Maximum Potential Difference In an oscillating L C circuit...

Maximum Potential Difference In an oscillating $L C$ circuit in which $C=4.00 \mu \mathrm{F}$, the maximum potential difference across the capacitor during the oscillations is $1.50 \mathrm{~V}$ and the maximum current through the inductor is $50.0 \mathrm{~mA}$. (a) What is the inductance $L$ ? (b) What is the frequency of the oscillations? (c) How much time is required for the charge on the capacitor to rise from zero to its maximum value?

Maximum Potential Difference In an oscillating $L C$ circuit in which $C=4.00 \mu \mathrm{F}$, the maximum potential difference across the capacitor during the oscillations is $1.50 \mathrm{~V}$ and the maximum current through the inductor is $50.0 \mathrm{~mA}$. (a) What is the inductance $L$ ?
(b) What is the frequency of the oscillations? (c) How much time is required for the charge on the capacitor to rise from zero to its maximum value?

Key Concepts

LC Circuit Oscillations

LC circuits, which consist of an inductor (L) and a capacitor (C), exhibit energy oscillations where energy is continually exchanged between the electric field of the capacitor and the magnetic field of the inductor. This process results in undamped, harmonic oscillations in an ideal circuit, and it forms the basis for resonant circuits used in various applications such as filters and oscillators.

Angular Frequency in LC Circuits

The angular frequency (?) of an LC circuit is determined by the relationship ? = 1/?(LC), which shows how the values of the inductance and capacitance set the natural frequency of the oscillation. This fundamental relationship allows for the prediction of system behavior and is central to understanding oscillatory dynamics in many physical systems.

Relationship Between Maximum Current and Maximum Charge

In LC oscillations, the maximum current in the inductor and the maximum charge on the capacitor are related by the equation I_max = ?Q_max, where Q_max = CV_max. This relationship stems from the conversion of stored electric energy in the capacitor into magnetic energy in the inductor and vice versa, highlighting the direct connection between voltage, capacitance, current, and the frequency of oscillation.

Phase and Time Relationships in LC Oscillations

The sinusoidal nature of the charge and current in an LC circuit means that there is a phase difference between them, typically 90° (or ?/2 radians). This phase difference implies that the time taken for the capacitor’s charge to rise from zero to its maximum value is one-quarter of the oscillation period, a key concept in analyzing the temporal behavior of LC circuits.

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