Electricity and Magnetism Charge and Conduction Coulomb's Law

Electricity and MagnetismCharge and Conduction

Coulomb’s Law

Lana Sheridan

De Anza College

Jan 9, 2018

Last time

• course structure

• introduced charge

Overview

• conductors

• insulators

• induced charge

• quantization of charge

• charge conservation

• force on interacting charges

Electric Charge

Charge is an intrinsic property of subatomic particles.

Charge can be positive or negative.

Particles can also be “chargeless”, ie. have zero net charge.

The unit for charge is the Coulomb, written with the symbol C.

Electric Charge on larger objects

Before there was any knowledge of atoms, charge was imagined asa kind of continuous fluid.

A large scale effect: In dry weather, it is easy to get a shock fromstatic electricity.

This is due to a charge imbalance.

Charge on larger objects

Most large objects around us have (approximately) zero net charge.

Objects can become charged when rubbed against one another.562 CHAPTE R 21 E LECTR IC CHARG E

HALLIDAY REVISED

These examples reveal that we have electric charge in our bodies, sweaters,carpets, doorknobs, faucets, and computers. In fact, every object contains a vastamount of electric charge. Electric charge is an intrinsic characteristic of thefundamental particles making up those objects; that is, it is a property that comesautomatically with those particles wherever they exist.

The vast amount of charge in an everyday object is usually hidden becausethe object contains equal amounts of the two kinds of charge: positive charge andnegative charge. With such an equality—or balance—of charge, the object is saidto be electrically neutral; that is, it contains no net charge. If the two types ofcharge are not in balance, then there is a net charge. We say that an object ischarged to indicate that it has a charge imbalance, or net charge.The imbalance isalways much smaller than the total amounts of positive charge and negativecharge contained in the object.

Charged objects interact by exerting forces on one another.To show this, we firstcharge a glass rod by rubbing one end with silk.At points of contact between the rodand the silk, tiny amounts of charge are transferred from one to the other, slightly up-setting the electrical neutrality of each. (We rub the silk over the rod to increase thenumber of contact points and thus the amount, still tiny,of transferred charge.)

Suppose we now suspend the charged rod from a thread to electrically isolateit from its surroundings so that its charge cannot change. If we bring a second,similarly charged, glass rod nearby (Fig. 21-2a), the two rods repel each other; thatis, each rod experiences a force directed away from the other rod. However, if werub a plastic rod with fur and then bring the rod near the suspended glass rod(Fig. 21-2b), the two rods attract each other; that is, each rod experiences a forcedirected toward the other rod.

We can understand these two demonstrations in terms of positive andnegative charges. When a glass rod is rubbed with silk, the glass loses some of itsnegative charge and then has a small unbalanced positive charge (represented bythe plus signs in Fig. 21-2a). When the plastic rod is rubbed with fur, the plasticgains a small unbalanced negative charge (represented by the minus signs inFig. 21-2b). Our two demonstrations reveal the following:

Fig. 21-2 (a) Two charged rods of thesame sign repel each other. (b) Twocharged rods of opposite signs attract eachother. Plus signs indicate a positive netcharge, and minus signs indicate a negativenet charge.

Glass

Glass

(a)

Glass

Plastic

(b)

+++++++++++++++++

++++++++++

+++

++++++++++

+++

––––––––––––––––F –F

F

–F

Fig. 21-3 A carrier bead from a photo-copying machine; the bead is covered withtoner particles that cling to it by electrosta-tic attraction.The diameter of the bead isabout 0.3 mm. (Courtesy Xerox)

Charges with the same electrical sign repel each other, and charges with oppositeelectrical signs attract each other.

In Section 21-4, we shall put this rule into quantitative form as Coulomb’s law ofelectrostatic force (or electric force) between charges. The term electrostatic isused to emphasize that, relative to each other, the charges are either stationary ormoving only very slowly.

The “positive” and “negative” labels and signs for electric charge werechosen arbitrarily by Benjamin Franklin. He could easily have interchanged thelabels or used some other pair of opposites to distinguish the two kinds of charge.(Franklin was a scientist of international reputation. It has even been said thatFranklin’s triumphs in diplomacy in France during the American War ofIndependence were facilitated, and perhaps even made possible, because he wasso highly regarded as a scientist.)

The attraction and repulsion between charged bodies have many industrial ap-plications, including electrostatic paint spraying and powder coating, fly-ash collec-tion in chimneys, nonimpact ink-jet printing, and photocopying. Figure 21-3 showsa tiny carrier bead in a photocopying machine, covered with particles of black pow-der called toner, which stick to it by means of electrostatic forces. The negativelycharged toner particles are eventually attracted from the carrier bead to a rotatingdrum, where a positively charged image of the document being copied has formed.A charged sheet of paper then attracts the toner particles from the drum to itself,after which they are heat-fused permanently in place to produce the copy.

halliday_c21_561-579v2.qxd 16-11-2009 10:54 Page 562

562 CHAPTE R 21 E LECTR IC CHARG E

HALLIDAY REVISED

These examples reveal that we have electric charge in our bodies, sweaters,carpets, doorknobs, faucets, and computers. In fact, every object contains a vastamount of electric charge. Electric charge is an intrinsic characteristic of thefundamental particles making up those objects; that is, it is a property that comesautomatically with those particles wherever they exist.

