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*Chapter Three Capacitance and Dielectrics 2020-03-24آ Capacitors mostly consist of two conductors...*

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Chapter Three

Capacitance and Dielectrics

You will learn:

The nature of capacitors, and their ability to store charges

Capacitors connection in a networks

Calculate the amount of energy stored in capacitors

Dielectric material

1

3.1 Capacitors and Capacitance:

• A capacitor is a device which stores electric charges. Capacitors mostly consist

of two conductors with an insulating medium placed between them (figure 3.1).

• The insulating medium can be either vacuum or an insulator material.

figure 3.1 Capacitor structure

• Capacitors vary in shape and size, but the basic configuration is two conductors

carrying equal but opposite charges.

• There are some common types of capacitors, parallel plate capacitors, cylindrical

capacitors, and concentric capacitors. In this chapter, parallel plate capacitors will

be discussed.

2

Continue

• In circuit diagrams, a capacitor is represented by either of below symbols:

Figure 3.2 Capacitor symbols (straight or curved)

• In most practical applications, each conductor initially has zero net charge (Q) and electrons

are transferred from the voltage source to the conductors; this is called charging the

capacitor. Then the two conductors have charges with equal magnitude and opposite sign,

and the net charge on the capacitor as a whole remains zero.

• A potential difference Δ𝑉 is created with the positively charged conductor at a higher potential than the negatively charged conductor.

• The amount of charge Q stored in a capacitor is linearly proportional to ΔV.

• To charge a capacitor, you need a power supply source or a battery.

• Once the charges Q and – Q are established on the conductors, the battery is disconnected.

This gives a fixed potential difference between the conductors (that is, the potential of the

positively charged conductor with respect to the negatively charged conductor ) that is just

equal to the voltage of the battery. 3

Capacitance (C):

• Each capacitor has capacitance. So, capacitance is a measure of the capacity of storing electric charges for a given potential difference. The SI unit of capacitance is the farad.

• 1 Farad (F) = 1 coulomb/volt

• Farad is a large quantity, A typical capacitance is in the picofarad (pF) to millifarad range (mF), microfarad (μF) or nanofarad (nF).

• Mathematically, we can say

• The electric field at any point in the region between the conductors is proportional to the magnitude Q of charge on each conductor. It follows that the potential difference Δ𝑉 (or written Vab) between the conductors is also proportional to Q.

• Note: Don’t confuse the symbol for capacitance C with the abbreviation of C for coulombs

4

Continue

• If we double the magnitude of charge on each conductor, the charge density at each point doubles, the electric field at each point doubles, and the potential difference between conductors doubles; however, the ratio of charge to potential difference does not change.

• The capacitance of the parallel plate capacitor:

Depends on the medium between the two plates.

Is directly proportional to the area of the plated.

Is inversely proportional to the plate separation.

• Mathematically,

Where:

A is the area of plates,

d is the distance between the two plates and

𝜀0 is the permittivity of a vacuum (free space).

Figure 3.3 A charged parallel-plate capacitor. 5

Some graphs

A commercial capacitor is labelled with the value of its capacitance. For these capacitors, C=2200, 1000 and 470 μF.

An assortment of commercially available capacitors.

6

Example 1

Solution (a)

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3.2 Connection of capacitors in Series and Parallel

3.2.1 Capacitors in series:

• Suppose two initially uncharged capacitors are connected in series, as shown in

Figure 3.4. A potential difference is then applied across both capacitors. The left

plate of capacitor 1 is connected to the positive terminal of the battery and becomes

positively charged with a charge +Q, while the right plate of capacitor 2 is

connected to the negative terminal and becomes negatively charged with charge –Q

as electrons flow in.

• What about the inner plates? They were initially uncharged; now the outside plates

each attract an equal and opposite charge. So the right plate of capacitor 1 will

acquire a charge –Q and the left plate of capacitor +Q.

Figure 3.4 Capacitors in series and an equivalent capacitor 8

Continue The potential differences across capacitors C1 and C2 are

and

The total potential difference is simply the sum of the two individual potential differences:

THUS

The generalization to any number of capacitors connected in series is

Note: The term of (total) is used instead of (equivalent) sometimes to show a total values of something.

9

3.2.2 Capacitors in Parallel: • Suppose we have two capacitors C1 with charge Q1 and C2 with charge Q2

which are connected in parallel, as shown in Figure 3.5. The left plates of

both capacitors C1 and C2 are connected to the positive terminal of the

battery and have the same electric potential as the positive terminal.

Similarly, both right plates are negatively charged and have the same

potential as the negative terminal.

Figure 3.5 A parallel connection of two capacitors.

(b) The equivalent single capacitor

10

Continue These two capacitors can be replaced by a single equivalent capacitor Ceq with a

total charge Q supplied by the battery. However, since Q is shared by the two

capacitors, we must have

And then

thus

The generalization to any number of capacitors is

11

3.2.2 Mixed connection • Capacitors can also be connected in both series and parallel simultaneously.

• To solve mixed connected capacitors mathematically, first the capacitors

connected in one type must be solved and then the rest of the connection

can be sorted out easily.

Figure 3.6 Connecting capacitors in mix

(a) (b) (c)

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3.3 Energy stored in a charged capacitor • Many of the most important applications of capacitors depend on their

ability to store energy.

• The electric potential energy stored in a charged capacitor is equal to

the amount of work required to charge the capacitor —that is, to

separate opposite charges and place them on different conductors. Or,

• electric potential energy stored in a charged capacitor is also equal to

the total work done by the electric field on the charge when the

capacitor discharged.

• When the capacitor is discharged, this stored energy is recovered as a work

done by electrical forces.

13

Continue • The total work needed to increase the capacitor’s charge from zero to Q is

• Consider a parallel plate capacitor that is initially uncharged, so that the

initial potential difference is zero. After charging it, the final potential

difference across the capacitor reaches to ∆V. The average potential

difference during the charging process is

• Thus

(work to charge a capacitor)

Then W is equal to the potential energy (U) of the charged capacitor. So we can

express U (which is equal to W) as below,

U is measured with joules 14

(c). Q1 + Q2 = Q0 Q1 = C1V and Q2 = C2V

Example 2

,Fig. below

15

3.4 Dielectrics

• Most capacitors have a non-conducting material, or a dielectric, between the two conducting plates.

• A common type of capacitor uses long strips of metal foil for the plates, separated by strips of plastic sheet.

• A sandwich of these materials is rolled up, forming a unit that can provide a capacitance of several microfarads in a compact package.

Figure 3.7 A common type of capacitor uses dielectric sheets to separate the conductors.

16

Continue

• Placing a solid dielectric between the plates of a capacitor serves three

functions:

First, it solves the mechanical problem of maintaining two large metal

sheets at a very small separation without actual contact.

Second, using a dielectric increases the maximum possible potential

difference between the capacitor plates.

Third, the capacitance of a capacitor of given dimensions is greater when

there is a dielectric material between the plates than when there is vacuum.

• Suppose a capacitor has a capacitance when there is no material between the plates. When a dielectric material is inserted to completely fill the space between the plates, the capacitance increases to

is called the dielectric constant

We shall show that electric field.

is a m