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 = 290 ?. The values for the inductances are: L₁ = 395 mH and L₂ = 172 mH. The capacitance is C = 91 ?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 L₁just 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 = 3.52 ms. Q(t₁) is defined to be positive if V(a)

Answer 1

a)

The current in the circuit before switch is opened

I=V/R =12/290

I=0.04138 A

the magnitude of the energy stored in inductor L₁ just after the switch is opened is

U_1(0)=(1/2)*L1*I^2

U_1(0) =(1/2)*0.395*0.04138^2

U_1(0)=3.38*10^-4 J

b)

L=L1+L2=0.395+0.172

L=0.567 H

Resonant freqency

Wo=1/sqrt[L*C]

Wo=1/sqrt[0.567*(91*10^-6)]

Wo=139.22 rad/s

c)

U_L =U_C

(1/2)*L*I^2=(1/2)*Qmax^2/C

0.567*0.04138^2 =Qmax^2/(91*10^-6)

Qmax=297.23 uC

d)

Q(t) =Qmax*sin(Wt)

The charge on the capacitor at t=3.52 ms

Q(t=3.52 ms) =297.23*sin(139.22*3.52*10^-3)

Q(t=3.52ms)=139.9 uC

e)

The capacitor has maximum energy when charge stored in capacitor at time t1 is equal maximum charge

Q=Qmax

sinWt2=1

Wt2=pi/2

t2=pi/2W =pi/2*139.22

t2=11.28 ms

f)

U=(1/2)Qmax^2/C =(1/2)*(297.23*10^-6)^2/(91*10^-6)

U=4.85*10^-4 J