Topic 1 Electric Charges and Forces

8/13/2019 Topic 1 Electric Charges and Forces

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INTRODUCTION

Do you know that lightning is an electrical phenomenon? The very mild electricshock that you experience upon touching a metallic door-knob after walkingacross a carpet room on a dry day is another example of this. In each of thesecases, sparks are created, even though the effect is momentary. In modernindustries, certain impurities (either solid particles or liquid droplets) are removedthrough electrostatic precipitation.These are all static electricity phenomena that canonly be explained by understanding the physics of electrostatics. There is a thusan overlap between the world of static electricity and the everyday world that welive.

Electrostatic forces are central to our existence. For example, the human body ismade up of atoms. Each atom, consisting of negative and positive charges, is heldtogether by these forces. Without electrostatic forces, life would be impossible.

In this chapter we will begin our study by examining the nature of electriccharges, which are carried by electrons and protons. Since electric charges arequantised, they obey the conservation principle. We then discuss interactionsbetween charges that are at rest, called electrostatic interactions. Such interactionsare responsible for holding atoms and molecules together in your body. Finally,

TTooppiicc11ElectricChargesandForces

LEARNING OUTCOMES

By the end of this topic, you should be able to:

1. Describe the basic properties of electric charges;

2. Explain that charging is the separation, not the creation, of charges anddistinguish the difference between conductors and insulators;

3. Describe the nature of electrostatic forces between charges; and

4. Solve electrostatic problems using Coulombs law.


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TOPIC 1 ELECTRIC CHARGES AND FORCES2

we will study Coulomb's law, which is the basic law governing the interactionbetween electric charges.

ELECTRIC CHARGEThere are a few simple experiments that you can try at home to demonstrate thenature of electrostatic charges.

For example, you will notice that when a glass rod is rubbed with a silk cloth, it isable to attract tiny bits of paper. A similar effect is also seen when a plastic combis run through dry hair and brought near tiny pieces of paper. In each of theseexamples, we say that the rod has become "electrified" or electrically charged.

Today we, know there exists only 2 kinds of electric charge; a positive charge (+)and a negative charge (-). How do these charges interact with one another?Experiments demonstrate that

Unlike charges attract, i.e. a positive charge and a negative charge attracteach other.

Like charge repel i.e. two positive charges or two negative charges and repeleach other.

This is the nature of electric charges.

Figure 1.1: The nature of charges

1.1.1 The structure of the atom

All objects are composed of atoms. The structure of an atom consists of a nucleus at the centre and a vast region of space outside the nucleus. The nucleus iscomposed of protons and neutrons. A proton has a positive charge, and theelectron a negative charge. The neutron carries no charge.

1.1


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The magnitude of the negative charge on the silk is equal to the magnitude of thepositive charge on the glass rod. See Figure 1.3.

This suggests that rubbing does not create new charges - it merely transfers themfrom one object to another. Thus, charge can neither be created nor destroyed.

Figure 1.3: Rubbing with silk produces positive charges in the glass rod. Note that an

equal amount of negative charge is produced in silk. The fact of the matter is that electric

charges are conserved, and can neither be created nor destroyed. They can only be

transferred from one object to another.

This important observation brings us to a very fundamental law in physics: thelaw of conservation of electric charge. This law states that the net charge of anisolated system remains constant.

An electroscope is an instrument used to detect the presence ofelectrostatic charges. Design a simple electroscope using everydaymaterials. You do not have to physically build the electroscope, but yourdesign should theoretically work based on the electrostatic principles youhave just learned.

ACTIVITY 1.1


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TOPIC 1 ELECTRIC CHARGES AND FORCES 5

THE UNIT OF CHARGE

The SI standard unit of charges is the Coulomb, which is represented by the

symbol C. The charge of an individual electron is --19

1.610 C. The magnitude ofthe charge of an individual electron is known as the elementary charge. Onecoulomb is the total charge of 186.2510 electrons. The charge that produces alarge lightning bolt is about 10 C.

1.2.1 Charge Quantisation

Experiments have shown that if an object is charged, its charge is always amultiple of the elementary charge, e. This implies that an object can have a chargeof e, 2e, 3e, 4e , and so on. But it can never have a value like 1.4e,

3.4e, 7.8e etc.

