Physics · Ch 7 — Alternating Current

AC Voltage Applied to a Pure Capacitor

7.6

AC Voltage Applied to a Pure Capacitor

Setting up the circuit. A pure capacitor $C$ is connected directly across an AC source supplying $v=v_0\sin\omega t$ .

The capacitor's defining relation. At every instant, the charge on the capacitor's plates is $q=Cv$ , since the full source voltage appears directly across it. The instantaneous current is the rate at which this charge is being delivered, $i=dq/dt$ :

i = \frac{dq}{dt} = C\frac{dv}{dt} = C\frac{d}{dt}(v_0\sin\omega t) = C v_0\omega\cos\omega t

Rewriting in sine form. Using the identity $\cos\theta=\sin(\theta+\pi/2)$ ,

i = \omega C v_0\sin\!\left(\omega t+\frac{\pi}{2}\right) = i_0\sin\!\left(\omega t+\frac{\pi}{2}\right), \qquad \text{where } i_0=\omega C v_0

Reading off the phase. Comparing with $v=v_0\sin\omega t$ , the current LEADS the voltage by exactly $\pi/2$ radians ( $90^\circ$ ) -- the current reaches its own peak a full quarter-cycle BEFORE the voltage does. Physically, the current must flow FIRST, to build up charge on the plates, before the voltage across the capacitor (which depends directly on how much charge has already accumulated) can rise to match it -- the current is always a step ahead of the voltage it is in the process of creating. …

Figure 1Phasor diagram and waveforms for a pure capacitive AC circuit

What this figure shows. The figure has two panels. The LEFT panel is a phasor diagram with two arrows from a common origin: the voltage phasor $V_C$ drawn pointing straight up (reference angle), and the current phasor $I_C$ drawn $90^\circ$ ANTICLOCKWISE (i.e. AHEAD OF/leading) from $V_C$ , with a small curved arc between the two arrows labelled $\pi/2$ , and a curved arrow showing both phasors rotating together anticlockwise at speed $\omega$ while always keeping the current $90^\circ$ ahead. The RIGHT panel is a waveform graph with $\omega t$ on the horizontal axis showing two sine curves, $v$ and $i$ , with the CURRENT curve shifted to the LEFT of the voltage curve by a quarter cycle (the current reaches its peak a quarter-cycle BEFORE the voltage reaches its peak), visually showing the current …

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