Chapter 31, Problem 31.67CP

CALC In an L-R-C series circuit the current is given by i = Icos ωt. The voltage amplitudes for the resistor, inductor, and capacitor are V_R, V_L, and V_C. (a) Show that the instantaneous power into the resistor is p_R = V_RIcos² ωt = $\frac{1}{2}$V_RI(1 + cos 2ωt). What does this expression give for the average power into the resistor? (b) Show that the instantaneous power into the inductor is p_L = −V_LIsin ωt cos ωt = − $\frac{1}{2}$V_LI sin 2ωt. What does this expression give for the average power into the inductor? (c) Show that the instantaneous power into the capacitor is p_C = V_CI sin ωt cos ωt = $\frac{1}{2}$V_CI sin 2ωt. 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 = VIcos ωt(cos ϕ cos ωt − sin ϕ sin ωt). Show that p_R + p_L + p_C equals p at each instant of time.