We can express this restriction mathematically as:

Q = ne (1.1)

where n = 1, 2, 3, ..and e = 191 60 10. C, is the elementary charge.

Equation 1.1 represents the quantised nature of electric charge. By quantised, weare just saying that any charge Qis just an integer multiple of e.

Example 1.1

How many electrons must you remove from an electrically-neutral 10 sen coin togive it a charge of + 1.6 -310 C?

Solution

In order to give the neutral 10 sen coin a charge of +1.6mC, we have to remove an

equal amount of negative charge ie 1.6 -310 C. We already know that the

charge of a single electron is. So the number of electrons that we have to remove

from the 10 sen coin is obtained by dividing 1.6 -310 C with the charge of a

single electron:

316

19

-1.610 C=110 electrons

1.6 10 C

1.2


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CONDUCTORS AND INSULATORS

Can you distinguish between conductors and insulators? In general, glass,ceramic, dry wood, most plastics, and dry air are all good insulators. Insulatorsdo not allow electric charge to move easily through them. In order words,virtually all the electrons in insulators are tightly bound. Consequently, charge

does not move easily through an insulator.

Materials such as metals can conduct electricity and they are good conductors. Inconductors, the outermost electrons (or the valence electron) in atoms are looselybound. Thus, these electrons can be easily removed from their parent atoms andbecome free electronsthat move freely within a conductor.

Another class of materials is called semiconductors. Their ability to conductelectricity is somewhere between those of insulators and conductors Theelectronic components in your hand-phones and computers are made from

semiconductor materials. Two well-known semiconductors are germanium andsilicon.

1.3

How many electrons are needed to give a charge of 2C?

EXERCISE 1.2

The usefulness of a conductor and its opposite, the insulator, is quiteobvious. How about a semi-conductor? If a semiconductor has propertiesthat are between that of a conductor and an insulator, then it is neither agood conductor nor a good insulator. Yet it is widely used in modernelectronics. How can a semiconductor be useful, considering its half-hearted properties? Explain.

SELF-CHECK 1.2


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CHARGING A CONDUCTOR

The charge distribution of a conductor can be changed by the presence of external

charges during a charging process. A metal may be charged in two ways:(a) By direct contact, and

(b) By induction.

(a) Charging by direct contactA metal sphere that is placed on an insulating stand is initially neutral. Whathappens when we touch the sphere with a positively charged plastic rod?Electrons are transferred from the sphere to the rod, leaving the sphere witha positive charge. Finally when the rod is removed, positive the chargespreads evenly over the metal sphere and remains there because theinsulating stand prevents its flow to the ground. See Figure 1.3.

Figure 1.3: Charging a metal sphere with a positively charged rod

(b) Charging by Induction:We are also able to charge a conductor through induction. This process isshown in Figure 1.4.

If we bring a negatively charged plastic rod near a neutral metal sphere thatis on an insulating stand, the charges on the sphere will be separated. Thepositive charges will be induced on the surface closest to the rod. At the

same time, the part furthest away from the rod will become negativelycharged.

As illustrated in Figure 1.4, when the metal is connected to the ground by awire, the free electrons on the sphere flow to the ground. When theconnection with the ground is removed followed by the rod, the sphere willbecome positively charged.

1.4


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Figure 1.4: Charging by induction

ELECTROSTATIC FORCES

Remember we have mentioned the word electrostatics earlier. Electrostaticsmeans that the electric charges that we are dealing with are at rest. The interactionbetween two charges is known as the electrostatic force. Lets recap the nature ofcharges:

Like charges,i.e. two positive charges (+Qand +Q) or two negative charges(Qand Q) always repel each other.

Unlike charges, i.e. a positive charge (+Q) and a negative charge (Q) alwaysattracteach other.

The properties of the electrostatic forces are shown in Figure 1.5.

(a) (b)

Figure 1.5: Properties of electrostatic forces (a) like charges repel (b) unlike charges

attract

Note that forces are vector quantities and require vector analysis (you havelearned this in the SBPH 2103 course). You will also recall that a force vector isrepresented by the symbol F, which has magnitude F and direction . Thehorizontal and vertical components of vector Fare given as Fcos and Fsin respectively.

