ELECTRIC CHARGES AND FIELDS

Transcription

1 Chapte One ELECTRIC CHARGES AND FIELDS 1.1 INTRODUCTION All of us have the expeience of seeing a spak o heaing a cackle when we take off ou synthetic clothes o sweate, paticulaly in dy weathe. This is almost inevitable with ladies gaments like a polyeste saee. Have you eve tied to find any explanation fo this phenomenon? Anothe common example of electic dischage is the lightning that we see in the sky duing thundestoms. We also expeience a sensation of an electic shock eithe while opening the doo of a ca o holding the ion ba of a bus afte sliding fom ou seat. The eason fo these expeiences is dischage of electic chages though ou body, which wee accumulated due to ubbing of insulating sufaces. You might have also head that this is due to geneation of static electicity. This is pecisely the topic we ae going to discuss in this and the next chapte. Static means anything that does not move o change with time. Electostatics deals with the study of foces, fields and potentials aising fom static chages. 1. ELECTRIC CHARGE Histoically the cedit of discovey of the fact that ambe ubbed with wool o silk cloth attacts light objects goes to Thales of Miletus, Geece, aound 6 BC. The name electicity is coined fom the Geek wod elekton meaning ambe. Many such pais of mateials wee known which

2 Physics on ubbing could attact light objects like staw, pith balls and bits of papes. You can pefom the following activity at home to expeience such an effect. Cut out long thin stips of white pape and lightly ion them. Take them nea a TV sceen o compute monito. You will see that the stips get attacted to the sceen. In fact they emain stuck to the sceen fo a while. It was obseved that if two glass ods ubbed with wool o silk cloth ae bought close to each othe, they epel each othe [Fig. 1.1(a)]. The two stands of wool o two pieces of silk cloth, with which the ods wee ubbed, also epel each othe. Howeve, the glass od and wool attacted each othe. Similaly, two plastic ods ubbed with cat s fu epelled each othe [Fig. 1.1(b)] but attacted the fu. On the othe hand, the plastic od attacts the glass od [Fig. 1.1(c)] and epel the silk o wool with which the glass od is ubbed. The glass od epels the fu. If a plastic od ubbed with fu is made to touch two small pith balls (now-a-days we can use polystyene balls) suspended by silk o nylon thead, then the balls epel each othe [Fig. 1.1(d)] and ae also epelled by the od. A simila effect is found if the pith balls ae touched with a glass od ubbed with silk [Fig. 1.1(e)]. A damatic obsevation is that a pith ball touched with glass od attacts anothe pith ball touched with plastic od [Fig. 1.1(f )]. These seemingly simple facts wee established fom yeas of effots and caeful expeiments and thei analyses. It was concluded, afte many caeful studies by diffeent scientists, that thee wee only two kinds of an entity which is called the electic chage. We say that the bodies like glass o plastic ods, silk, fu and pith balls ae electified. They acquie an electic chage on ubbing. The expeiments on pith balls suggested that thee ae two kinds of electification and we find that (i) like chages epel and (ii) unlike chages attact each othe. The expeiments also demonstated that the chages ae tansfeed fom the ods to the pith balls on contact. It is said that the pith balls ae electified o ae chaged by contact. The popety which diffeentiates the two kinds of chages is called the polaity of chage. When a glass od is ubbed with silk, the od acquies one kind of chage and the silk acquies the second kind of chage. This is tue fo any pai of objects that ae ubbed to be electified. Now if the electified glass od is bought in contact with silk, with which it was ubbed, they no longe attact each othe. They also do not attact o epel othe light objects as they did on being electified. Thus, the chages acquied afte ubbing ae lost when the chaged bodies ae bought in contact. What can you conclude fom these obsevations? It just tells us that unlike chages acquied by the objects FIGURE 1.1 Rods and pith balls: like chages epel and unlike chages attact each othe. Inteactive animation on simple electostatic expeiments:

