In a certain oscillating L C circuit, the total energy is...

In a certain oscillating $L C$ circuit, the total energy is converted from clectrical cnergy in the capacitor to magnetie cnergy in the inductor in $2.50 \mu \mathrm{s}$. What are (a) the period of oscillation and (b) the frequency of oscillation? (c) How long after the magnetic energy is a maximum will it be a maximum again? (d) In one full cycle, how many times will the electrical energy be maximum?

Key Concepts

Energy Maximums in Oscillatory Cycles

Throughout one full oscillation cycle, the maximum stored energy in the capacitor (electrical energy) or the inductor (magnetic energy) occurs at specific time intervals. In an LC circuit, the electrical energy typically attains its maximum value twice in each cycle, with these peaks corresponding to the beginning and midway points of the oscillation.

Quarter and Half-Period Intervals

In an LC oscillator, the energy conversion from one form to another, such as electrical to magnetic energy, typically occurs over a quarter of a full cycle. This is because the energy oscillates sinusoidally, reaching one form’s maximum when the other is zero. Additionally, the same type of energy reaches its maximum again after a half-period, reflecting the periodic nature of the system.

LC Circuit Oscillations

An LC circuit consists of an inductor (L) and a capacitor (C) that together create a system capable of oscillating energy between electric (capacitor) and magnetic (inductor) forms. This oscillatory behavior is analogous to a mechanical harmonic oscillator and is governed by parameters like the inductance and capacitance, which define the natural frequency of the circuit.

Period and Frequency

The period is the time taken for the oscillatory system to complete one full cycle of energy transfer, while the frequency is the number of cycles per unit time. In any oscillating system these two quantities are inversely related, with frequency being the reciprocal of the period.

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