An L-C circuit containing an 80.0-mH inductor and a 1.25-nF capacitor oscillates with a maximum

step 1 If i is the maximum current in the circuit, then we know that a half L i squared is equal to Q s

An L-C circuit containing an 80.0-mH inductor and a 1.25-nF...

An $L-C$ circuit containing an $80.0-\mathrm{mH}$ inductor and a $1.25-\mathrm{nF}$ capacitor oscillates with a maximum current of 0.750 $\mathrm{A}$ . Calculate: (a) the maximum charge on the capacitor and (b) the oscillation frequency of the circuit. (c) Assuming the capacitor had its maximum charge at time $t=0$ , calculate the energy stored in the inductor after 2.50 $\mathrm{ms}$ of oscillate the energy stored in.

Key Concepts

LC Circuit Oscillations

An LC circuit is an electrical circuit consisting of an inductor (L) and a capacitor (C) that can exchange energy back and forth. The capacitor stores energy in the electric field when charged, and the inductor stores energy in the magnetic field when current flows through it. In the absence of resistance, the energy oscillates sinusoidally between the capacitor and the inductor, resulting in undamped oscillations.

Resonant Frequency in LC Circuits

The oscillation or resonant frequency of an LC circuit is determined by the values of the inductance and capacitance and is given by the formula ? = 1/?(LC), where ? is the angular frequency. This natural frequency defines how quickly the circuit oscillates and is a fundamental property of the system that applies regardless of initial conditions.

Energy Storage in Capacitors and Inductors

Energy in an LC circuit is alternately stored in the capacitor and the inductor. The energy stored in a capacitor can be expressed as U_C = Q²/(2C) where Q is the charge, while the energy stored in an inductor is given by U_L = (1/2)L I², with I being the current. This relationship means that the maximum energy initially stored in the capacitor eventually becomes entirely stored in the inductor when the current is at its peak.

Phase Relationship and Time Evolution in LC Circuits

The oscillatory behavior in an LC circuit is characterized by a phase difference between the charge on the capacitor and the current in the circuit. At the moment when the capacitor has its maximum charge, the current is zero. As time evolves, the current increases, reaching a maximum when the capacitor's charge is zero. This phase shift is a result of the sinusoidal nature of the solutions to the circuit’s differential equations and is essential for determining how energy is exchanged over time.

Transcript

Please give Ace some feedback