A series circuit has an inductor of 0.4 henry, no resistance...

A series circuit has an inductor of 0.4 henry, no resistance and a capacitor of $10^{-4}$ farad. The initial charge on the capacitor is $10^{-5}$ 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. (See exercise 21 in section $15.1 .)$

Key Concepts

LC Circuit Dynamics

An LC circuit, consisting of an inductor and a capacitor connected in series or parallel, exhibits oscillatory behavior. When the resistor is absent, the circuit’s energy oscillates between the electric field of the capacitor and the magnetic field of the inductor, leading to undamped oscillations described by second-order differential equations.

Differential Equation for Oscillatory Systems

The behavior of the charge in an LC circuit is governed by a second-order linear homogeneous differential equation similar to that governing simple harmonic motion. This equation is derived from Kirchhoff's voltage law and takes the form d²q/dt² + (1/LC) q = 0, which characterizes the time evolution of the charge on the capacitor.

Angular Frequency

In the context of LC circuits, the angular frequency ? of the oscillations is given by the formula ? = 1/?(LC). This frequency determines the rate of oscillation and is a fundamental property that characterizes the dynamics of the circuit under ideal, energy-conserving conditions.

Initial Conditions and Phase Shift

The specific solution to the differential equation for the LC circuit is determined by the initial conditions, such as the initial charge on the capacitor and the initial current. These conditions are used to set the amplitude and phase shift in the sinusoidal solution form (typically expressed as q(t) = Q cos(?t + ?)), ensuring that the solution accurately reflects the circuit's starting state.

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