$A$ circuit containing an $80 \, mH$ inductor and a $60 \, \mu F$ capacitor in series is connected to a $230 \, V, 50 \, Hz$ supply. The resistance of the circuit is negligible.
$(a)$ Obtain the current amplitude and $rms$ values.
$(b)$ Obtain the $rms$ values of potential drops across each element.
$(c)$ What is the average power transferred to the inductor?
$(d)$ What is the average power transferred to the capacitor?
$(e)$ What is the total average power absorbed by the circuit? ('Average' implies 'averaged over one cycle'.)

(A) Given: $L = 80 \, mH = 0.08 \, H$,$C = 60 \, \mu F = 60 \times 10^{-6} \, F$,$V_{rms} = 230 \, V$,$f = 50 \, Hz$.
Angular frequency $\omega = 2 \pi f = 100 \pi \approx 314.16 \, rad/s$.
Inductive reactance $X_L = \omega L = 100 \pi \times 0.08 = 8 \pi \approx 25.13 \, \Omega$.
Capacitive reactance $X_C = \frac{1}{\omega C} = \frac{1}{100 \pi \times 60 \times 10^{-6}} = \frac{10^6}{6000 \pi} \approx 53.05 \, \Omega$.
Net reactance $X = |X_L - X_C| = |25.13 - 53.05| = 27.92 \, \Omega$.
$(a)$ $I_{rms} = \frac{V_{rms}}{X} = \frac{230}{27.92} \approx 8.24 \, A$. Current amplitude $I_0 = I_{rms} \sqrt{2} \approx 8.24 \times 1.414 \approx 11.65 \, A$.
$(b)$ $V_{L,rms} = I_{rms} X_L = 8.24 \times 25.13 \approx 207.07 \, V$. $V_{C,rms} = I_{rms} X_C = 8.24 \times 53.05 \approx 437.13 \, V$.
$(c)$ Average power in inductor $P_L = V_{L,rms} I_{rms} \cos(90^\circ) = 0 \, W$.
$(d)$ Average power in capacitor $P_C = V_{C,rms} I_{rms} \cos(90^\circ) = 0 \, W$.
$(e)$ Total average power $P = P_L + P_C = 0 \, W$.