Current as Function of Time In an oscillating L C circuit with C=64.0 μF, the current as a funct

Step 1: The current in an oscillating LC circuit is given by the equation $i = I_m sin(omega t +

Current as Function of Time In an oscillating L C circuit...

Current as Function of Time In an oscillating $L C$ circuit with $C=64.0 \mu \mathrm{F}$, the current as a function of time is given by $i=(1.60 \mathrm{~A})$ $\sin \left[\begin{array}{llll}(2500 & \mathrm{rad} / \mathrm{s}) & t+0.680 & \mathrm{rad}]\end{array}\right.$ where $t$ is in seconds. (a) How soon after $t=0 \mathrm{~s}$ will the current reach its maximum value? What are (b) the inductance $L$ and $(\mathrm{c})$ the total energy?

Current as Function of Time In an oscillating $L C$ circuit with $C=64.0 \mu \mathrm{F}$, the current as a function of time is given by $i=(1.60 \mathrm{~A})$ $\sin \left[\begin{array}{llll}(2500 & \mathrm{rad} / \mathrm{s}) & t+0.680 & \mathrm{rad}]\end{array}\right.$
where $t$ is in seconds. (a) How soon after $t=0 \mathrm{~s}$ will the current reach its maximum value? What are
(b) the inductance $L$ and $(\mathrm{c})$ the total energy?

Key Concepts

Oscillating LC Circuit

An LC circuit consists of an inductor (L) and a capacitor (C) connected together, which can exchange energy between the magnetic field in the inductor and the electric field in the capacitor. This energy exchange leads to oscillations, much like in a mechanical harmonic oscillator, and is fundamental in understanding resonance phenomena and transient responses in electrical circuits.

Resonant Frequency in LC Circuits

The resonant or natural frequency of an LC circuit is determined by the formula ? = 1/?(LC), where ? is the angular frequency. This frequency represents the rate at which the circuit oscillates, assuming no energy losses. It is crucial for designing circuits in applications such as radio transmitters, filters, and oscillators.

Phase Angle and Sinusoidal Functions

In oscillatory motion, sinusoidal functions such as sine and cosine are used to describe the time-dependent behavior of current or voltage. The phase angle in these functions indicates the shift in the waveform relative to a reference point in time. For instance, determining when the current attains its maximum value often involves analyzing the phase shift in the sine function.

Energy Conservation in LC Circuits

In an LC circuit, the total energy remains constant (ignoring losses) as it is continuously exchanged between the capacitor, where energy is stored as electric potential energy, and the inductor, where energy is stored as magnetic field energy. This conservation of energy principle is key to analyzing the oscillatory behavior and calculating quantities like peak energies stored in the circuit components.

Relationship Between Inductance and Capacitance

The interplay between inductance and capacitance in an LC circuit governs its oscillation properties. Specifically, the inductance and capacitance determine the natural resonant frequency and influence the energy storage capability of the circuit. Understanding this relationship is essential for tailoring circuits to achieve desired oscillatory behaviors and for the analysis of transient phenomena in electronic systems.

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