Voltage Applied to a Capacitor:
Let a source of alternating emf be connected across a capacitor. The instantaneous value of alternating emf is given by
When a d.c. source is connected across a capacitor the current flows across it for a short time interval till the capacitor is charged. As the charges start accumulating on the capacitor plates a potential difference across them also increases which opposes the current. This phenomenon continues till the potential difference across the plates becomes equal to the potential difference across the terminals of the battery. As this happens the charging of the capacitor stops and the current in the circuit falls to zero.
When capacitor is connected to a source of alternating emf as shown in figure, the capacitor gets charged for first half cycle in one direction and then gets discharged. It again gets charged for the second half cycle but now in the opposite direction and again gets discharged and so on. Thus, a capacitor connected across an a.c. source limits the current but not completely stop the flow of charge. Thus, there is a continuous alternating current in the circuit. If q is charge on the capacitor at any instant t.
Let I be the instantaneous current flowing through the capacitor, then
Comparing equations (5) and (6), it is clear that in a capacitive a.c. circuit, current leads the voltage by nl 2. This is shown in figure and the corresponding phasor diagram is shown in figure.
As is clear from the figure that the current phasor I is π / 2 ahead of voltage phasor V. When these phasors are rotated counter clockwise, they generate the curves given in figure. Comparing equation Im = \(\frac{V_{m}}{1 / \omega C}\) with Ohm’s law, we find out that the quantity \(\frac{1}{\omega C}\) plays the same role in this circuit as is done by the resistance in the purely resistive circuit thus, it is the effective reactance offered by the capacitor C and is called capacitive reactance. It is denoted by XC .
Thus, XC = \(\frac{1}{\omega C}=\frac{1}{2 \pi f C}\) ……………… (7)
where f is the frequency of the a.c. supply. As is clear from equation (7) that unlike inductive reactance, capacitive reactance decreases with an increase in the frequency of a.c. It is also measured in ohms (Ω).
In a d.c. circuit f = 0, thus XC = ∞. Thus, a capacitor offers infinite resistance to d.c. current.
As charge q = CV
∴ Power in circuit P = VI
Voltage, current, charge and power are plotted in figure as shown below.
