Circuits with Capacitors and Inductors (LC Circuits)

A charged capacitor across an ideal inductor obeys $L\,d^2Q/dt^2 + Q/C = 0$, the mass-spring equation with $L$ as mass and $1/C$ as stiffness, so $Q = Q_0\cos(\omega t)$ with $\omega = 1/\sqrt{LC}$ and $T = 2\pi\sqrt{LC}$. The energy $Q^2/2C + LI^2/2$ is constant, which forces the charge maximum and the current maximum a quarter period apart: $I = 0$ when $Q = \pm Q_0$, and $I = I_{\max} = \omega Q_0$ when $Q = 0$. With no resistance the amplitude never shrinks; the inductor carries the charge through zero and refills the capacitor with reversed polarity.

Three errors account for nearly every miss. Peaking charge and current together, which would double the energy the circuit was given. Mangling $\omega = 1/\sqrt{LC}$ by inverting it, dropping the root, or scaling the period linearly, so that quadrupling $C$ is read as quadrupling the frequency when it halves it. And drawing the ideal LC discharge as an RC exponential that slides to zero and stops, when the current is at its maximum exactly then and the oscillation goes on unless a resistor takes a share.