In an oscillating L C circuit in which C=6.00 μF, the maximum potential difference across the ca

step 1 31 .20 we have an LC circuit. We have a four micro -tharid capacitor. The maximum value of the p

In an oscillating L C circuit in which C=6.00 μF, the...

In an oscillating $L C$ circuit in which $C=6.00 \mu \mathrm{F}$, the maximum potential difference across the capacitor during the oscillations is $1.50 \mathrm{~V}$ and the maximum current through the inductor is $50.0 \mathrm{~mA}$. What are (a) the inductance $L$ and (b) 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

Angular Frequency in LC Circuits

The angular frequency (?) in an LC circuit is given by the formula ? = 1/?(LC). This relationship is central in understanding the speed of oscillation in the circuit, as it directly connects the physical properties of the inductor (L) and the capacitor (C) with the circuit's natural frequency, underpinning the dynamic behavior of energy exchange.

LC Circuit Oscillations

An LC circuit, comprising an inductor and a capacitor, exhibits oscillatory behavior where energy continuously shifts between the magnetic field of the inductor and the electric field of the capacitor. This oscillation, absent any resistive losses, is a fundamental example of harmonic motion in electrical circuits and illustrates key principles of energy conservation in oscillatory systems.

Relationship between Capacitance, Voltage, and Charge

The relationship Q = CV ties the charge stored in a capacitor to its capacitance (C) and the voltage (V) across it. This fundamental equation is crucial in oscillatory circuits, where the maximum potential difference defines the peak charge, which in turn influences other quantities like current when energy shifts between the capacitor and the inductor.

Relationship between Inductance, Current, and Oscillation

In an LC circuit, the maximum current is intrinsically linked to the rate of change of charge, related by the equation I_max = Q_max ?. This concept connects the inductive properties of the circuit with the dynamic oscillatory process, showing how the inductance and the sinusoidal variation of charge determine the current's amplitude during the energy oscillation.

Time Period and Phase Relationships in Sinusoidal Oscillations

The period of oscillation in an LC circuit is given by T = 2??(LC), with the frequency of oscillation f = 1/T. Additionally, the sinusoidal nature of the charge and current distributions means that they are out of phase by a quarter cycle. For example, the time it takes for the charge to rise from zero to its maximum value is T/4, a direct consequence of the mathematical properties of sine functions in describing oscillatory motion.

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