CALC In an L - R - C series circuit the current is given by i = I cos 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 R I cos 2 ωt = 1 2 V R I (1 + cos 2 ωt ). 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 L I sin ωt cos ωt = − 1 2 V L I sin 2 ωt . 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 C I sin ωt cos ωt = 1 2 V C I sin 2 ωt . 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 = VI cos ωt (cos ϕ cos ωt − sin ϕ sin ωt ). Show that p R + p L + p C equals p at each instant of time.
CALC In an L - R - C series circuit the current is given by i = I cos 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 R I cos 2 ωt = 1 2 V R I (1 + cos 2 ωt ). 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 L I sin ωt cos ωt = − 1 2 V L I sin 2 ωt . 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 C I sin ωt cos ωt = 1 2 V C I sin 2 ωt . 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 = VI cos ωt (cos ϕ cos ωt − sin ϕ sin ωt ). Show that p R + p L + p C equals p at each instant of time.
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 VR, VL, and VC. (a) Show that the instantaneous power into the resistor is pR = VRIcos2ωt =
1
2
VRI(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 pL = −VLIsin ωt cos ωt = −
1
2
VLI sin 2ωt. What does this expression give for the average power into the inductor? (c) Show that the instantaneous power into the capacitor is pC = VCI sin ωt cos ωt =
1
2
VCI 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 pR + pL + pC equals p at each instant of time.
A resistor with resistance R and a capacitor with capacitance C are connected in series to an AC voltage source. The time-dependent voltage across the capacitor is given by VC(t) = VC0sin(omegat). What is the amplitude VR of the voltage across the resistor?
Find the charge on the capacitor in an LRC-series circuit when L =
-h, R = 10 N, C = 0.01 f, E(t) = 150 V, q(0) = 1 C, and i(0) = 0 A.
-10t
e
3
q(t) =
(cos 10t + sin 10t) +
What is the charge on the capacitor after a long time?
Chapter 31 Solutions
University Physics with Modern Physics (14th Edition)
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