3 neutalise o nullify each othe s effect. Theefoe the chages wee named as positive and negative by the Ameican scientist Benjamin Fanklin. We know that when we add a positive numbe to a negative numbe of the same magnitude, the sum is zeo. This might have been the philosophy in naming the chages as positive and negative. By convention, the chage on glass od o cat s fu is called positive and that on plastic od o silk is temed negative. If an object possesses an electic chage, it is said to be electified o chaged. When it has no chage it is said to be neutal. Electic Chages and Fields UNIFICATION OF ELECTRICITY AND MAGNETISM In olden days, electicity and magnetism wee teated as sepaate subjects. Electicity dealt with chages on glass ods, cat s fu, batteies, lightning, etc., while magnetism descibed inteactions of magnets, ion filings, compass needles, etc. In 18 Danish scientist Oested found that a compass needle is deflected by passing an electic cuent though a wie placed nea the needle. Ampee and Faaday suppoted this obsevation by saying that electic chages in motion poduce magnetic fields and moving magnets geneate electicity. The unification was achieved when the Scottish physicist Maxwell and the Dutch physicist Loentz put fowad a theoy whee they showed the intedependence of these two subjects. This field is called electomagnetism. Most of the phenomena occuing aound us can be descibed unde electomagnetism. Vitually evey foce that we can think of like fiction, chemical foce between atoms holding the matte togethe, and even the foces descibing pocesses occuing in cells of living oganisms, have its oigin in electomagnetic foce. Electomagnetic foce is one of the fundamental foces of natue. Maxwell put foth fou equations that play the same ole in classical electomagnetism as Newton s equations of motion and gavitation law play in mechanics. He also agued that light is electomagnetic in natue and its speed can be found by making puely electic and magnetic measuements. He claimed that the science of optics is intimately elated to that of electicity and magnetism. The science of electicity and magnetism is the foundation fo the moden technological civilisation. Electic powe, telecommunication, adio and television, and a wide vaiety of the pactical appliances used in daily life ae based on the pinciples of this science. Although chaged paticles in motion exet both electic and magnetic foces, in the fame of efeence whee all the chages ae at est, the foces ae puely electical. You know that gavitational foce is a long-ange foce. Its effect is felt even when the distance between the inteacting paticles is vey lage because the foce deceases invesely as the squae of the distance between the inteacting bodies. We will lean in this chapte that electic foce is also as pevasive and is in fact stonge than the gavitational foce by seveal odes of magnitude (efe to Chapte 1 of Class XI Physics Textbook). A simple appaatus to detect chage on a body is the gold-leaf electoscope [Fig. 1.(a)]. It consists of a vetical metal od housed in a box, with two thin gold leaves attached to its bottom end. When a chaged object touches the metal knob at the top of the od, chage flows on to the leaves and they divege. The degee of divegance is an indicato of the amount of chage. 3

4 4 Physics FIGURE 1. Electoscopes: (a) The gold leaf electoscope, (b) Schematics of a simple electoscope. FIGURE 1.3 Pape stip expeiment. Students can make a simple electoscope as follows [Fig. 1.(b)]: Take a thin aluminium cutain od with ball ends fitted fo hanging the cutain. Cut out a piece of length about cm with the ball at one end and flatten the cut end. Take a lage bottle that can hold this od and a cok which will fit in the opening of the bottle. Make a hole in the cok sufficient to hold the cutain od snugly. Slide the od though the hole in the cok with the cut end on the lowe side and ball end pojecting above the cok. Fold a small, thin aluminium foil (about 6 cm in length) in the middle and attach it to the flattened end of the od by cellulose tape. This foms the leaves of you electoscope. Fit the cok in the bottle with about 5 cm of the ball end pojecting above the cok. A pape scale may be put inside the bottle in advance to measue the sepaation of leaves. The sepaation is a ough measue of the amount of chage on the electoscope. To undestand how the electoscope woks, use the white pape stips we used fo seeing the attaction of chaged bodies. Fold the stips into half so that you make a mak of fold. Open the stip and ion it lightly with the mountain fold up, as shown in Fig Hold the stip by pinching it at the fold. You would notice that the two halves move apat. This shows that the stip has acquied chage on ioning. When you fold it into half, both the halves have the same chage. Hence they epel each othe. The same effect is seen in the leaf electoscope. On chaging the cutain od by touching the ball end with an electified body, chage is tansfeed to the cutain od and the attached aluminium foil. Both the halves of the foil get simila chage and theefoe epel each othe. The divegence in the leaves depends on the amount of chage on them. Let us fist ty to undestand why mateial bodies acquie chage. You know that all matte is made up of atoms and/o molecules. Although nomally the mateials ae electically neutal, they do contain chages; but thei chages ae exactly balanced. Foces that hold the molecules togethe, foces that hold atoms togethe in a solid, the adhesive foce of glue, foces associated with suface tension, all ae basically electical in natue, aising fom the foces between chaged paticles. Thus the electic foce is all pevasive and it encompasses almost each and evey field associated with ou life. It is theefoe essential that we lean moe about such a foce. To electify a neutal body, we need to add o emove one kind of chage. When we say that a body is chaged, we always efe to this excess chage o deficit of chage. In solids, some of the electons, being less tightly bound in the atom, ae the chages which ae tansfeed fom one body to the othe. A body can thus be chaged positively by losing some of its electons. Similaly, a body can be chaged negatively

