Conservation of Energy in LC Oscillations

The total energy of an LC circuit at any instant is the sum of its electrical (capacitor) and magnetic (inductor) energies, $U=U_E+U_B=q^2/2C+\tfrac12Li^2$; evaluating this sum at three different, representative instants of the oscillation shows that it always comes out to exactly the SAME constant value, proving that total energy is genuinely conserved throughout an LC oscillation.

Case (i): when charge $q=Q_m$ (maximum) and current $i=0$, $U=Q_m^2/2C+0=Q_m^2/2C \qquad (4.56)$ -- wholly electrical. Case (ii): when charge = 0 and current $i=I_m$ (maximum), $U=0+\tfrac12LI_m^2=\tfrac12LI_m^2 \qquad (4.57)$ -- wholly magnetic; since (by energy conservation, taken as already established) this must equal the case-(i) value, $\tfrac12LI_m^2=Q_m^2/2C$, giving the useful relation $I_m=Q_m/\sqrt{LC}=Q_m\omega$ (with $\omega=1/\sqrt{LC}$ the LC circuit's own natural angular frequency, derived by analogy in section 4.9.3). Case (iii): at a GENERAL instant with charge $q=Q_m\cos\omega t$ and current $i=dq/dt=-Q_m\omega\sin\omega t$ (the negative sign showing the capacitor's charge is decreasing at this point in the cycle), the total energy is $U = \dfrac{Q_m^2\cos^2\omega t}{2C}+\dfrac{1}{2}LQ_m^2\omega^2\sin^2\omega t$; substituting $\omega^2=1/LC$ into the second term turns it into $\dfrac{Q_m^2}{2C}\sin^2\omega t$, so the whole expression collapses to $U = \dfrac{Q_m^2}{2C}\left(\cos^2\omega t+\sin^2\omega t\right) = \dfrac{Q_m^2}{2C} \qquad (4.58)$, using $\cos^2\theta+\sin^2\theta=1$. …