Question

The voltage across a 100-μF capacitor takes the followingvalues. Calculate the expression for the current through thecapacitor in each case.

A. vc(t) = 40 cos (20t -π/2) V
B. vc(t) = 20 sin 100t V
C. vc(t) = -60 sin (80t + π/6) V
D. vc(t) = 30 cos (100t + π/4) V

Answer 1

The current through a capacitor in an AC circuit is calculated by differentiating the voltage function across the capacitor with respect to time, and multiplying the result by the capacitor's capacitance. The current leads the voltage by a quarter cycle or a phase difference of π/2.

Step-by-step explanation:

The question concerns the calculation of the current through a capacitor in an AC circuit for various voltage functions. Current i(t) through a capacitor can be found using the derivative of the charge Q, which itself is given by Cv(t) where C is capacitance and v(t) is the voltage across the capacitor at any time t. Thus, by differentiating the voltage function v(t) and multiplying by the capacitor's capacitance C, we can find the instantaneous current i(t). We can express it as i(t) = C dv(t)/dt, where dv(t)/dt is the derivative of the voltage function with respect to time.

For example, given v(t) = V0 sin(ωt), the current i(t) would be:

i(t) = CωV0 cos(ωt), where ω is the angular frequency. Noting that cos(ωt) leads sin(ωt) by π/2 or a quarter of a cycle, this reflects that in an AC circuit, the current through the capacitor leads the voltage across it by π/2 or 90 degrees phase difference.