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Capacitor

Now let’s consider a capacitor    connected across an ac voltage source. From Kirchhoff’s loop rule, the instantaneous voltage across the capacitor of [link] (a) is

v C ( t ) = V 0 sin ω t .

Recall that the charge in a capacitor is given by Q = C V . This is true at any time measured in the ac cycle of voltage. Consequently, the instantaneous charge on the capacitor is

q ( t ) = C v C ( t ) = C V 0 sin ω t .

Since the current in the circuit is the rate at which charge enters (or leaves) the capacitor,

i C ( t ) = d q ( t ) d t = ω C V 0 cos ω t = I 0 cos ω t ,

where I 0 = ω C V 0 is the current amplitude. Using the trigonometric relationship cos ω t = sin ( ω t + π / 2 ) , we may express the instantaneous current as

i C ( t ) = I 0 sin ( ω t + π 2 ) .

Dividing V 0 by I 0 , we obtain an equation that looks similar to Ohm’s law:

V 0 I 0 = 1 ω C = X C .

The quantity X C is analogous to resistance in a dc circuit in the sense that both quantities are a ratio of a voltage to a current. As a result, they have the same unit, the ohm. Keep in mind, however, that a capacitor stores and discharges electric energy, whereas a resistor dissipates it. The quantity X C is known as the capacitive reactance    of the capacitor, or the opposition of a capacitor to a change in current. It depends inversely on the frequency of the ac source—high frequency leads to low capacitive reactance.

Figure a shows a circuit with an AC voltage source connected to a capacitor. The source is labeled V0 sine omega t. Figure b shows sine waves of AC voltage and current on the same graph. Voltage has a greater amplitude than current and its maximum value is marked V0 on the y axis. The maximum value of current is marked I0. The two curves have the same wavelength but are out of phase by one quarter wavelength. The voltage curve is labeled V subscript C parentheses t parentheses equal to V0 sine omega t. The current curve is labeled I subscript C parentheses t parentheses equal to I0 sine parentheses omega t plus pi by 2 parentheses.
(a) A capacitor connected across an ac generator. (b) The current i C ( t ) through the capacitor and the voltage v C ( t ) across the capacitor. Notice that i C ( t ) leads v C ( t ) by π / 2 rad.

A comparison of the expressions for v C ( t ) and i C ( t ) shows that there is a phase difference of π / 2 rad between them. When these two quantities are plotted together, the current peaks a quarter cycle (or π / 2 rad ) ahead of the voltage, as illustrated in [link] (b). The current through a capacitor leads the voltage across a capacitor by π / 2 rad , or a quarter of a cycle.

The corresponding phasor diagram is shown in [link] . Here, the relationship between i C ( t ) and v C ( t ) is represented by having their phasors rotate at the same angular frequency, with the current phasor leading by π / 2 rad .

Figure shows the coordinate axes. An arrow labeled V0 starts from the origin and points up and right making an angle omega t with the x axis. An arrow labeled omega is shown near its tip, perpendicular to it, pointing up and left. The tip of the arrow V0 makes a y-intercept labeled V subscript C parentheses t parentheses. An arrow labeled I0 starts at the origin and points up and left. It is perpendicular to V0. It makes a y intercept labeled i subscript C parentheses t parentheses. A arrow labeled omega is shown near its tip, perpendicular to it, pointing down and left.
The phasor diagram for the capacitor of [link] . The current phasor leads the voltage phasor by π / 2 rad as they both rotate with the same angular frequency.

To this point, we have exclusively been using peak values of the current or voltage in our discussion, namely, I 0 and V 0 . However, if we average out the values of current or voltage, these values are zero. Therefore, we often use a second convention called the root mean square value, or rms value, in discussions of current and voltage. The rms operates in reverse of the terminology. First, you square the function, next, you take the mean, and then, you find the square root. As a result, the rms values of current and voltage are not zero. Appliances and devices are commonly quoted with rms values for their operations, rather than peak values. We indicate rms values with a subscript attached to a capital letter (such as I rms ).

Although a capacitor is basically an open circuit, an rms current    , or the root mean square of the current, appears in a circuit with an ac voltage applied to a capacitor. Consider that

Practice Key Terms 4

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Source:  OpenStax, University physics volume 2. OpenStax CNX. Oct 06, 2016 Download for free at http://cnx.org/content/col12074/1.3
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