An ideal capacitor of capacitance (no leakage

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Electromagnetic induction and alternating-current circuits · Electromagnetic induction and alternating-current circuits

Electricity, Magnetism and Electromagnetism, Ch AC circuits: phasors, complex impedance, resonance and power

Problem

An ideal capacitor of capacitance

C

(no leakage, no series resistance) carries a charge

q = CV

on its plates, so the current into it is

I = C\,dV/dt

. The voltage across it is held at

V(t) = V_0\cos\omega t .

((a)) By differentiating, show that

I(t) = \omega C V_0\cos(\omega t + 90^\circ)

, and hence that the ratio of the voltage amplitude to the current amplitude is

X_C = 1/\omega C

while the current leads the voltage by a quarter cycle. Show the same result in the complex representation, where

d/dt

becomes multiplication by

i\omega

. ((b)) Evaluate

X_C

for

C = 0.47\,\mu\mathrm{F}

f = 1.2\,\mathrm{kHz}

. ((c)) With

V_0 = 5.0\,\mathrm{V}

at that frequency, find the current amplitude. ((d)) What is the instantaneous current at an instant when the voltage is at a crest?

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