At t=0, let Q=Q0, and I=0 in an L C circuit. (a) At the...

At $t=0$, let $Q=Q_{0}$, and $I=0$ in an $L C$ circuit. (a) At the first moment when the energy is shared equally by the inductor and the capacitor, what is the charge on the capacitor? (b) How much time has elapsed (in terms of the period $T$ )?

Key Concepts

Phase Relationship and Energy Equality

At specific moments during oscillation, the energy in the capacitor and the inductor may become equal. By analyzing the sinusoidal functions of charge and current, one can determine the precise phase (or time) at which this occurs. Typically, this involves solving for the point where the squared cosine and squared sine functions equal one another, indicating an equal split of energy.

Energy Storage in Capacitors and Inductors

Capacitors store energy in the form of an electric field, with energy given by the formula (1/2)(charge)^2/capacitance, while inductors store energy in a magnetic field, which is expressed as (1/2) times inductance times the square of the current. These expressions are essential for understanding how energy is partitioned during oscillations.

Sinusoidal Time Dependence and Angular Frequency

The charge on the capacitor and the current in the inductor in an LC circuit can be described by sinusoidal functions, where the angular frequency is defined as ? = 1/?(LC). This relation leads to a period of oscillation T = 2??(LC), which is fundamental in analyzing the timing of energy transfer between the components.

LC Circuit Oscillations

An LC circuit is a simple harmonic oscillator where energy alternates between the electric field of the capacitor and the magnetic field of the inductor. The charge and current in the circuit vary sinusoidally with time, and the angular frequency of oscillation is determined solely by the inductance and capacitance.

Energy Conservation in LC Circuits

In an ideal LC circuit, the total energy remains constant over time. Initially, all the energy may reside in the capacitor, and as the system oscillates, energy is periodically swapped between the capacitor and the inductor. At any point during the oscillation, the sum of the energies in the capacitor and inductor equals the initial stored energy.

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