The vast amount of charge in an everyday object is usually hidden becausethe object contains equal amounts of the two kinds of charge: positive charge andnegative charge. With such an equality—or balance—of charge, the object is saidto be electrically neutral; that is, it contains no net charge. If the two types ofcharge are not in balance, then there is a net charge. We say that an object ischarged to indicate that it has a charge imbalance, or net charge.The imbalance isalways much smaller than the total amounts of positive charge and negativecharge contained in the object.

Charged objects interact by exerting forces on one another.To show this, we firstcharge a glass rod by rubbing one end with silk.At points of contact between the rodand the silk, tiny amounts of charge are transferred from one to the other, slightly up-setting the electrical neutrality of each. (We rub the silk over the rod to increase thenumber of contact points and thus the amount, still tiny,of transferred charge.)

Suppose we now suspend the charged rod from a thread to electrically isolateit from its surroundings so that its charge cannot change. If we bring a second,similarly charged, glass rod nearby (Fig. 21-2a), the two rods repel each other; thatis, each rod experiences a force directed away from the other rod. However, if werub a plastic rod with fur and then bring the rod near the suspended glass rod(Fig. 21-2b), the two rods attract each other; that is, each rod experiences a forcedirected toward the other rod.

We can understand these two demonstrations in terms of positive andnegative charges. When a glass rod is rubbed with silk, the glass loses some of itsnegative charge and then has a small unbalanced positive charge (represented bythe plus signs in Fig. 21-2a). When the plastic rod is rubbed with fur, the plasticgains a small unbalanced negative charge (represented by the minus signs inFig. 21-2b). Our two demonstrations reveal the following:

Fig. 21-2 (a) Two charged rods of thesame sign repel each other. (b) Twocharged rods of opposite signs attract eachother. Plus signs indicate a positive netcharge, and minus signs indicate a negativenet charge.

Glass

Glass

(a)

Glass

Plastic

(b)

+++++++++++++++++

++++++++++

+++

++++++++++

+++

––––––––––––––––F –F

F

–F

Fig. 21-3 A carrier bead from a photo-copying machine; the bead is covered withtoner particles that cling to it by electrosta-tic attraction.The diameter of the bead isabout 0.3 mm. (Courtesy Xerox)

Charges with the same electrical sign repel each other, and charges with oppositeelectrical signs attract each other.

In Section 21-4, we shall put this rule into quantitative form as Coulomb’s law ofelectrostatic force (or electric force) between charges. The term electrostatic isused to emphasize that, relative to each other, the charges are either stationary ormoving only very slowly.

The “positive” and “negative” labels and signs for electric charge werechosen arbitrarily by Benjamin Franklin. He could easily have interchanged thelabels or used some other pair of opposites to distinguish the two kinds of charge.(Franklin was a scientist of international reputation. It has even been said thatFranklin’s triumphs in diplomacy in France during the American War ofIndependence were facilitated, and perhaps even made possible, because he wasso highly regarded as a scientist.)

The attraction and repulsion between charged bodies have many industrial ap-plications, including electrostatic paint spraying and powder coating, fly-ash collec-tion in chimneys, nonimpact ink-jet printing, and photocopying. Figure 21-3 showsa tiny carrier bead in a photocopying machine, covered with particles of black pow-der called toner, which stick to it by means of electrostatic forces. The negativelycharged toner particles are eventually attracted from the carrier bead to a rotatingdrum, where a positively charged image of the document being copied has formed.A charged sheet of paper then attracts the toner particles from the drum to itself,after which they are heat-fused permanently in place to produce the copy.


1Diagrams from Halliday, Resnick, Walker, 9th ed.

Electrostatic force

Charged objects exert a force on one another.

Charges with the same electrical sign repel each other.

Charges with opposite electrical signs attract each other.

Some Vocabulary

electrically neutral

An object is electrically neutral if its net charge is zero.

electrically isolated

An object is electrically isolated if it cannot exchange charge withits surroundings.

Conductors and Insulators

Some materials allow charges to flow through them easily, some donot.

Conductors

materials through which charge can move readily

Insulators

(also called nonconductors) are materials that charge cannot movethrough freely

Conductors and Insulators

Some materials allow charges to flow through them easily, some donot.

Conductors

materials through which charge can move readily

Insulators

(also called nonconductors) are materials that charge cannot movethrough freely

Induced Charge Polarization

If a conductor is brought close to a charged object, positive andnegative charges in the conductor start to separate and we say acharge is induced on one side of the conductor, or the conductor’scharge is polarized.

