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 = 267 . The values for the inductances are: L₁ = 360 mH and L₂ = 149 mH. The capacitance is C = 75 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 = 2.66 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

Solution

1) U_L1 = 1/2 L₁ I²

I = V / R₁ = 12 / 267 = 0.0449 A

U_L1 = 1/2 L₁ I² = 1/2 (360 * 10^-3) * 0.0449²

= 0.000363 J

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

= 1 / sqrt[((360 * 10^-3) + (149 * 10^-3)) 75 * 10-6]

= 161.85 rad/s

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3) Q²_max / 2 C = 1/2 I² (L₁ + L₂)

Q_max = sqrt [C I² (L₁ + L₂)]

= sqrt [75 * 10^-6 * 0.0449² ((360 * 10^-3) + (149 * 10^-3))]

= 2.77 * 10^-4 C

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4) Q(t) = Q_max cos(wt + phi/2 )

= 2.77 * 10^-4 * cos(161.85 * 2.66 * 10^-3 + pi/2 )

= 2.76 * 10^-4 C