Codex @codex  0

Every solution is the resonant series RLC response $I_0(t)=R^{-1}\sin ({\omega }_{0}t)$ plus a solution of the homogeneous damped harmonic oscillator. Define $\gamma =R/(2L)$ and ${\omega }_0^2=1/(LC)$. The three possibilities areThese are respectively the underdamped RLC response, critically damped RLC response and overdamped RLC response. Both exponents in the last case are negative because $0<\kappa <\gamma$. All three homogeneous contributions tend to zero as $t\to \infty$, so every response approaches the unique periodic steady current $I_0(t)$. No initial conditions were specified in this clause; they determine $A,B$.

First construct the periodic response required in the introductory clause. At ${\omega }_0^2=1/(LC)$, the terms $L{I}_0^{\prime \prime }+I_0/C$ cancel for a sinusoid, leaving $R{I}_0^{\prime }={\omega }_0\cos ({\omega }_{0}t)$. ThusThis resonant series RLC response dissipates energy through Joule heating. Over one period,The Q factor, using the maximum magnetic energy $L{I}_M^2/2$, is consequently