## Chapter 12

## Q. 12.16

The voltage across a capacitor having capacitance C=100 pF is given by \nu(t)=V_0 \cos (2 \pi f t) with \nu_0=5 V and f=500 kHz . Find the current i(t) through the capacitor and draw a phasor diagram depicting the current and voltage, with the voltage as the phase reference.

## Step-by-Step

## Verified Solution

The phasor \tilde{V} for the voltage and the admittance Y of the capacitor are

\begin{aligned} &\tilde{V}=V_0 \angle 0=5 \angle 0 V, \\ &Y=j \omega C=j 314 \mu S =314 \mu S \angle(\pi / 2). \end{aligned}

From (12.24), the phasor for the current is

\begin{aligned} \tilde{I} &=Y \tilde{V}=(5 V \angle 0)[314 \mu S \angle(\pi / 2)] \\ &=1.57 mA \angle(\pi / 2). \end{aligned}

It follows that

i(t)=1.57 \cos (2 \pi f t+\pi / 2) mA .

Figure 12.10 shows a phasor diagram for the current and voltage, where the voltage is taken as the reference. The current leads the voltage by \pi / 2 . Equivalently, the voltage lags the current by \pi / 2.