Oscillation in an LC Circuit

A charged capacitor wired to an inductor does not just dump its charge through the wire. As the capacitor drains, the growing current builds a magnetic field in the inductor, and that inductor fights any change in current. By the time the capacitor is empty, all its energy sits in the inductor's magnetic field, and the current is at a maximum. That current keeps flowing, now charging the capacitor with opposite polarity, until the field collapses and the cycle reverses. Nothing dissipates energy, so the system rings forever between electric and magnetic storage.

Kirchhoff's voltage law around the loop gives L(dI/dt) + Q/C = 0. Since I = dQ/dt, this becomes L(d^2Q/dt²) + Q/C = 0, or d^2Q/dt² = -(1/LC)Q. Compare that to a mass on a spring: d^2x/dt² = -(k/m)x. Same equation, different letters. Charge Q plays the role of position x, inductance L plays the role of mass m, and 1/C plays the role of spring constant k. The solution is Q(t) = Q0 cos(omega t), with angular frequency omega = 1/√LC, so the ordinary frequency is f = 1/(2 π √LC).

Current is the derivative of charge, so I(t) = -omega Q0 sin(omega t), peaking at I_max = Q0 omega = Q0/√LC. You can also get that peak from energy conservation: all the energy starts as (1/2)Q0²/C in the capacitor and all of it ends up as (1/2)L I_max² in the inductor, since total energy is conserved. This topic is the electrical twin of the mechanical oscillators from earlier: same math, new physical quantities, and it sets up driven and damped circuits next.