# Capacitor in an AC Circuit

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Physik
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Capacitor in an AC Circuit

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Consider a capacitor connected across an alternating current voltage source. Recalling Kirchhoff's loop rule, the instantaneous voltage across and the charge on the capacitor can be determined.

The rate at which charge enters or exits the capacitor is equivalent to the current flowing through the circuit, and the trigonometric relationship can be used to determine the instantaneous current.

When voltage and current are plotted together, the current through the capacitor leads the voltage across the capacitor by a quarter of a cycle.

The relationship between instantaneous current and voltage can be represented using phasor diagrams, where both phasors rotate at the same angular frequency, with the current phasor leading the voltage phasor by π by 2 radians.

The ratio of peak voltage to peak current gives the capacitive reactance of the capacitor, expressed in ohms.

The capacitive reactance of the capacitor depends inversely on the frequency of the alternating current source, where a high frequency leads to a low capacitive reactance and vice versa.

## Capacitor in an AC Circuit

A capacitor is charged by passing an electric current through it, which causes the plates to start accumulating an electrostatic charge. Since the strength of the charging current is maximum when the capacitor plates are uncharged and gradually decreases exponentially until the capacitor is fully charged, the charging process is neither instantaneous nor linear. The property of a capacitor to store a charge on its plates is called its capacitance.

Consider a purely capacitive circuit consisting of a capacitor directly connected across an AC supply voltage. The capacitor charges and discharges in response to changes in the supply voltage as they occur.  The rate of change of the voltage across the plates is directly proportional to the charging current, and this rate of change is greatest when the supply voltage switches from its positive to its negative half cycle or vice versa at specific points (0 degrees and 180 degrees) along the sinusoidal wave, as shown in Figure 1.

The maximum charging current occurs at 0 degrees, where the rate of change of the supply voltage increases in a positive direction. For a brief instant in time, there is no current flowing through the circuit as the applied voltage briefly reaches its maximum peak value at 90 degrees, because the supply voltage is neither increasing nor decreasing.

Additionally, the capacitor discharges negatively as the applied voltage starts to fall to zero due to the voltage's negative slope.  At 180 degrees, the maximum current flows because the rate of change of the voltage is again at its maximum.

Hence, for capacitors in AC circuits, the instantaneous current is at its minimum or zero whenever the applied voltage is at its maximum, and vice versa. A comparison of instantaneous voltage and current reveals that there is a phase difference of 90 degrees between them, suggesting that the current through a capacitor leads the voltage across the capacitor by 90 degrees, or a quarter of a cycle. The opposition to current flow in purely capacitive circuits is known as capacitive reactance, which is measured in Ohms and denoted by XC.