5 by gaining electons. When we ub a glass od with silk, some of the electons fom the od ae tansfeed to the silk cloth. Thus the od gets positively chaged and the silk gets negatively chaged. No new chage is ceated in the pocess of ubbing. Also the numbe of electons, that ae tansfeed, is a vey small faction of the total numbe of electons in the mateial body. Also only the less tightly bound electons in a mateial body can be tansfeed fom it to anothe by ubbing. Theefoe, when a body is ubbed with anothe, the bodies get chaged and that is why we have to stick to cetain pais of mateials to notice chaging on ubbing the bodies. 1.3 CONDUCTORS AND INSULATORS A metal od held in hand and ubbed with wool will not show any sign of being chaged. Howeve, if a metal od with a wooden o plastic handle is ubbed without touching its metal pat, it shows signs of chaging. Suppose we connect one end of a coppe wie to a neutal pith ball and the othe end to a negatively chaged plastic od. We will find that the pith ball acquies a negative chage. If a simila expeiment is epeated with a nylon thead o a ubbe band, no tansfe of chage will take place fom the plastic od to the pith ball. Why does the tansfe of chage not take place fom the od to the ball? Some substances eadily allow passage of electicity though them, othes do not. Those which allow electicity to pass though them easily ae called conductos. They have electic chages (electons) that ae compaatively fee to move inside the mateial. Metals, human and animal bodies and eath ae conductos. Most of the non-metals like glass, pocelain, plastic, nylon, wood offe high esistance to the passage of electicity though them. They ae called insulatos. Most substances fall into one of the two classes stated above*. When some chage is tansfeed to a conducto, it eadily gets distibuted ove the entie suface of the conducto. In contast, if some chage is put on an insulato, it stays at the same place. You will lean why this happens in the next chapte. This popety of the mateials tells you why a nylon o plastic comb gets electified on combing dy hai o on ubbing, but a metal aticle like spoon does not. The chages on metal leak though ou body to the gound as both ae conductos of electicity. When we bing a chaged body in contact with the eath, all the excess chage on the body disappeas by causing a momentay cuent to pass to the gound though the connecting conducto (such as ou body). This pocess of shaing the chages with the eath is called gounding o eathing. Eathing povides a safety measue fo electical cicuits and appliances. A thick metal plate is buied deep into the eath and thick wies ae dawn fom this plate; these ae used in buildings fo the pupose of eathing nea the mains supply. The electic wiing in ou houses has thee wies: live, neutal and eath. The fist two cay Electic Chages and Fields * Thee is a thid categoy called semiconductos, which offe esistance to the movement of chages which is intemediate between the conductos and insulatos. 5

6 Physics electic cuent fom the powe station and the thid is eathed by connecting it to the buied metal plate. Metallic bodies of the electic appliances such as electic ion, efigeato, TV ae connected to the eath wie. When any fault occus o live wie touches the metallic body, the chage flows to the eath without damaging the appliance and without causing any injuy to the humans; this would have othewise been unavoidable since the human body is a conducto of electicity. FIGURE 1.4 Chaging by induction CHARGING BY INDUCTION When we touch a pith ball with an electified plastic od, some of the negative chages on the od ae tansfeed to the pith ball and it also gets chaged. Thus the pith ball is chaged by contact. It is then epelled by the plastic od but is attacted by a glass od which is oppositely chaged. Howeve, why a electified od attacts light objects, is a question we have still left unansweed. Let us ty to undestand what could be happening by pefoming the following expeiment. (i) Bing two metal sphees, A and B, suppoted on insulating stands, in contact as shown in Fig. 1.4(a). (ii) Bing a positively chaged od nea one of the sphees, say A, taking cae that it does not touch the sphee. The fee electons in the sphees ae attacted towads the od. This leaves an excess of positive chage on the ea suface of sphee B. Both kinds of chages ae bound in the metal sphees and cannot escape. They, theefoe, eside on the sufaces, as shown in Fig. 1.4(b). The left suface of sphee A, has an excess of negative chage and the ight suface of sphee B, has an excess of positive chage. Howeve, not all of the electons in the sphees have accumulated on the left suface of A. As the negative chage stats building up at the left suface of A, othe electons ae epelled by these. In a shot time, equilibium is eached unde the action of foce of attaction of the od and the foce of epulsion due to the accumulated chages. Fig. 1.4(b) shows the equilibium situation. The pocess is called induction of chage and happens almost instantly. The accumulated chages emain on the suface, as shown, till the glass od is held nea the sphee. If the od is emoved, the chages ae not acted by any outside foce and they edistibute to thei oiginal neutal state. (iii) Sepaate the sphees by a small distance while the glass od is still held nea sphee A, as shown in Fig. 1.4(c). The two sphees ae found to be oppositely chaged and attact each othe. (iv) Remove the od. The chages on sphees eaange themselves as shown in Fig. 1.4(d). Now, sepaate the sphees quite apat. The chages on them get unifomly distibuted ove them, as shown in Fig. 1.4(e). In this pocess, the metal sphees will each be equal and oppositely chaged. This is chaging by induction. The positively chaged glass od does not lose any of its chage, contay to the pocess of chaging by contact. When electified ods ae bought nea light objects, a simila effect takes place. The ods induce opposite chages on the nea sufaces of the objects and simila chages move to the fathe side of the object.