1.5

FF F

Q Q + Q Q


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COULOMB'S LAW

Electrostatic forces are often referred to as Coulomb forces. In 1785, Charles

Augustin de Coulomb (1736-1806) found that the magnitude of the force (F)between two electrically point charges is directly proportional to the product ofthe charges and inversely proportional to the square of the distance (r

2) between

them. We can write these relationships as follows:

1 2F Q Q (1.2)

2

1F

r (1.3)

or1 2

2Q QF = k

r (1.4)

Equation (1.4) is called Coulombs Law.

kis a constant that depends on the medium in which the charges are situated. (N).

For charges that are placed in free-space or vacuum,0

1

4k=

, where 0 is

called the permittivity of vacuum.

Therefore, Coulombs law can also be written as

1 2

2

0

1

4

Q QF

r=

(1.5)

The value of 0 is12 2 2

8 85 10. C / N m , and so k= 9 2 29 0 10. Nm / C .

If the charges are placed in a medium other than vacuum, we must replace 0 inEquation (1.5) with , where is called the permittivity of that medium. Notethe relation:

0 = (1.6)

1.6


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In Equation (1.6), is a number that depends on the medium and 1.Forexample, the values of for air, benzene, acetone and water are 1.005, 2.3, 27and 80 respectively.

Example 1.2

What is the type and magnitude of the Coulomb force between two point charges2 C and + 5C having a distance of 0.03 m apart in vacuum?

SolutionSince they are unlike charges, the force is an attractive one.

The magnitude force can be calculated according to equation (1.5). Thus

-6 -6

1 2 9

2 2

0

Q Q1 (2.010 )(5.010 )F = = (9.010 ) = 100 N.

4 r (0.03 )

Example 1.3

Two identical negative charges repel each other with a force of magnitude 9 N.The distance between the charges is 1 cm. What are the values on each charge?

Solution

F= 9 N. k= 9 2 2910 Nm / C , r= 1 cm =0.01 m

From Coulombs Law: 1 22

kq qF =

r

But since 1 2q = q ,2

2

kqF =

r

Therefore the value on each charge is2

7

9

9 0.013.16 10 C

9 10

= =

2Frq =

k


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THE SUPERPOSITION PRINCIPLE

Coulombs law gives us the force which two charges,Q1,and Q2 exert on eachother when there are no other charges present around them. Suppose a third

charge, Q3 is now introduced. How would you then find the total electrostaticforce acting on Q1 due toQ2 andQ3?

In order to work the force acting on Q1, we need to apply the Superpositionprinciple. According to the superposition principle, the resultant force on Q3 isgiven by:

F=F12+F13 (1.7)

whereF12is the force on q1due to the presence of charge q2andF13is the force

on q1due to charge q3.

While solving a problem, it is useful to resolve the individual force vectors intotheir components to find the resultant force.

The following examples will help illustrate this technique.

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1. Two positive charges of 6.0 C are 3 cm apart. What is themagnitude of the force that exists between the charges? What is thenature of this force?

2. A negative charge of 2 C and a positive charge of + 10 C areseparated by 0.03 m. What is the magnitude of the force betweenthe two charges?

3. Two charges 1.5 m apart in air, each experiences a force of 2.0 N.

(a) Find the force between them if the separation between them is

increased to 2.0 m.(b) Find the force between them if the relative permittivity of the

medium separating them is 5.

EXERCISE 1.3


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Example 1.4:

The figure below shows three small charges, Q1= 2 C, Q2=5 C, Q3=3 C,

located along the positive x-axis. What is the (i) direction, and (ii) magnitude ofthe resultant force exerted by these two charges on Q2?

Q1 Q2 Q3

Solution

Q1 Q2 Q3

(i) The force, 12F

exerted by Q1 on Q2 is attractive because these two charges

have opposite signs. Similarly, the force 32F exerted by Q3 on Q2 is

repulsive because both these two charges have the same sign.

