In an oscillating L C circuit, when 75.0% of the total...

In an oscillating $L C$ circuit, when 75.0$\%$ of the total energy is stored in the inductor's magnetic field, (a) what multiple of the maximum charge is on the capacitor and (b) what multiple of the maximum current is in the inductor?

Key Concepts

LC Circuit Oscillations

An LC circuit is a resonant system where energy continually transfers between the electric field of a capacitor and the magnetic field of an inductor. The circuit oscillates at its natural resonant frequency, which is determined by the values of the inductance and capacitance. This behavior is analogous to a simple harmonic oscillator in mechanics.

Energy Storage in Reactive Components

The capacitor stores energy in its electric field, quantified by the expression (1/2) Q²/C, while the inductor stores energy in its magnetic field, given by (1/2) L I². These formulas allow us to understand how energy is partitioned in the circuit at different points in the oscillation cycle.

Conservation of Energy in LC Circuits

In an ideal LC circuit with no resistive losses, the total energy remains constant over time. This energy conservation principle means that as energy moves from the capacitor to the inductor and back, the sum of the energies in both components is always equal to the maximum energy of the system, which occurs when either the capacitor or the inductor is fully energized.

Phase Relationship and Amplitude Scaling

The oscillatory behavior of charge and current in an LC circuit manifests as sinusoidal functions with a phase difference of 90 degrees. The charge on the capacitor and the current in the inductor reach their maximum values at different times, with the current peaking when the charge is zero. This phase relationship allows for the determination of the instantaneous amplitudes as fractions or multiples of their maximum values based on the energy distribution at any given moment.

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