56321-3 CON DUCTORS AN D I N S U LATORSPART 3

HALLIDAY REVISED

21-3 Conductors and InsulatorsWe can classify materials generally according to the ability of charge to movethrough them. Conductors are materials through which charge can move ratherfreely; examples include metals (such as copper in common lamp wire), thehuman body, and tap water. Nonconductors — also called insulators — are ma-terials through which charge cannot move freely; examples include rubber(such as the insulation on common lamp wire), plastic, glass, and chemicallypure water. Semiconductors are materials that are intermediate between con-ductors and insulators; examples include silicon and germanium in computerchips. Superconductors are materials that are perfect conductors, allowingcharge to move without any hindrance. In these chapters we discuss only con-ductors and insulators.

Here is an example of how conduction can eliminate excess charge on anobject. If you rub a copper rod with wool, charge is transferred from the wool tothe rod. However, if you are holding the rod while also touching a faucet, youcannot charge the rod in spite of the transfer. The reason is that you, the rod, andthe faucet are all conductors connected, via the plumbing, to Earth’s surface,which is a huge conductor. Because the excess charges put on the rod by the woolrepel one another, they move away from one another by moving first through therod, then through you, and then through the faucet and plumbing to reachEarth’s surface, where they can spread out.The process leaves the rod electricallyneutral.

In thus setting up a pathway of conductors between an object and Earth’ssurface, we are said to ground the object, and in neutralizing the object (by elimi-nating an unbalanced positive or negative charge), we are said to discharge theobject. If instead of holding the copper rod in your hand, you hold it by aninsulating handle, you eliminate the conducting path to Earth, and the rod canthen be charged by rubbing (the charge remains on the rod), as long as you donot touch it directly with your hand.

The properties of conductors and insulators are due to the structure andelectrical nature of atoms.Atoms consist of positively charged protons, negativelycharged electrons, and electrically neutral neutrons. The protons and neutrons arepacked tightly together in a central nucleus.

The charge of a single electron and that of a single proton have the samemagnitude but are opposite in sign. Hence, an electrically neutral atom containsequal numbers of electrons and protons. Electrons are held near the nucleusbecause they have the electrical sign opposite that of the protons in the nucleusand thus are attracted to the nucleus.

When atoms of a conductor like copper come together to form the solid,some of their outermost (and so most loosely held) electrons become free towander about within the solid, leaving behind positively charged atoms ( positiveions). We call the mobile electrons conduction electrons. There are few (if any)free electrons in a nonconductor.

The experiment of Fig. 21-4 demonstrates the mobility of charge in a conduc-tor. A negatively charged plastic rod will attract either end of an isolated neutral

Fig. 21-4 A neutral copper rod is electrically iso-lated from its surroundings by being suspended on anonconducting thread. Either end of the copper rodwill be attracted by a charged rod. Here, conductionelectrons in the copper rod are repelled to the far endof that rod by the negative charge on the plastic rod.Then that negative charge attracts the remaining posi-tive charge on the near end of the copper rod, rotatingthe copper rod to bring that near end closer to theplastic rod.

Neutral copper

Charged plastic

++++++ + + + +

–––––––

––––––––––

–––––– – –F –F


Induced Charge Polarization

23.2 Charging Objects by Induction 693

the side of the sphere near the rod with an effective positive charge because of the diminished number of electrons as in Figure 23.3b. (The left side of the sphere in Figure 23.3b is positively charged as if positive charges moved into this region, but remember that only electrons are free to move.) This process occurs even if the rod never actually touches the sphere. If the same experiment is performed with a conducting wire connected from the sphere to the Earth (Fig. 23.3c), some of the electrons in the conductor are so strongly repelled by the presence of the negative charge in the rod that they move out of the sphere through the wire and into the Earth. The symbol at the end of the wire in Figure 23.3c indicates that the wire is connected to ground, which means a reservoir, such as the Earth, that can accept or provide electrons freely with negligible effect on its electrical characteristics. If the wire to ground is then removed (Fig. 23.3d), the conducting sphere contains an excess of induced positive charge because it has fewer electrons than it needs to can-cel out the positive charge of the protons. When the rubber rod is removed from the vicinity of the sphere (Fig. 23.3e), this induced positive charge remains on the ungrounded sphere. Notice that the rubber rod loses none of its negative charge during this process. Charging an object by induction requires no contact with the object inducing the charge. That is in contrast to charging an object by rubbing (that is, by conduc-tion), which does require contact between the two objects. A process similar to induction in conductors takes place in insulators. In most neutral molecules, the center of positive charge coincides with the center of nega-tive charge. In the presence of a charged object, however, these centers inside each molecule in an insulator may shift slightly, resulting in more positive charge on one side of the molecule than on the other. This realignment of charge within individ-ual molecules produces a layer of charge on the surface of the insulator as shown in Figure 23.4a. The proximity of the positive charges on the surface of the object and the negative charges on the surface of the insulator results in an attractive force between the object and the insulator. Your knowledge of induction in insulators should help you explain why a charged rod attracts bits of electrically neutral paper as shown in Figure 23.4b.