7 [This happens even when the light object is not a conducto. The mechanism fo how this happens is explained late in Sections 1.1 and.1.] The centes of the two types of chages ae slightly sepaated. We know that opposite chages attact while simila chages epel. Howeve, the magnitude of foce depends on the distance between the chages and in this case the foce of attaction oveweighs the foce of epulsion. As a esult the paticles like bits of pape o pith balls, being light, ae pulled towads the ods. Electic Chages and Fields Example 1.1 How can you chage a metal sphee positively without touching it? Solution Figue 1.5(a) shows an unchaged metallic sphee on an insulating metal stand. Bing a negatively chaged od close to the metallic sphee, as shown in Fig. 1.5(b). As the od is bought close to the sphee, the fee electons in the sphee move away due to epulsion and stat piling up at the fathe end. The nea end becomes positively chaged due to deficit of electons. This pocess of chage distibution stops when the net foce on the fee electons inside the metal is zeo. Connect the sphee to the gound by a conducting wie. The electons will flow to the gound while the positive chages at the nea end will emain held thee due to the attactive foce of the negative chages on the od, as shown in Fig. 1.5(c). Disconnect the sphee fom the gound. The positive chage continues to be held at the nea end [Fig. 1.5(d)]. Remove the electified od. The positive chage will spead unifomly ove the sphee as shown in Fig. 1.5(e). Inteactive animation on chaging a two-sphee system by induction: FIGURE 1.5 In this expeiment, the metal sphee gets chaged by the pocess of induction and the od does not lose any of its chage. Simila steps ae involved in chaging a metal sphee negatively by induction, by binging a positively chaged od nea it. In this case the electons will flow fom the gound to the sphee when the sphee is connected to the gound with a wie. Can you explain why? EXAMPLE 1.1 7

8 Physics BASIC PROPERTIES OF ELECTRIC CHARGE We have seen that thee ae two types of chages, namely positive and negative and thei effects tend to cancel each othe. Hee, we shall now descibe some othe popeties of the electic chage. If the sizes of chaged bodies ae vey small as compaed to the distances between them, we teat them as point chages. All the chage content of the body is assumed to be concentated at one point in space Additivity of chages We have not as yet given a quantitative definition of a chage; we shall follow it up in the next section. We shall tentatively assume that this can be done and poceed. If a system contains two point chages q 1 and q, the total chage of the system is obtained simply by adding algebaically q 1 and q, i.e., chages add up like eal numbes o they ae scalas like the mass of a body. If a system contains n chages q 1, q, q 3,, q n, then the total chage of the system is q 1 + q + q q n. Chage has magnitude but no diection, simila to the mass. Howeve, thee is one diffeence between mass and chage. Mass of a body is always positive wheeas a chage can be eithe positive o negative. Pope signs have to be used while adding the chages in a system. Fo example, the total chage of a system containing five chages +1, +, 3, +4 and 5, in some abitay unit, is (+1) + (+) + ( 3) + (+4) + ( 5) = 1 in the same unit Chage is conseved We have aleady hinted to the fact that when bodies ae chaged by ubbing, thee is tansfe of electons fom one body to the othe; no new chages ae eithe ceated o destoyed. A pictue of paticles of electic chage enables us to undestand the idea of consevation of chage. When we ub two bodies, what one body gains in chage the othe body loses. Within an isolated system consisting of many chaged bodies, due to inteactions among the bodies, chages may get edistibuted but it is found that the total chage of the isolated system is always conseved. Consevation of chage has been established expeimentally. It is not possible to ceate o destoy net chage caied by any isolated system although the chage caying paticles may be ceated o destoyed in a pocess. Sometimes natue ceates chaged paticles: a neuton tuns into a poton and an electon. The poton and electon thus ceated have equal and opposite chages and the total chage is zeo befoe and afte the ceation Quantisation of chage Expeimentally it is established that all fee chages ae integal multiples of a basic unit of chage denoted by e. Thus chage q on a body is always given by q = ne

