A series circuit has an inductor of 0.2 henry, no resistance...

A series circuit has an inductor of 0.2 henry, no resistance and a capacitor of $10^{-5}$ farad. The initial charge on the capacitor is $10^{-6}$ coulomb and there is no initial current. Find the charge on the capacitor and the current at any time $t$. Find the amplitude and phase shift of the charge function.

Key Concepts

Second-Order Linear Differential Equations

The dynamic behavior of an LC circuit is governed by a second-order linear differential equation that arises from Kirchhoff’s voltage law. This equation has solutions that take the form of sinusoidal functions, reflecting the oscillatory nature of the circuit. The solution of this differential equation in the context of LC circuits reveals how initial conditions, such as initial charge on the capacitor and initial current in the inductor, determine the evolution of the system over time.

Amplitude and Phase Shift Determination in Oscillatory Systems

In oscillatory systems, the amplitude and phase shift of the sinusoidal response are determined by the initial conditions imposed on the system. For an LC circuit, applying the initial charge and current conditions allows the complete determination of the sinusoidal functions describing charge and current over time. The amplitude represents the maximum value of the charge or current, while the phase shift accounts for any horizontal displacement of the waveform relative to a reference point in time.

Resonant (Angular) Frequency

The resonant or natural frequency of an LC circuit is given by the expression ?? = 1/sqrt(LC). This angular frequency is central to the analysis of the circuit because it sets the pace at which the charge and current oscillate. It plays a crucial role in the sinusoidal solutions of the differential equation associated with the circuit.

LC Circuit Oscillations

An LC circuit, made up solely of an inductor (L) and a capacitor (C) in series or parallel, exhibits natural oscillatory behavior when energy is exchanged between its magnetic field (in the inductor) and its electric field (in the capacitor). This energy exchange creates an oscillation in charge and current at a frequency determined by the values of L and C, and the system models simple harmonic motion.

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