everyone is familiar with electricity as a source of power

8/3/2019 Everyone is Familiar With Electricity as a Source of Power

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Everyone is familiar with electricity as a source of power. Pressing a switch will turn on a light or heat an

oven. Energy is continuously being produced in these processes, energy that is carried by an electric

current through the metal wires connected to the electricity supply. The electric current is made up of a

ow of moving electrons. We cannot see the movement because the electrons are very small and are

able to move through a metal without disturbing the structure of the metal. Electrons are pushed along

a wire by forces that act on them because they carry electric charge. These forces are called electric

forces. The electric force on a charged particle is the same whether the charge is stationary or moving.

There are additional forces that act only on moving charges, which are called magnetic forces. Moving

charges and magnetic forces are discussed in Chapter 16. The subject of this chapter is electrostatics,

which is the study of the electric forces acting on stationary charges, and of how these forces are

modied in the presence of matter. Like gravitational forces, electrostatic forces act at a distancethere

is an electrostatic force between two charged particles even if they are separated by a vacuum. The

magnitude of the electrostatic force also has the same inverse square variation with distance as the

gravitational force. There are, however, two very important differences between gravitational and

electrostatic forces. The rst is that the gravitational force between two masses is always attractive,

whereas charges may attract or repel one another. The other difference is that, on an atomic scale,

electrostatic forces are enormously strong compared with gravitational forces. In the discussion of the

internal motion of individual atoms and molecules, which is the subject of Chapter 11, only electrostatic

forces were considered and the effects of gravitational forces were completely neglected. This seems

paradoxical, because in everyday life we are well aware of gravitational force, but do not often notice

electrical forces. The reason is that electrostatic attractions and repulsions tend to cancel out, whereas

gravitational forces are always attractive and, in particular, the whole of the Earth attracts everything on

its surface. Although the laws governing the forces between charges are introduced in this chapter in

the context of electrostatics, these laws always apply, even when magnetic effects or electromagnetic

waves are present. The chapter starts by discussing the forces between very small idealized electric

charges in order to explain the concepts of electric eld and potential. Later on, in Sections 15.5 and

15.6, we are concerned with objects containing very large numbers of atoms. Only average electrical

properties are then of interest: it will be shown how these averages can be obtained without having to

consider the electrical forces within each atom in turn.

542 15: ELECTROSTATICS15.1 Forces between charged particles

Positive and negative charges

It was mentioned above that electrostatic forces are sometimes attractive and sometimes repulsive.

This is because there are two different kinds of charge, which are called positive and negative. Just like

positive and negative numbers, positive and negative charges are described as being of opposite sign.

Electrons carry a negative charge. Like numbers, positive and negative charges may cancel one another

out. For example, as is described more fully in Chapter 9, an atom consists of a number of electrons

bound to a positively charged nucleus. The charge on the nucleus has exactly the same magnitude as the

charge of all the electrons in the atom. Since the nuclear charge is of opposite sign to the charges on the

electrons, the net charge carried by the atom, which is the algebraic sum of all its charges, is zero. The

atom is said to be electrically neutral. In SI units, charge is measured in coulombs (symbol C). The


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coulomb is dened with reference to the force between wires carrying electric current: this is discussed

in Chapter 16. The charge carried by a single electron is written as e, and its magnitude is e 1:602 10

19

coulombs

to four signicant gures. Because the coulomb is a very large unit,

charges are often measured in microcoulombs (symbol mC:

1mC 106

C).

Ordinary matter, made up of electrons and nuclei, may be electrically

neutral or may have a charge that is e times an integer. Other particles

besides electrons and atomic nuclei are found in cosmic rays, or may be

created in high-energy collisions in accelerators. All these particles also

have charges that are zero or e times an integer. Within a nucleus there

are thought to be particles called quarks that carry an amount of charge

that is a fraction of e. However, quarks have a property called

connement, which means that they are never observed singly but go

around in packets that do not have fractional charge. It is thus a universal

rule that any object is either electrically neutral or has a positive or

negative charge with magnitude that is an integral multiple of e.

Coulomb's law

Coulomb's law tells us the strength and direction of the forces acting

between two charges. This is the simplest possible case, but on the basis of

Coulomb's law it is possible to work out the electric force on any

distribution of charges. Consider two charges that we shall label q1 and q2:

q1 and q2 are numbers representing the magnitudes of the charges in

z All observable charges are


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multiples of the electronic

charge

FORCES BETWEEN CHARGED PARTICLES 543coulombs, and these numbers are positive or negative

depending on

whether the charges are positive or negative. We shall idealize the

problem by supposing that q1 and q2 occupy such a small volume that

they may be treated as point charges with no spatial extent at all.

Let us dene F12 to be the magnitude of the force exerted by a charge q1

on a charge q2. According to Coulomb's law, F12 depends on the product

q1q2; doubling the magnitude of either charge doubles the strength of the

force. How do the forces between q1 and q2 vary with distance? Just like

the gravitational force between two masses, the electrostatic force between

two charges varies as the inverse square of their distance apart. This

inverse square law is known to hold with great precision, not from direct

measurement of the force between charges, but from the observation of

other phenomena that are deduced from the inverse square law. The

magnitude of the force between the charges q1 and q2 is expressed

mathematically as

F12 /

q1q2

r

2

12

: 15:1

A constant of proportionality is required to give the strength of the force

in newtons when q1 and q2 are in coulombs and r12 in metres. In the


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SI system the constant is written as 1=4p0including the factor 4p

simplies other equations in electricity and magnetism. The equation for

the magnitude F12 of the force becomes

F12

q1q2

4p0r

2

12

: 15:2

The direction of the force between the two charges depends on their

signs. The force acts on the line joining the charges and, if the charges

have opposite sign, like the negatively charged electron and the positively

charged nucleus in the hydrogen atom, the force is attractive: it is the

electrical attraction that binds an electron to the nucleus and ensures that

the atom is stable. On the other hand, if both charges are positive, or both

are negative, the force between them is repulsive.