Q uick Quiz 23.2 Three objects are brought close to one another, two at a time. When objects A and B are brought together, they attract. When objects B and C are brought together, they repel. Which of the following are necessarily true? (a) Objects A and C possess charges of the same sign. (b) Objects A and C pos-sess charges of opposite sign. (c) All three objects possess charges of the same sign. (d) One object is neutral. (e) Additional experiments must be performed to determine information about the charges on the objects.

Figure 23.4 (a) A charged bal-loon is brought near an insulating wall. (b) A charged rod is brought close to bits of paper.

Wall

Charged Inducedcharge

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The charged balloon induces a charge separation on the surface of the wall due to realignment of charges in the molecules of the wall.

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Question

A,B, and D are charged pieces of plastic. C is an electricallyneutral copper plate.


HALLIDAY REVISED

copper rod. What happens is that many of the conduction electrons in the closerend of the copper rod are repelled by the negative charge on the plastic rod.Some of the conduction electrons move to the far end of the copper rod, leavingthe near end depleted in electrons and thus with an unbalanced positive charge.This positive charge is attracted to the negative charge in the plastic rod.Although the copper rod is still neutral, it is said to have an induced charge, whichmeans that some of its positive and negative charges have been separated due tothe presence of a nearby charge.

Similarly, if a positively charged glass rod is brought near one end of aneutral copper rod, conduction electrons in the copper rod are attracted to thatend. That end becomes negatively charged and the other end positively charged,so again an induced charge is set up in the copper rod.Although the copper rod isstill neutral, it and the glass rod attract each other.

Note that only conduction electrons, with their negative charges, can move;positive ions are fixed in place. Thus, an object becomes positively charged onlythrough the removal of negative charges.

Blue Flashes from a Wintergreen LifeSaverIndirect evidence for the attraction of charges with opposite signs can be seenwith a wintergreen LifeSaver (the candy shaped in the form of a marine lifesaver). If you adapt your eyes to darkness for about 15 minutes and then havea friend chomp on a piece of the candy in the darkness, you will see a faint blueflash from your friend’s mouth with each chomp. Whenever a chomp breaks asugar crystal into pieces, each piece will probably end up with a different numberof electrons. Suppose a crystal breaks into pieces A and B, with A ending up withmore electrons on its surface than B (Fig. 21-5). This means that B has positiveions (atoms that lost electrons to A) on its surface. Because the electrons on Aare strongly attracted to the positive ions on B, some of those electrons jumpacross the gap between the pieces.

As A and B fall away from each other, air (primarily nitrogen, N2) flows intothe gap, and many of the jumping electrons collide with nitrogen molecules in theair, causing the molecules to emit ultraviolet light. You cannot see this type oflight. However, the wintergreen molecules on the surfaces of the candy piecesabsorb the ultraviolet light and then emit blue light, which you can see—it is theblue light coming from your friend’s mouth.

Fig. 21-5 Two pieces of a wintergreen LifeSaver candy asthey fall away from each other. Electrons jumping from thenegative surface of piece A to the positive surface of piece Bcollide with nitrogen (N2) molecules in the air.

A

B + + ++ +++

–– –

– ––

–

N2

CHECKPOINT 1

The figure shows fivepairs of plates: A, B, andD are charged plasticplates and C is an elec-trically neutral copperplate. The electrostaticforces between the pairsof plates are shown forthree of the pairs. For the remaining two pairs, do the plates repel or attract each other?

A C C D B

B A D A D


Plates C and D

(A) attract each other

(B) repel each other

1Page 564, Halliday, Resnick, Walker, 9th ed.

Question

A,B, and D are charged pieces of plastic. C is an electricallyneutral copper plate.


HALLIDAY REVISED

copper rod. What happens is that many of the conduction electrons in the closerend of the copper rod are repelled by the negative charge on the plastic rod.Some of the conduction electrons move to the far end of the copper rod, leavingthe near end depleted in electrons and thus with an unbalanced positive charge.This positive charge is attracted to the negative charge in the plastic rod.Although the copper rod is still neutral, it is said to have an induced charge, whichmeans that some of its positive and negative charges have been separated due tothe presence of a nearby charge.

Similarly, if a positively charged glass rod is brought near one end of aneutral copper rod, conduction electrons in the copper rod are attracted to thatend. That end becomes negatively charged and the other end positively charged,so again an induced charge is set up in the copper rod.Although the copper rod isstill neutral, it and the glass rod attract each other.

Note that only conduction electrons, with their negative charges, can move;positive ions are fixed in place. Thus, an object becomes positively charged onlythrough the removal of negative charges.