9 Electic Chages and Fields whee n is any intege, positive o negative. This basic unit of chage is the chage that an electon o poton caies. By convention, the chage on an electon is taken to be negative; theefoe chage on an electon is witten as e and that on a poton as +e. The fact that electic chage is always an integal multiple of e is temed as quantisation of chage. Thee ae a lage numbe of situations in physics whee cetain physical quantities ae quantised. The quantisation of chage was fist suggested by the expeimental laws of electolysis discoveed by English expeimentalist Faaday. It was expeimentally demonstated by Millikan in 191. In the Intenational System (SI) of Units, a unit of chage is called a coulomb and is denoted by the symbol C. A coulomb is defined in tems the unit of the electic cuent which you ae going to lean in a subsequent chapte. In tems of this definition, one coulomb is the chage flowing though a wie in 1 s if the cuent is 1 A (ampee), (see Chapte of Class XI, Physics Textbook, Pat I). In this system, the value of the basic unit of chage is e = C Thus, thee ae about electons in a chage of 1C. In electostatics, chages of this lage magnitude ae seldom encounteed and hence we use smalle units 1 μc (mico coulomb) = 1 6 C o 1 mc (milli coulomb) = 1 3 C. If the potons and electons ae the only basic chages in the univese, all the obsevable chages have to be integal multiples of e. Thus, if a body contains n 1 electons and n potons, the total amount of chage on the body is n e + n 1 ( e) = (n n 1 ) e. Since n 1 and n ae integes, thei diffeence is also an intege. Thus the chage on any body is always an integal multiple of e and can be inceased o deceased also in steps of e. The step size e is, howeve, vey small because at the macoscopic level, we deal with chages of a few μc. At this scale the fact that chage of a body can incease o decease in units of e is not visible. The gainy natue of the chage is lost and it appeas to be continuous. This situation can be compaed with the geometical concepts of points and lines. A dotted line viewed fom a distance appeas continuous to us but is not continuous in eality. As many points vey close to each othe nomally give an impession of a continuous line, many small chages taken togethe appea as a continuous chage distibution. At the macoscopic level, one deals with chages that ae enomous compaed to the magnitude of chage e. Since e = C, a chage of magnitude, say 1 μc, contains something like 1 13 times the electonic chage. At this scale, the fact that chage can incease o decease only in units of e is not vey diffeent fom saying that chage can take continuous values. Thus, at the macoscopic level, the quantisation of chage has no pactical consequence and can be ignoed. At the micoscopic level, whee the chages involved ae of the ode of a few tens o hundeds of e, i.e., 9

10 Physics they can be counted, they appea in discete lumps and quantisation of chage cannot be ignoed. It is the scale involved that is vey impotant. EXAMPLE 1.3 EXAMPLE 1. Example 1. If 1 9 electons move out of a body to anothe body evey second, how much time is equied to get a total chage of 1 C on the othe body? Solution In one second 1 9 electons move out of the body. Theefoe the chage given out in one second is C = C. The time equied to accumulate a chage of 1 C can then be estimated to be 1 C ( C/s) = s = ( ) yeas = 198 yeas. Thus to collect a chage of one coulomb, fom a body fom which 1 9 electons move out evey second, we will need appoximately yeas. One coulomb is, theefoe, a vey lage unit fo many pactical puposes. It is, howeve, also impotant to know what is oughly the numbe of electons contained in a piece of one cubic centimete of a mateial. A cubic piece of coppe of side 1 cm contains about electons. Example 1.3 How much positive and negative chage is thee in a cup of wate? Solution Let us assume that the mass of one cup of wate is 5 g. The molecula mass of wate is 18g. Thus, one mole (= molecules) of wate is 18 g. Theefoe the numbe of molecules in one cup of wate is (5/18) Each molecule of wate contains two hydogen atoms and one oxygen atom, i.e., 1 electons and 1 potons. Hence the total positive and total negative chage has the same magnitude. It is equal to (5/18) C = C. 1.6 COULOMB S LAW Coulomb s law is a quantitative statement about the foce between two point chages. When the linea size of chaged bodies ae much smalle than the distance sepaating them, the size may be ignoed and the chaged bodies ae teated as point chages. Coulomb measued the foce between two point chages and found that it vaied invesely as the squae of the distance between the chages and was diectly popotional to the poduct of the magnitude of the two chages and acted along the line joining the two chages. Thus, if two point chages q 1, q ae sepaated by a distance in vacuum, the magnitude of the foce (F) between them is given by q q 1 F = k (1.1) How did Coulomb aive at this law fom his expeiments? Coulomb used a tosion balance* fo measuing the foce between two chaged metallic 1 * A tosion balance is a sensitive device to measue foce. It was also used late by Cavendish to measue the vey feeble gavitational foce between two objects, to veify Newton s Law of Gavitation.

