An L C circuit includes a 0.025-μF capacitor and a 340-μH inductor. (a) If the peak capacitor vo

step 1 Hi, here in this given problem in the LC circuit, which is having inductor and capacitor in seri

An L C circuit includes a 0.025-μF capacitor and a 340-μH...

An $L C$ circuit includes a $0.025-\mu \mathrm{F}$ capacitor and a $340-\mu \mathrm{H}$ inductor. (a) If the peak capacitor voltage is $190 \mathrm{~V}$, what's the peak inductor current? (b) How long after the voltage peak does the current peak occur?

Key Concepts

LC Circuit Dynamics

An LC circuit consists of an inductor and a capacitor that exchange energy, creating oscillations. Energy stored in the capacitor’s electric field (1/2 C V²) is converted to magnetic energy in the inductor (1/2 L I²) and vice versa. This interplay, with no energy losses in an ideal circuit, results in a natural oscillatory behavior, which is essential for understanding resonant circuits and filters.

Energy Conservation in Oscillatory Systems

In an ideal LC circuit, the total energy remains constant as it oscillates between the capacitor and the inductor. At the moment when the capacitor voltage is at its maximum, all the energy is stored in the capacitor. As the circuit oscillates, this energy is transferred to the inductor, reaching a maximum when the current is at its peak. Equating the energy expressions allows us to relate the peak capacitor voltage to the peak inductor current.

Resonant Frequency and Angular Frequency

The natural frequency of oscillation in an LC circuit is determined by its inductance and capacitance. The angular frequency, given by ? = 1/?(LC), defines the rate at which the circuit oscillates. This frequency is crucial for calculating the period of the oscillation (T = 2?/?), which in turn helps to determine the timing relationships between voltage and current peaks.

Phase Relationship in LC Circuits

In an LC circuit, there is a phase difference between the capacitor voltage and the inductor current. Specifically, the current lags the voltage by 90 degrees (or ?/2 radians), meaning that the peak current occurs one quarter of a cycle after the peak voltage. This phase shift is inherent to the sinusoidal nature of the oscillatory motion in the circuit and is a key concept in alternating current (AC) analysis.

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