In an L-R-C series circuit, the current is given by i = I cost. The voltage amplitudes

In an L-R-C series circuit, the current is given by i = I cosωt. The voltage amplitudes for the resistor, inductor, and capacitor are VR, VL, and VC.

 (a) Show that the instantaneous power into the resistor is PR = VR I cos2ωt = ½ VR I (1 + cos2ωt). What does this expression give for the average power into the resistor? 

(b) Show that the instantaneous power into the inductor is PL = -VL I sinωt cosωt = -½ VL I sin2ωt. What does this expression give for the average power into the inductor? 

(c) Show that the instantaneous power into the capacitor is PC = VC I sinωt cosωt = ½ VC I sin2ωt. What does this expression give for the average power into the capacitor? 

(d) The instantaneous power delivered by the source is shown in Section 31.4 to be P = V I cosωt (cosΦ cosωt – sinΦ sinωt). Show that PR + PL + PC equals P at each instant of time.