11 Electic Chages and Fields sphees. When the sepaation between two sphees is much lage than the adius of each sphee, the chaged sphees may be egaded as point chages. Howeve, the chages on the sphees wee unknown, to begin with. How then could he discove a elation like Eq. (1.1)? Coulomb thought of the following simple way: Suppose the chage on a metallic sphee is q. If the sphee is put in contact with an identical unchaged sphee, the chage will spead ove the two sphees. By symmety, the chage on each sphee will be q/*. Repeating this pocess, we can get chages q/, q/4, etc. Coulomb vaied the distance fo a fixed pai of chages and measued the foce fo diffeent sepaations. He then vaied the chages in pais, keeping the distance fixed fo each pai. Compaing foces fo diffeent pais of chages at diffeent distances, Coulomb aived at the elation, Eq. (1.1). Coulomb s law, a simple mathematical statement, was initially expeimentally aived at in the manne descibed above. While the oiginal expeiments established it at a macoscopic scale, it has also been established down to subatomic level ( ~ 1 1 m). Coulomb discoveed his law without knowing the explicit magnitude of the chage. In fact, it is the othe way ound: Coulomb s law can now be employed to funish a definition fo a unit of chage. In the elation, Eq. (1.1), k is so fa abitay. We can choose any positive value of k. The choice of k detemines the size of the unit of chage. In SI units, the value of k is about The unit of chage that esults fom this choice is called a coulomb which we defined ealie in Section 1.4. Putting this value of k in Eq. (1.1), we see that fo q 1 = q = 1 C, = 1 m F = N That is, 1 C is the chage that when placed at a distance of 1 m fom anothe chage of the same magnitude in vacuum expeiences an electical foce of epulsion of magnitude N. One coulomb is evidently too big a unit to be used. In pactice, in electostatics, one uses smalle units like 1 mc o 1 μc. The constant k in Eq. (1.1) is usually put as k = 1/4πε fo late convenience, so that Coulomb s law is witten as 1 q q 1 F = (1.) 4 π ε ε is called the pemittivity of fee space. The value of ε in SI units is ε = C N 1 m * Implicit in this is the assumption of additivity of chages and consevation: two chages (q/ each) add up to make a total chage q. Chales Augustin de Coulomb ( ) Coulomb, a Fench physicist, began his caee as a militay enginee in the West Indies. In 1776, he etuned to Pais and etied to a small estate to do his scientific eseach. He invented a tosion balance to measue the quantity of a foce and used it fo detemination of foces of electic attaction o epulsion between small chaged sphees. He thus aived in 1785 at the invese squae law elation, now known as Coulomb s law. The law had been anticipated by Piestley and also by Cavendish ealie, though Cavendish neve published his esults. Coulomb also found the invese squae law of foce between unlike and like magnetic poles. 11 CHARLES AUGUSTIN DE COULOMB ( )

12 Physics FIGURE 1.6 (a) Geomety and (b) Foces between chages. Since foce is a vecto, it is bette to wite Coulomb s law in the vecto notation. Let the position vectos of chages q 1 and q be 1 and espectively [see Fig.1.6(a)]. We denote foce on q 1 due to q by F 1 and foce on q due to q 1 by F 1. The two point chages q 1 and q have been numbeed 1 and fo convenience and the vecto leading fom 1 to is denoted by 1 : 1 = 1 In the same way, the vecto leading fom to 1 is denoted by 1 : 1 = 1 = 1 The magnitude of the vectos 1 and 1 is denoted by 1 and 1, espectively ( = ). The 1 1 diection of a vecto is specified by a unit vecto along the vecto. To denote the diection fom 1 to (o fom to 1), we define the unit vectos: ˆ 1 1, 1 = ˆ =, ˆ = ˆ Coulomb s foce law between two point chages q 1 and q located at 1 and is then expessed as 1 q q F = ˆ (1.3) π εo 1 Some emaks on Eq. (1.3) ae elevant: Equation (1.3) is valid fo any sign of q 1 and q whethe positive o negative. If q 1 and q ae of the same sign (eithe both positive o both negative), F 1 is along ˆ 1, which denotes epulsion, as it should be fo like chages. If q 1 and q ae of opposite signs, F 1 is along ˆ 1 (= ˆ 1 ), which denotes attaction, as expected fo unlike chages. Thus, we do not have to wite sepaate equations fo the cases of like and unlike chages. Equation (1.3) takes cae of both cases coectly [Fig. 1.6(b)]. The foce F 1 on chage q 1 due to chage q, is obtained fom Eq. (1.3), by simply intechanging 1 and, i.e., 1 q q F = ˆ = F π ε 1 1 Thus, Coulomb s law agees with the Newton s thid law. Coulomb s law [Eq. (1.3)] gives the foce between two chages q 1 and q in vacuum. If the chages ae placed in matte o the intevening space has matte, the situation gets complicated due to the pesence of chaged constituents of matte. We shall conside electostatics in matte in the next chapte.

