= 3.5x10^-3 s

   = 3.5 ms

6) u = 1/2 Q_max²/ c

= 1/2 [(5.19 * 10-4 C )² / (142 * 10-6)]

= 9.48x10^-4 J