Monday, October 27, 2014

Capacitance : Capacitance, Parallel-Plate Capacitor, Capacitors in Combination, Energy of a Charged Capacitor, Charging a Capacitor and Discharging a Capacitor

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:

Capacitance equations 6-01-53 PM

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, whereCapacitance equations 6-02-21 PMA charge of 10−6 C on each plate of 1-mF 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 ofCapacitance equations 6-02-32 PMThe constant ε0 is the permittivity of free space; its value isCapacitance equations 6-02-42 PMThe 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!
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

Capacitance ]_Page_096_Image_0001

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) isCapacitance equations 6-03-05 PM

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:Capacitance equations 6-03-21 PMCapacitance ]_Page_096_Image_0002

Capacitance equations 6-03-36 PM

(b) n parallel, the equivalent capacitance can be found by:Capacitance equations 6-03-49 PM

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 byCapacitance equations 6-04-03 PM

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 thenCapacitance equations 6-04-14 PM

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 Q0 = CV. The product RC is called the time constant T of the circuit. Af-

Capacitance ]_Page_098_Image_0001

Figure 14-3

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 isCapacitance equations 6-04-29 PM

where Qo 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 Qo in time T. When t = T, t/T = 1 andCapacitance equations 6-04-39 PM

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

Capacitance equations 6-04-51 PM

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.

Capacitance ]_Page_099_Image_0002

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?

Solution.Capacitance equations 6-05-08 PM Capacitance ]_Page_099_Image_0001

Figure 14-5

Capacitance ]_Page_100_Image_0001

Figure 14-6