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Significance

The energy relationship set up in part (b) is not the only way we can equate energies. At most times, some energy is stored in the capacitor and some energy is stored in the inductor. We can put both terms on each side of the equation. By examining the circuit only when there is no charge on the capacitor or no current in the inductor, we simplify the energy equation.

Check Your Understanding The angular frequency of the oscillations in an LC circuit is 2.0 × 10 3 rad/s. (a) If L = 0.10 H , what is C ? (b) Suppose that at t = 0 , all the energy is stored in the inductor. What is the value of ϕ ? (c) A second identical capacitor is connected in parallel with the original capacitor. What is the angular frequency of this circuit?

a. 2.5 μ F ; b. π / 2 rad or 3 π / 2 rad ; c. 1.4 × 10 3 rad/s

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Summary

  • The energy transferred in an oscillatory manner between the capacitor and inductor in an LC circuit occurs at an angular frequency ω = 1 L C .
  • The charge and current in the circuit are given by
    q ( t ) = q 0 cos ( ω t + ϕ ) , i ( t ) = ω q 0 sin ( ω t + ϕ ) .

Conceptual questions

Do Kirchhoff’s rules apply to circuits that contain inductors and capacitors?

yes

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Can a circuit element have both capacitance and inductance?

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In an LC circuit, what determines the frequency and the amplitude of the energy oscillations in either the inductor or capacitor?

The amplitude of energy oscillations depend on the initial energy of the system. The frequency in a LC circuit depends on the values of inductance and capacitance.

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Problems

A 5000-pF capacitor is charged to 100 V and then quickly connected to an 80-mH inductor. Determine (a) the maximum energy stored in the magnetic field of the inductor, (b) the peak value of the current, and (c) the frequency of oscillation of the circuit.

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The self-inductance and capacitance of an LC circuit are 0.20 mH and 5.0 pF. What is the angular frequency at which the circuit oscillates?

ω = 3.2 × 10 −7 rad/s

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What is the self-inductance of an LC circuit that oscillates at 60 Hz when the capacitance is 10 μ F ?

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In an oscillating LC circuit, the maximum charge on the capacitor is 2.0 × 10 −6 C and the maximum current through the inductor is 8.0 mA. (a) What is the period of the oscillations? (b) How much time elapses between an instant when the capacitor is uncharged and the next instant when it is fully charged?

a. 7.9 × 10 −4 s ; b. 4.0 × 10 −4 s

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The self-inductance and capacitance of an oscillating LC circuit are L = 20 mH and C = 1.0 μ F , respectively. (a) What is the frequency of the oscillations? (b) If the maximum potential difference between the plates of the capacitor is 50 V, what is the maximum current in the circuit?

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In an oscillating LC circuit, the maximum charge on the capacitor is q m . Determine the charge on the capacitor and the current through the inductor when energy is shared equally between the electric and magnetic fields. Express your answer in terms of q m , L , and C .

q = q m 2 , I = q m 2 L C

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In the circuit shown below, S 1 is opened and S 2 is closed simultaneously. Determine (a) the frequency of the resulting oscillations, (b) the maximum charge on the capacitor, (c) the maximum current through the inductor, and (d) the electromagnetic energy of the oscillating circuit.

A 12 volt battery is connected to a 4 microfarad capacitor and a 100 millihenry inductor which are both connected in parallel with each other. There are two switches in the circuit. Switch S1 is closed. If opened, it would open the whole circuit. Switch S2 is open and hence the inductor is currently disconnected.
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An LC circuit in an AM tuner (in a car stereo) uses a coil with an inductance of 2.5 mH and a variable capacitor. If the natural frequency of the circuit is to be adjustable over the range 540 to 1600 kHz (the AM broadcast band), what range of capacitance is required?

C = 1 4 π 2 f 2 L f 1 = 540 Hz; C 1 = 3.5 × 10 −11 F f 2 = 1600 Hz; C 2 = 4.0 × 10 −12 F

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Source:  OpenStax, University physics volume 2. OpenStax CNX. Oct 06, 2016 Download for free at http://cnx.org/content/col12074/1.3
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