Initially a Maximum In an oscillating L C circuit with L=50 mH and C=4.0 μF, the current is init

step 1 In this problem, we have an oscillating LC circuit with the inductance equal to 50 millie hemorr

Initially a Maximum In an oscillating L C circuit with L=50...

Initially a Maximum In an oscillating $L C$ circuit with $L=50 \mathrm{mH}$ and $C=4.0 \mu \mathrm{F}$, the current is initially a maximum. How long will it take before the capacitor is fully charged for the first time?

Key Concepts

LC Circuit Oscillations

An LC circuit, composed of an inductor (L) and a capacitor (C), exhibits oscillatory behavior where energy continuously transfers between the magnetic field of the inductor and the electric field of the capacitor. This oscillatory dynamics is a classic example of simple harmonic motion in electrical systems and serves as a fundamental concept in understanding transient and resonant responses in circuits.

Resonant Frequency

The resonant (or natural) frequency of an LC circuit is determined by the values of the inductance and capacitance, and it represents the frequency at which the circuit naturally oscillates. This frequency is given by ? = 1/sqrt(LC), and it is central to predicting the timing and behavior of the circuit’s oscillations, including how long it takes for the capacitor to become fully charged in phase with the oscillation.

Phase Relationships in Oscillations

In the context of an LC circuit, the current and voltage across the capacitor are out of phase due to the periodic exchange of energy. Understanding the phase relationship is important because a maximum current corresponds to a zero voltage across the capacitor, and the capacitor reaches its maximum charge when the current first reaches zero. This phase shift, typically a quarter period of the oscillation, is crucial for determining the time intervals at which particular circuit states occur.

Time Period of Oscillation

The time period of the oscillations in an LC circuit is directly related to its resonant frequency by the relation T = 2?/?. Knowing the time period allows calculation of specific time intervals for events such as the capacitor being fully charged. Since the capacitor is fully charged after a quarter of the oscillation period following a maximum current, leveraging the time period becomes essential in solving timing problems in LC circuits.

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