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https://texercises.com/exercise/capacitive-reactance/
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Exercise:
Show that the reactance and phase shift for a capacitor are given by X_C fracomega C quad textand quad tanDeltaphi -fracpi

Solution:
For a capacitor directly connected to a sinusoidal ac voltage it follows using the loop rule that vt V_ cosomega t fracqtC where pm qt is the charge on the capacitor plates at time t. Taking the derivative with respect to time on both sides of the yields dotvt -V_ omega sinomega t fraqtC fracitC where we have used that itdotqt. Solving this for it yields it -V_ omega C sinomega t labeleq The general expression for the ac current is it I_ cosomega t-Delta phi labeleq By comparing s refeq and refeq we find that I_ V_ omega C labeleq and -sinomega t cosomega t-Deltaphi labeleq Using refeq we find the expression for the reactance: X_C fracV_I_ fracomega C Using the trigonometric identity -sinomega t cosomega t + pi/ and refeq we find that Deltaphi -fracpi
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Exercise:
Show that the reactance and phase shift for a capacitor are given by X_C fracomega C quad textand quad tanDeltaphi -fracpi

Solution:
For a capacitor directly connected to a sinusoidal ac voltage it follows using the loop rule that vt V_ cosomega t fracqtC where pm qt is the charge on the capacitor plates at time t. Taking the derivative with respect to time on both sides of the yields dotvt -V_ omega sinomega t fraqtC fracitC where we have used that itdotqt. Solving this for it yields it -V_ omega C sinomega t labeleq The general expression for the ac current is it I_ cosomega t-Delta phi labeleq By comparing s refeq and refeq we find that I_ V_ omega C labeleq and -sinomega t cosomega t-Deltaphi labeleq Using refeq we find the expression for the reactance: X_C fracV_I_ fracomega C Using the trigonometric identity -sinomega t cosomega t + pi/ and refeq we find that Deltaphi -fracpi
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Branches
Electrodynamics
Tags
ac circuit, capacitor, reactance
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Difficulty
(3, default)
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0 (default)
Language
ENG (English)
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