13 Electic Chages and Fields Example 1.4 Coulomb s law fo electostatic foce between two point chages and Newton s law fo gavitational foce between two stationay point masses, both have invese-squae dependence on the distance between the chages/masses. (a) Compae the stength of these foces by detemining the atio of thei magnitudes (i) fo an electon and a poton and (ii) fo two potons. (b) Estimate the acceleations of electon and poton due to the electical foce of thei mutual attaction when they ae 1 Å (= 1-1 m) apat? (m p = kg, m e = kg) Solution (a) (i) The electic foce between an electon and a poton at a distance apat is: 1 e Fe = 4 πε whee the negative sign indicates that the foce is attactive. The coesponding gavitational foce (always attactive) is: mp me FG = G whee m p and m e ae the masses of a poton and an electon espectively. Fe e F = 4πε Gm m = G p e (ii) On simila lines, the atio of the magnitudes of electic foce to the gavitational foce between two potons at a distance apat is : Fe e = = FG 4πεGmpmp Howeve, it may be mentioned hee that the signs of the two foces ae diffeent. Fo two potons, the gavitational foce is attactive in natue and the Coulomb foce is epulsive. The actual values of these foces between two potons inside a nucleus (distance between two potons is ~ 1-15 m inside a nucleus) ae F e ~ 3 N wheeas F G ~ N. The (dimensionless) atio of the two foces shows that electical foces ae enomously stonge than the gavitational foces. (b) The electic foce F exeted by a poton on an electon is same in magnitude to the foce exeted by an electon on a poton; howeve the masses of an electon and a poton ae diffeent. Thus, the magnitude of foce is 1 e F = = πε 9 Nm /C ( C) / (1 1 m) = N Using Newton s second law of motion, F = ma, the acceleation that an electon will undego is a = N / kg =.5 1 m/s Compaing this with the value of acceleation due to gavity, we can conclude that the effect of gavitational field is negligible on the motion of electon and it undegoes vey lage acceleations unde the action of Coulomb foce due to a poton. The value fo acceleation of the poton is N / kg = m/s EXAMPLE 1.4 Inteactive animation on Coulomb s law: 13

14 Physics Example 1.5 A chaged metallic sphee A is suspended by a nylon thead. Anothe chaged metallic sphee B held by an insulating handle is bought close to A such that the distance between thei centes is 1 cm, as shown in Fig. 1.7(a). The esulting epulsion of A is noted (fo example, by shining a beam of light and measuing the deflection of its shadow on a sceen). Sphees A and B ae touched by unchaged sphees C and D espectively, as shown in Fig. 1.7(b). C and D ae then emoved and B is bought close to A to a distance of 5. cm between thei centes, as shown in Fig. 1.7(c). What is the expected epulsion of A on the basis of Coulomb s law? Sphees A and C and sphees B and D have identical sizes. Ignoe the sizes of A and B in compaison to the sepaation between thei centes. 14 EXAMPLE 1.5 FIGURE 1.7

15 Electic Chages and Fields Solution Let the oiginal chage on sphee A be q and that on B be q. At a distance between thei centes, the magnitude of the electostatic foce on each is given by 1 qq F = 4 π ε neglecting the sizes of sphees A and B in compaison to. When an identical but unchaged sphee C touches A, the chages edistibute on A and C and, by symmety, each sphee caies a chage q/. Similaly, afte D touches B, the edistibuted chage on each is q /. Now, if the sepaation between A and B is halved, the magnitude of the electostatic foce on each is 1 ( q/)( q /) 1 ( qq ) F = = = F 4πε 4πε ( /) Thus the electostatic foce on A, due to B, emains unalteed. EXAMPLE FORCES BETWEEN MULTIPLE CHARGES The mutual electic foce between two chages is given by Coulomb s law. How to calculate the foce on a chage whee thee ae not one but seveal chages aound? Conside a system of n stationay chages q 1, q, q 3,..., q n in vacuum. What is the foce on q 1 due to q, q 3,..., q n? Coulomb s law is not enough to answe this question. Recall that foces of mechanical oigin add accoding to the paallelogam law of addition. Is the same tue fo foces of electostatic oigin? Expeimentally it is veified that foce on any chage due to a numbe of othe chages is the vecto sum of all the foces on that chage due to the othe chages, taken one at a time. The individual foces ae unaffected due to the pesence of othe chages. This is temed as the pinciple of supeposition. To bette undestand the concept, conside a system of thee chages q 1, q and q 3, as shown in Fig. 1.8(a). The foce on one chage, say q 1, due to two othe chages q, q 3 can theefoe be obtained by pefoming a vecto addition of the foces due to each one of these chages. Thus, if the foce on q 1 due to q is denoted by F 1, F 1 is given by Eq. (1.3) even though othe chages ae pesent. 1 qq 1 Thus, F 1 = ˆ 1 4 π ε 1 In the same way, the foce on q 1 due to q 3, denoted FIGURE 1.8 A system of (a) thee by F 13, is given by chages (b) multiple chages. F 1 qq = πε ˆ