(ii) The magnitudesof these forces are given by Coulombs Law:6 6

91 212 2 2

12

(2.0 10 )(5.0 10 )(9.0 10 ) 100 N,

(0.03)

Q QF k

r

= = =

6 693 2

32 2 2

32

(3.0 10 )(5.0 10 )(9.0 10 ) 150 N,

(0.03)

Q QF k

r

= = =

Note that both these forces act towards the left

The magnitude of the resultant force is thus 150+100=50 N. This resultant forcepoints in the x direction.

32F

12F

0.03m 0.03m


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Example 1.5

An equilateral triangle has sides of 0.03 m. Consider three charges that are placed

at the corners of this triangle as shown below in Fig. 1.6(a), where 1 3 CQ = ,2 1 CQ = and 3 4 CQ = . Calculate the magnitude and direction of the resultant

electric force acting on Q2.

Figure 1.6 (a) Figure 1.6 (b)

Solution

The force, 12F

exerted by Q1 on Q2 is attractive because these two charges have

opposite signs.

Similarly, the force 32F

exerted by Q3 on Q2 is also attractive because these two

charges have opposite signs. Let F

be the resultant force. See Figure 1.6 (b)

The magnitudesof these forces are given by Coulombs Law:

6 691 2

12 2 2

12

(3.0 10 )(1.0 10 )(9.0 10 ) 30 N

(0.03)

Q QF k

r

= = =

6 6

93 232 2 2

12

(4.0 10 )(1.0 10 )(9.0 10 ) 40 N(0.03)

Q QF k r

= = =

Notice that the electrostatic force is a vector quantity with 2 components, whichlie along the x and y axes, respectively:

Force x-component y-component

12F

12F cos60 = 15N 12F sin60 = 25.98 N

32F

40N 0

Q3Q2

Q1

12F

32F

F

0.03 m0.03 m

0.03 m


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We can now add up the two forces vectorially to find the x and y components of

the resultant force, F

:

x =15+ 40 = 55N

= 25.98 + 0 26Ny

F

F

The magnitude of the resultant force is2 2 255 26 3701 60.8x yF F F N= + = + = =

and it makes an angle of with the x-xis that is given by

-1 -1tan tan 25.3( ) =y o

x

F 26= = =

F 55

There are two kinds of electric charges, positive and negative. Electrons arethe negatives and protons are the positives.

Electric charges are conserved; they cannot be created or destroyed.

Transferring electrons to or from an object is the way to charge the object. Anobject is charged negatively by adding electrons to it or it can be chargedpositively by removing electrons from it.

Charges added to one part of an insulator remains there but charges added to aconductor very quickly spreads over the body.

1. Two point charges 2 C and 5 C are located at 2 cm and 3 cmrespectively from the origin on the positive x-axis. What is the totalforce exerted by these two charges on a third charge of 3 mC at theorigin? What is the direction?

2. Two point charges 4 C and +4 C are located at coordinates (0,3 cm) and (0, 3 cm) respectively. A third point charge +3 C islocated at (4cm, 0). Find the magnitude and direction of the totalforce on the third charge

EXERCISE 1.3


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A neutral body has an equal number of positive and negative charges. Acharged body of either sign can produce a separation of charge in a neutralbody.

There is an electric force between two charged bodies. Like charges repel andunlike charges attract.

Coulomb's law states that the force between two charged objects isproportional to the product of the charges

1 2Q Q and inversely proportional to

the square of the distance 2r between the centres of the objects

ConductorCoulombs Law

Electric Charges

ElectronsSuperposition Principle

1. (a) How many electrons are contained in 2C of charge ?

(b) What is the total mass of these electrons?

2. Two positive charges, +Q and +10Q exert a repulsive force F on each other.If the distance between them is tripled, what is the new repulsive force interms of F?

1. The figure below shows three small positive charges, Q1= 2 C, Q2=5 C,Q3=3 C, located along the positive x-axis. What is the magnitude anddirection of the resultant force exerted by these two charges on Q3?

Q1 Q2 Q3

2. Two small positively charged objects experience a repulsive force ofmagnitude 2 N when they are 0.2 m apart. The sum of the charges on theobjects is 2 C. Find the charge on each object.

0.03m 0.03m

Topic 1 Electric Charges and Forces

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