__Capacitance__

__Capacitance__

In This Chapter:

✔ *Capacitance*

✔ *Parallel-Plate Capacitor*

✔ *Capacitors in Combination*

✔ *Energy of a Charged Capacitor*

✔ *Charging a Capacitor*

✔ *Discharging a Capacitor*

__Capacitance__

A *capacitor *is a system that stores energy in the form of an electric ﬁeld. In its simplest form, a capacitor consists of a pair of parallel metal plates separated by air or other insulating material.

The potential difference *V *between the plates of

a capacitor is directly proportional to the charge *Q *on either of them, so the ratio *Q*/*V *is always the same for a particular capacitor. This ratio is called the *capacitance C *of the capacitor:

The unit of capacitance is the *farad *(F), where 1 farad = 1 coulomb/ volt. Since the farad is too large for practical purposes, the *microfarad** *and *picofarad *are commonly used, whereA charge of 10−6 C on each plate of 1-*m*F capacitor will produce a potential difference of *V *= *Q*/*C *= 1 V between the plates.

###### Parallel-Plate Capacitor

A capacitor that consists of parallel plates each of area *A *separated by the distance *d *has a capacitance ofThe constant ε0 is the permittivity of free space; its value isThe quantity *K *is the *dielectric constant *of the material between the capacitor plates; the greater *K *is, the more effective the material is in diminishing an electric ﬁeld.

__Note!__

__Note!__

###### For free space, *K *= 1; for air, *K *= 1.0006; a typical value for glass is *K *= 6; and for water, *K *= 80.

__Capacitors in Combination__

The *equivalent capacitance *of a set of connected capacitors is the capacitance of the single capacitor that can replace the set without changing the

Figure 14-1

properties of any circuit it is part of. The equivalent capacitance of a set of capacitors joined in series (Figure 14-1) is

**Solved Problem 14**.**1 **Find the equivalent capacitance of three capaci- tors whose capacitances are 1, 2, and 3 mF that are connected in: (*a*) se- ries and (b) parallel.

Solution.

(a) n series, the equivalent capacitance can be found by:

(b) n parallel, the equivalent capacitance can be found by:

__Energy of a Charged Capacitor__

To produce the electric ﬁeld in a charged capacitor, work must be done to separate the positive and negative charges. This work is stored as electric potential energy in the capacitor. The potential energy *W *of a capacitor of capacitance *C *whose charge is *Q *and whose potential difference is *V *given by

*Charging a Capacitor*

When a capacitor is being charged in a circuit such as that of Figure 14-3, at any moment the voltage *Q*/*C *across it is in the opposite direction to the battery voltage *V *and thus tends to oppose the ﬂow of additional charge. For this reason, a capacitor does not acquire its ﬁnal charge the instant it is connected to a battery or other source of emf. The net potential difference when the charge on the capacitor is *Q *is *V *− (*Q*/*C*), and the current is then

As *Q *increases, its rate of increase *I *= ∆*Q*/∆*t *decreases. Figure 14-4 shows how *Q*, measured in percent of ﬁnal charge, varies with time when a capacitor is being charged; the switch of Figure 14-3 is closed at *t *= 0.

The product *RC *of the resistance *R *in the circuit and the capacitance *C *governs the rate at which the capacitor reaches its ultimate charge of *Q*0 = *CV*. The product *RC *is called the *time constant T *of the circuit. Af-

ter a time equal to *T*, the charge on the capacitor is 63 percent of its ﬁnal value.

The formula that governs the growth of charge in the circuit of Fig- ure 14-3 is

where *Q*o is the ﬁnal charge *CV *and *T *is the time constant *RC*. Figure 14-4 is a graph of that formula. It is easy to see why *Q *reaches 63 percent of *Q*o in time *T*. When *t *= *T*, *t*/*T *= 1 and

__Discharging a Capacitor__

__Discharging a Capacitor__

When a charged capacitor is discharged through a resistance, as in Fig- ure 14-5, the decrease in charge is governed by the formula

where again *T *= *RC *is the time constant. The charge will fall to 37 per- cent of its original value after time *T *(Figure 14-6). The smaller the time constant *T*, the more rapidly a capacitor can be charged or discharged.

Figure 14-4

**Solved Problem 14**.**2 **A 20-mF capacitor is connected to a 45-V battery through a circuit whose resistance is 2000 W. (*a*) What is the ﬁnal charge on the capacitor? (*b*) How long does it take for the charge to reach 63 per- cent of its ﬁnal value?

**Figure 14-5**

**Figure 14-6**