Blue Flashes from a Wintergreen LifeSaverIndirect evidence for the attraction of charges with opposite signs can be seenwith a wintergreen LifeSaver (the candy shaped in the form of a marine lifesaver). If you adapt your eyes to darkness for about 15 minutes and then havea friend chomp on a piece of the candy in the darkness, you will see a faint blueflash from your friend’s mouth with each chomp. Whenever a chomp breaks asugar crystal into pieces, each piece will probably end up with a different numberof electrons. Suppose a crystal breaks into pieces A and B, with A ending up withmore electrons on its surface than B (Fig. 21-5). This means that B has positiveions (atoms that lost electrons to A) on its surface. Because the electrons on Aare strongly attracted to the positive ions on B, some of those electrons jumpacross the gap between the pieces.

As A and B fall away from each other, air (primarily nitrogen, N2) flows intothe gap, and many of the jumping electrons collide with nitrogen molecules in theair, causing the molecules to emit ultraviolet light. You cannot see this type oflight. However, the wintergreen molecules on the surfaces of the candy piecesabsorb the ultraviolet light and then emit blue light, which you can see—it is theblue light coming from your friend’s mouth.

Fig. 21-5 Two pieces of a wintergreen LifeSaver candy asthey fall away from each other. Electrons jumping from thenegative surface of piece A to the positive surface of piece Bcollide with nitrogen (N2) molecules in the air.

A

B + + ++ +++

–– –

– ––

–

N2

CHECKPOINT 1

The figure shows fivepairs of plates: A, B, andD are charged plasticplates and C is an elec-trically neutral copperplate. The electrostaticforces between the pairsof plates are shown forthree of the pairs. For the remaining two pairs, do the plates repel or attract each other?

A C C D B

B A D A D


Plates B and D

(A) attract each other

(B) repel each other

1Page 564, Halliday, Resnick, Walker, 9th ed.

Charging by Induction

692 Chapter 23 Electric Fields

as negative charges (electrons). Conservation of electric charge for an isolated system is like conservation of energy, momentum, and angular momentum, but we don’t identify an analysis model for this conservation principle because it is not used often enough in the mathematical solution to problems. In 1909, Robert Millikan (1868–1953) discovered that electric charge always occurs as integral multiples of a fundamental amount of charge e (see Section 25.7). In modern terms, the electric charge q is said to be quantized, where q is the standard symbol used for charge as a variable. That is, electric charge exists as dis-crete “packets,” and we can write q 5 6Ne, where N is some integer. Other experi-ments in the same period showed that the electron has a charge 2e and the proton has a charge of equal magnitude but opposite sign 1e. Some particles, such as the neutron, have no charge.

Q uick Quiz 23.1 Three objects are brought close to each other, two at a time. When objects A and B are brought together, they repel. When objects B and C are brought together, they also repel. Which of the following are true? (a) Objects A and C possess charges of the same sign. (b) Objects A and C possess charges of opposite sign. (c) All three objects possess charges of the same sign. (d) One object is neutral. (e) Additional experiments must be performed to determine the signs of the charges.

23.2 Charging Objects by InductionIt is convenient to classify materials in terms of the ability of electrons to move through the material:

Electrical conductors are materials in which some of the electrons are free electrons1 that are not bound to atoms and can move relatively freely through the material; electrical insulators are materials in which all electrons are bound to atoms and cannot move freely through the material.

Materials such as glass, rubber, and dry wood fall into the category of electrical insulators. When such materials are charged by rubbing, only the area rubbed becomes charged and the charged particles are unable to move to other regions of the material. In contrast, materials such as copper, aluminum, and silver are good electrical conductors. When such materials are charged in some small region, the charge readily distributes itself over the entire surface of the material. Semiconductors are a third class of materials, and their electrical properties are somewhere between those of insulators and those of conductors. Silicon and ger-manium are well-known examples of semiconductors commonly used in the fabri-cation of a variety of electronic chips used in computers, cellular telephones, and home theater systems. The electrical properties of semiconductors can be changed over many orders of magnitude by the addition of controlled amounts of certain atoms to the materials. To understand how to charge a conductor by a process known as induction, con-sider a neutral (uncharged) conducting sphere insulated from the ground as shown in Figure 23.3a. There are an equal number of electrons and protons in the sphere if the charge on the sphere is exactly zero. When a negatively charged rubber rod is brought near the sphere, electrons in the region nearest the rod experience a repulsive force and migrate to the opposite side of the sphere. This migration leaves

1A metal atom contains one or more outer electrons, which are weakly bound to the nucleus. When many atoms combine to form a metal, the free electrons are these outer electrons, which are not bound to any one atom. These electrons move about the metal in a manner similar to that of gas molecules moving in a container.

Electrons redistribute when a charged rod is brought close.

The excess positive charge is nonuniformly distributed.

Some electrons leave the grounded sphere through the ground wire.

The neutral sphere has equal numbers of positive and negative charges.

The remaining electrons redistribute uniformly, and there is a net uniform distribution of positive charge on the sphere.