We must be careful to get the direction of the force correct when

setting up the nal equation for Coulomb's law. To do this we must use

the vector notation, and in particular we shall use unit vectors, which are

vectors pointing in any direction, but which always have unit length.

Since we are interested in the direction between the two charges, we

introduce the vector ^r12, which is a vector of unit length pointing from an

origin at the centre of charge q1 towards charge q2.

Unit vectors are used in Section 20.2 (in the mathematical review at the

end of the book) to dene the directions of the axes of a Cartesian


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coordinate system. In that context, the unit vectors i, j, and k (all of unit

544 15: ELECTROSTATICSlength) are pointing in the xed directions chosen for the x-, y-, and z-

axes. Any vector a that has components ax , ay

, and az

along the x-, y-,

and z-axes can then be written as ax i ay j az k. This expression

species both the magnitude and direction of the vector a. Here it is more

convenient to use a different notation, allowing unit vectors to point in

any direction. The symbol ^is used to indicate that a vector has unit length:

thus ^a is a vector of unit length pointing in the same direction as a.

The force exerted by charge q1 on charge q2 is denoted by the vector

F12. This force points along the line joining the charges, in the direction

away from q1 for a repulsive force (q1 and q2 having the same sign) and

towards q1 for an attractive force (q1 and q2 having different signs).

Different possibilities are illustrated in Fig 15.1, which shows both F12 and

^r12 for different signs of the charges. In Fig 15.1(c), where the signs are

different, the force is towards q1, which is in the opposite direction to ^r12.

However, the unit vector ^r12 is also in the opposite direction to ^r12. The

sign required in front of the unit vector ^r12 is thus the same as the sign of

the product q1q2.

The force between two charges q1 and q2 may now be expressed in

mathematical terms, using the same notation as in Fig 15.1 for the

position vector of q2 with respect to q1. The magnitude of the force is

given by eqn (15.2) and in SI units Coulomb's law is

F12


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q1q2

40r

2

12

^r12: 15:3

Similarly, the force F21 exerted by q2 on q1 is

F21

q1q2

4p0r

2

12

^r21: 15:4

Since ^r21 is a unit vector pointing from q2 towards q1, in the opposite

direction to ^r12, the forces exerted on the two charges according to

eqns (15.3) and (15.4) are equal and opposite, as they should be (compare

Figs 15.1(a) and (d)).

In words, Coulomb's law states that

The force between two charges acts along the line between the

charges, and is proportional to the product of the charges and to the

inverse square of the distance between them. The force is repulsive for

charges of the same sign and attractive for charges of opposite sign.

The dimensions of all the quantities in eqn (15.3) are dened

independently of Coulomb's law. The unit of force, the newton, is dened

by Newton's second law. The unit of charge, the coulomb, is dened with

reference to the magnetic force between wires carrying current. To satisfy


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eqn (15.3), the units of the constant 0 are C

2

N1

m2

. Its value is related

Fig. 15.1 The force on a charge due

to the presence of another charge is

in the same direction as the unit

vector on the line joining the

charges. (a), (b), and (c) show the

force on charge q2 caused by q1 for

different combinations of the sign of

the charges. (d) shows the force on

q1 caused by q2 when both are

positive.

FORCES BETWEEN CHARGED PARTICLES 545to the speed of light, which is the distance light travels in a

vacuum in one

second. Since the the unit of length is itself dened in terms of the time

taken for light to travel a distance of one metre, the speed of light is also

dened to be a particular number of metres per second. Scientists all over

the world have agreed that the value of the speed of light is exactly

2:997 924 58 m s1

. Because the constant 0, which is called the

permittivity of free space, is derived from the speed of light, it is also

in principle known exactly. However, it is not a rational number, but

when expressed in decimals it can be calculated to any number of places.


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To four signicant gures, its value is

0 8:854 10

12

C

2

N

1

m

2

: 15:5

Worked Example 15.1 Two small particles of carbon, each weighing 1 mg

and each carrying a charge of 106

C, are one centimetre apart. Calculate

the electrostatic force between them.

Answer The force between the particles is found directly by substitution

in eqn (15.2). It is

106

106

4p0 104

108

1:1 1010

90 N:

This example illustrates the enormous strength of electrostatic forces.

The force of 90 N is nearly equal to the weight of a 10 kg mass, acting


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between two tiny particles. If the particles were free to move, their initial

acceleration would be 90 km s2

. In practice it is not possible to

accumulate as much charge as 1 mC on such small pieces of matter, even

though only a small fraction of the atoms need to gain or lose an electron

to reach this value. The number of atoms in one mole of carbon is

Avogadro's number, NA 6 10

23

, and, since the mass number of

carbon is 12, the number of atoms in 1 mg is about 5 10

19

. The number

of electronic charges in 1 mC is 106

=e 10

13

=1:6. The fraction of carbon

atoms that must lose one electron to charge the particles with 1mC is

10

13

=5 10

19

1:6, or a little more than one in a million.

15.2 The electric eld

Coulomb's law in the form given in eqn (15.3) enables us to work out the

forces that two point charges exert on each other. Most practical electrical


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problems involve not just two charged particles, but vast numbers of

them. This section introduces the idea of the electric eld, which describes

the force on a charged particle due to all the other charges in its

neighbourhood

everyone is familiar with electricity as a source of power

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