An L-C circuit containing an 80.0-mH inductor and a 1.25-nF...

An $L$-$C$ circuit containing an 80.0-mH inductor and a 1.25-nF capacitor oscillates with a maximum current of 0.750 A. Calculate: (a) the maximum charge on the capacitor and (b) the oscillation frequency of the circuit. (c) Assuming the capacitor had its maximum charge at time $t = 0$, calculate the energy stored in the inductor after 2.50 ms of oscillation.

Key Concepts

Energy Storage in the Inductor

The energy stored in an inductor is expressed by the formula U = ½ L I², where L is the inductance and I is the instantaneous current. This concept is essential in understanding how magnetic energy in the circuit evolves with the oscillating current, especially when analyzing the circuit at different points in its cycle.

Charge-Current Relationship in Oscillatory Circuits

The maximum charge on the capacitor and the maximum current through the inductor are intrinsically linked through the conservation of energy in the LC circuit. The peak energy can be expressed in terms of the capacitor's charge (U = Q²/(2C)) or the inductor's current (U = ½ L I²), allowing one to derive one quantity if the other is specified.

Energy Conservation in LC Circuits

In an ideal LC circuit, the total energy remains constant as it is cyclically converted between the electric potential energy in the capacitor and the magnetic energy in the inductor. This principle of energy conservation is fundamental when analyzing the oscillatory behavior of the circuit and calculating various parameters such as charge and current at different times.

Resonant Frequency

The resonant frequency of an LC circuit is the natural frequency at which it oscillates, determined by the inductance and capacitance of the circuit. It is given by the formula f = 1/(2??(LC)) and indicates how fast the energy transfers back and forth between the capacitor and the inductor.

LC Circuit Oscillations

An LC circuit consists of an inductor and a capacitor in a configuration where energy continuously oscillates between the magnetic field in the inductor and the electric field in the capacitor. This oscillatory behavior is analogous to simple harmonic motion, where the current and charge vary sinusoidally over time.

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