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Figure 23.3 Charging a metallic object by induction. (a) A neutral metallic sphere. (b) A charged rub-ber rod is placed near the sphere. (c) The sphere is grounded. (d) The ground connection is removed. (e) The rod is removed.

A conductor is initially neutrally charged.

1Figures from Serway & Jewett, 9th ed.














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Excess (negative) charge on the far side is drawn off the conductorby grounding it.















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The conductor is isolated again.















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The conductor is now (positively) charged.


Charge is Quantized

quantization

A physical quantity is said to be quantized if if can only takediscrete values.

Originally, charge was thought to be a continuous fluid, but it isnot.

Just like water has a smallest unit, the H2O molecule, charge has asmallest unit, written e, the elementary charge.

e = 1.602× 10−19 C

Any charge must be

q = ne , n ∈ Z

Charge is Quantized

quantization

A physical quantity is said to be quantized if if can only takediscrete values.

Originally, charge was thought to be a continuous fluid, but it isnot.

Just like water has a smallest unit, the H2O molecule, charge has asmallest unit, written e, the elementary charge.

e = 1.602× 10−19 C

Any charge must be

q = ne , n ∈ Z

Question

Initially, sphere A has a charge of −50e and sphere B has a chargeof 20e. The spheres are made of conducting material and areidentical in size. If the spheres then touch, what is the resultingcharge on sphere A?

(A) −50e

(B) −30e

(C) −15e

(D) 20e

Conservation of Charge

Charge can move from one body to another but the net charge ofan isolated system never changes.

This is called charge conservation.

What other quantities are conserved?

Noether’s Theorem ⇒ the corresponding












One interesting phenomenon that shows the conservation of chargeis pair production.

A gamma ray (very high energy photon) converts into an electronand a positron (anti-electron):

γ→ e− + e+

New mass is created out of light, but charge is still conserved!

Electrostatic Forces

Charged objects interact via the electrostatic force.

The force that one charge exerts on another can be attractive orrepulsive, depending on the signs of the charges.

• Charges with the same electrical sign repel each other.

• Charges with opposite electrical signs attract each other.

Charge is written with the symbol q or Q.

Electrostatic Forces

For a pair of point-particles with charges q1 and q2, the magnitudeof the force on each particle is given by Coulomb’s Law:

F1,2 =ke q1q2

r2

ke is the electrostatic constant and r is the distance between thetwo charged particles.

ke = 14πε0

= 8.99× 109 N m2/C2

How Coulomb’s Law was found: Torsion Balance694 Chapter 23 Electric Fields

23.3 Coulomb’s LawCharles Coulomb measured the magnitudes of the electric forces between charged objects using the torsion balance, which he invented (Fig. 23.5). The operating prin-ciple of the torsion balance is the same as that of the apparatus used by Cavendish to measure the density of the Earth (see Section 13.1), with the electrically neutral spheres replaced by charged ones. The electric force between charged spheres A and B in Figure 23.5 causes the spheres to either attract or repel each other, and the resulting motion causes the suspended fiber to twist. Because the restoring torque of the twisted fiber is proportional to the angle through which the fiber rotates, a measurement of this angle provides a quantitative measure of the electric force of attraction or repulsion. Once the spheres are charged by rubbing, the electric force between them is very large compared with the gravitational attraction, and so the gravitational force can be neglected. From Coulomb’s experiments, we can generalize the properties of the electric force (sometimes called the electrostatic force) between two stationary charged par-ticles. We use the term point charge to refer to a charged particle of zero size. The electrical behavior of electrons and protons is very well described by modeling them as point charges. From experimental observations, we find that the magni-tude of the electric force (sometimes called the Coulomb force) between two point charges is given by Coulomb’s law.

Fe 5 ke 0 q1 0 0 q2 0

r 2 (23.1)

where ke is a constant called the Coulomb constant. In his experiments, Coulomb was able to show that the value of the exponent of r was 2 to within an uncertainty of a few percent. Modern experiments have shown that the exponent is 2 to within an uncertainty of a few parts in 1016. Experiments also show that the electric force, like the gravitational force, is conservative. The value of the Coulomb constant depends on the choice of units. The SI unit of charge is the coulomb (C). The Coulomb constant ke in SI units has the value

ke 5 8.987 6 3 109 N ? m2/C2 (23.2)

This constant is also written in the form

ke 51

4pP0 (23.3)

where the constant P0 (Greek letter epsilon) is known as the permittivity of free space and has the value

P0 5 8.854 2 3 10212 C2/N ? m2 (23.4)

The smallest unit of free charge e known in nature,2 the charge on an electron (2e) or a proton (1e), has a magnitude

e 5 1.602 18 3 10219 C (23.5)

Therefore, 1 C of charge is approximately equal to the charge of 6.24 3 1018 elec-trons or protons. This number is very small when compared with the number of free electrons in 1 cm3 of copper, which is on the order of 1023. Nevertheless, 1 C is a substantial amount of charge. In typical experiments in which a rubber or glass rod is charged by friction, a net charge on the order of 1026 C is obtained. In other

Coulomb’s law X

Coulomb constant X

2No unit of charge smaller than e has been detected on a free particle; current theories, however, propose the exis-tence of particles called quarks having charges 2e/3 and 2e/3. Although there is considerable experimental evidence for such particles inside nuclear matter, free quarks have never been detected. We discuss other properties of quarks in Chapter 46.

Charles CoulombFrench physicist (1736–1806)Coulomb’s major contributions to sci-ence were in the areas of electrostatics and magnetism. During his lifetime, he also investigated the strengths of materials, thereby contributing to the field of structural mechanics. In ergonomics, his research provided an understanding of the ways in which people and animals can best do work.

© IN

TERF

OTO/

Alam

y

Fiber

B

A

Suspensionhead

Figure 23.5 Coulomb’s balance, used to establish the inverse-square law for the electric force.

1Figure from Serway & Jewett, Physics for Scientists and Engineers, 9th ed.

Electrostatic Forces: Coulomb’s Law

F1,2 =ke q1q2

r2

Remember however, forces are vectors. The vector version of thelaw is:

F1→2 =ke q1q2

r2r1→2

where F1→2 is the force that particle 1 exerts on particle 2, andr1→2 is a unit vector pointing from particle 1 to particle 2.

Coulomb’s Law

Coulomb’s Law:

F1→2 =k q1q2r2

r1→2

Does this look a bit familiar?

Similar to this?

F1→2 = −G m1m2

r2r1→2

Coulomb’s Law

Coulomb’s Law:

F1→2 =k q1q2r2

r1→2

Does this look a bit familiar?

Similar to this?

F1→2 = −G m1m2

r2r1→2

Coulomb’s Law

F1→2 =k q1q2r2

r1→2696 Chapter 23 Electric Fields

same sign as in Figure 23.6a, the product q1q2 is positive and the electric force on one particle is directed away from the other particle. If q1 and q2 are of opposite sign as shown in Figure 23.6b, the product q1q2 is negative and the electric force on one par-ticle is directed toward the other particle. These signs describe the relative direction of the force but not the absolute direction. A negative product indicates an attractive force, and a positive product indicates a repulsive force. The absolute direction of the force on a charge depends on the location of the other charge. For example, if an x axis lies along the two charges in Figure 23.6a, the product q1q2 is positive, but F

S12

points in the positive x direction and FS

21 points in the negative x direction. When more than two charges are present, the force between any pair of them is given by Equation 23.6. Therefore, the resultant force on any one of them equals the vector sum of the forces exerted by the other individual charges. For example, if four charges are present, the resultant force exerted by particles 2, 3, and 4 on particle 1 is

FS

1 5 FS

21 1 FS

31 1 FS

41

Q uick Quiz 23.3 Object A has a charge of 12 mC, and object B has a charge of 16 mC. Which statement is true about the electric forces on the objects? (a) F

SAB 5 23 F

SBA (b) F

SAB 5 2 F

SBA (c) 3 F

SAB 5 2 F

SBA (d) F

SAB 5 3 F

SBA

(e) FS

AB 5 FS

BA (f) 3 FS

AB 5 FS

BA

Example 23.2 Find the Resultant Force

Consider three point charges located at the corners of a right triangle as shown in Figure 23.7, where q1 5 q3 5 5.00 mC, q2 5 22.00 mC, and a 5 0.100 m. Find the resultant force exerted on q3.

Conceptualize Think about the net force on q3. Because charge q3 is near two other charges, it will experience two electric forces. These forces are exerted in dif-ferent directions as shown in Figure 23.7. Based on the forces shown in the figure, estimate the direction of the net force vector.

Categorize Because two forces are exerted on charge q3, we categorize this exam-ple as a vector addition problem.

Analyze The directions of the individual forces exerted by q1 and q2 on q3 are shown in Figure 23.7. The force F

S23 exerted by q2 on q3 is attractive because q2

and q3 have opposite signs. In the coordinate system shown in Figure 23.7, the attractive force F

S23 is to the left (in the negative x direction).

The force FS

13 exerted by q1 on q3 is repulsive because both charges are positive. The repulsive force FS

13 makes an angle of 45.08 with the x axis.

S O L U T I O N

Figure 23.6 Two point charges separated by a distance r exert a force on each other that is given by Coulomb’s law. The force F

S21

exerted by q2 on q1 is equal in mag-nitude and opposite in direction to the force F

S12 exerted by q1 on q2.

r

q1

q2

r12ˆ

When the charges are of the same sign, the force is repulsive.

a b

F12S

F21S

!

!

q1

q2

When the charges are of opposite signs, the force is attractive.

F12S

F21S

!

"

!

!

"

F13S

F23S

q3

q1

q2

a

a

y

x

2a!

Figure 23.7 (Example 23.2) The force exerted by q1 on q3 is F

S13. The

force exerted by q2 on q3 is FS

23. The resultant force F

S3 exerted on q3

is the vector sum FS

13 1 FS

23.

1Figure from Serway & Jewett, Physics for Scientists and Engineers, 9th ed.

Electrostatic Constant

The electrostatic constant is:

k =1

4πε0= 8.99× 109 N m2 C−2

ε0 is called the permittivity constant or the electricalpermittivity of free space.

ε0 = 8.85× 10−12 C2 N−1 m−2

Example

The diagram a shows two positively charged particles fixed in placeon the x axis.The charges are q1 = 1.60× 10−19 C andq2 = 3.20× 10−19 C, and the particle separation is R = 0.0200 m.What are the magnitude and direction of the electrostatic forceF1→2 from particle 1 on particle 2?

56721-4 COU LOM B’S LAWPART 3

HALLIDAY REVISED

Sample Problem

(b) Figure 21-8c is identical to Fig. 21-8a except that particle3 now lies on the x axis between particles 1 and 2. Particle 3has charge q3 ! "3.20 # 10"19 C and is at a distance fromparticle 1. What is the net electrostatic force on particle1 due to particles 2 and 3?

The presence of particle 3 does not alter the electrostaticforce on particle 1 from particle 2. Thus, force still acts onF

:12

F:

1,net

34 R

Finding the net force due to two other particles

(a) Figure 21-8a shows two positively charged particles fixed inplace on an x axis.The charges are q1 ! 1.60 # 10"19 C and q2 !3.20 # 10"19 C, and the particle separation is R ! 0.0200 m.What are the magnitude and direction of the electrostatic force

on particle 1 from particle 2?

Because both particles are positively charged, particle 1 is re-pelled by particle 2, with a force magnitude given by Eq. 21-4.Thus, the direction of force on particle 1 is away from parti-cle 2, in the negative direction of the x axis, as indicated in thefree-body diagram of Fig. 21-8b.

Two particles: Using Eq. 21-4 with separation R substitutedfor r, we can write the magnitude F12 of this force as

Thus, force has the following magnitude and direction(relative to the positive direction of the x axis):

1.15 # 10"24 N and 180°. (Answer)

We can also write in unit-vector notation as

. (Answer)F:

12 ! "(1.15 # 10 "24 N)i

F:

12

F:

12

! 1.15 # 10 "24 N.

#(1.60 # 10 "19 C)(3.20 # 10 "19 C)

(0.0200 m)2

! (8.99 # 10 9 N $m2/C2)

F12 !1

4%&0

!q1!!q2!R2

F:

12

F:

12

KEY I DEAS KEY I DEA

Rx

q2q1

(a)

x(b)

F12

R3__4

xq2q3q1

(c)

x(d)

F12 F13

This is the firstarrangement.

This is the secondarrangement.

This is the thirdarrangement.

This is the particleof interest.

This is still theparticle of interest.

It is pushed awayfrom particle 2.


It is pulled towardparticle 3.


It is pulled towardparticle 4.

This is still theparticle of interest.

x

y

q2q1

q4

3__4 R

(e)

( f )

θ

x

y

θF12

F14

Fig. 21-8 (a)Two charged parti-cles of charges q1

and q2 are fixed inplace on an x axis.(b) The free-bodydiagram for particle1, showing the elec-trostatic force on itfrom particle 2. (c)Particle 3 included.(d) Free-body dia-gram for particle 1.(e) Particle 4included. (f ) Free-body diagram forparticle 1.

particle 1. Similarly, the force that acts on particle 1 dueto particle 3 is not affected by the presence of particle 2.Because particles 1 and 3 have charge of opposite signs,particle 1 is attracted to particle 3. Thus, force is di-rected toward particle 3, as indicated in the free-body dia-gram of Fig. 21-8d.

Three particles: To find the magnitude of , we canrewrite Eq. 21-4 as

We can also write in unit-vector notation:

F:

13 ! (2.05 # 10 "24 N)i .

F:

13

! 2.05 # 10 "24 N.

#(1.60 # 10 "19 C)(3.20 # 10 "19 C)

(34)

2(0.0200 m)2

! (8.99 # 10 9 N $m2/C2)

F13 !1

4%&0

!q1!!q3!

(34R)2

F:

13

F:

13

F:

13

A


k = 8.99× 109 N m2 C−2 or ε0 = 8.85× 10−12 C2 N−1 m−2

1Question from Halliday, Resnick, Walker, 9th ed

Summary

• charge

• conductors and insulators

• induced charge

• quantization of charge

• charge conservation

• Coulomb force

Homework• Get the textbook: Physics for Scientists and Engineers, 9th

Edition, Serway & Jewett

• Read Ch 23.

• Ch 23, onward from page 716. Objective Qs: 7, 9; ConceptualQs: 1, 5; Probs: 1, 3, 9, 16, 17

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