16 Physics which again is the Coulomb foce on q 1 due to q 3, even though othe chage q is pesent. Thus the total foce F 1 on q 1 due to the two chages q and q 3 is given as F 1 qq 1 qq = F + F = ˆ + ˆ (1.4) πε 1 4πε 13 The above calculation of foce can be genealised to a system of chages moe than thee, as shown in Fig. 1.8(b). The pinciple of supeposition says that in a system of chages q 1, q,..., q n, the foce on q 1 due to q is the same as given by Coulomb s law, i.e., it is unaffected by the pesence of the othe chages q 3, q 4,..., q n. The total foce F 1 on the chage q 1, due to all othe chages, is then given by the vecto sum of the foces F 1, F 13,..., F 1n : i.e., 1 qq 1 qq 1 3 qq 1 n F1 = F1 + F ˆ ˆ ˆ F1n = n 4πε n = q q 4 (1.5) π n 1 i ˆ 1i ε i = 1 i The vecto sum is obtained as usual by the paallelogam law of addition of vectos. All of electostatics is basically a consequence of Coulomb s law and the supeposition pinciple. Example 1.6 Conside thee chages q 1, q, q 3 each equal to q at the vetices of an equilateal tiangle of side l. What is the foce on a chage Q (with the same sign as q) placed at the centoid of the tiangle, as shown in Fig. 1.9? 16 EXAMPLE 1.6 FIGURE 1.9 Solution In the given equilateal tiangle ABC of sides of length l, if we daw a pependicula AD to the side BC, AD = AC cos 3º = ( 3/) l and the distance AO of the centoid O fom A is (/3) AD = (1/ 3 ) l. By symmaty AO = BO = CO.

17 Electic Chages and Fields Thus, 3 Qq Foce F 1 on Q due to chage q at A = 4πε l 3 Qq Foce F on Q due to chage q at B = 4πε l 3 Qq Foce F 3 on Q due to chage q at C = 4πε l 3 Qq The esultant of foces F and F 3 is 4πε l along AO along BO along CO along OA, by the 3 Qq 4πε l paallelogam law. Theefoe, the total foce on Q = ( ˆ ˆ ) =, whee ˆ is the unit vecto along OA. It is clea also by symmety that the thee foces will sum to zeo. Suppose that the esultant foce was non-zeo but in some diection. Conside what would happen if the system was otated though 6º about O. EXAMPLE 1.6 Example 1.7 Conside the chages q, q, and q placed at the vetices of an equilateal tiangle, as shown in Fig What is the foce on each chage? FIGURE 1.1 Solution The foces acting on chage q at A due to chages q at B and q at C ae F 1 along BA and F 13 along AC espectively, as shown in Fig By the paallelogam law, the total foce F 1 on the chage q at A is given by F 1 = F ˆ 1 whee ˆ 1 is a unit vecto along BC. The foce of attaction o epulsion fo each pai of chages has the same magnitude q F = 4 π ε l The total foce F on chage q at B is thus F = F ˆ, whee ˆ is a unit vecto along AC. EXAMPLE

18 Physics EXAMPLE 1.7 Similaly the total foce on chage q at C is F 3 = 3 F ˆn, whee ˆn is the unit vecto along the diection bisecting the BCA. It is inteesting to see that the sum of the foces on the thee chages is zeo, i.e., F 1 + F + F 3 = The esult is not at all supising. It follows staight fom the fact that Coulomb s law is consistent with Newton s thid law. The poof is left to you as an execise. FIGURE 1.11 Electic field (a) due to a chage Q, (b) due to a chage Q ELECTRIC FIELD Let us conside a point chage Q placed in vacuum, at the oigin O. If we place anothe point chage q at a point P, whee OP =, then the chage Q will exet a foce on q as pe Coulomb s law. We may ask the question: If chage q is emoved, then what is left in the suounding? Is thee nothing? If thee is nothing at the point P, then how does a foce act when we place the chage q at P. In ode to answe such questions, the ealy scientists intoduced the concept of field. Accoding to this, we say that the chage Q poduces an electic field eveywhee in the suounding. When anothe chage q is bought at some point P, the field thee acts on it and poduces a foce. The electic field poduced by the chage Q at a point is given as E ( 1 Q 1 Q ) = ˆ = ˆ 4πε 4πε (1.6) whee ˆ = /, is a unit vecto fom the oigin to the point. Thus, Eq.(1.6) specifies the value of the electic field fo each value of the position vecto. The wod field signifies how some distibuted quantity (which could be a scala o a vecto) vaies with position. The effect of the chage has been incopoated in the existence of the electic field. We obtain the foce F exeted by a chage Q on a chage q, as 1 Qq F = ˆ 4 πε (1.7) Note that the chage q also exets an equal and opposite foce on the chage Q. The electostatic foce between the chages Q and q can be looked upon as an inteaction between chage q and the electic field of Q and vice vesa. If we denote the position of chage q by the vecto, it expeiences a foce F equal to the chage q multiplied by the electic field E at the location of q. Thus, F() = q E() (1.8) Equation (1.8) defines the SI unit of electic field as N/C*. (i) Some impotant emaks may be made hee: Fom Eq. (1.8), we can infe that if q is unity, the electic field due to a chage Q is numeically equal to the foce exeted by it. Thus, the electic field due to a chage Q at a point in space may be defined as the foce that a unit positive chage would expeience if placed * An altenate unit V/m will be intoduced in the next chapte.

19 at that point. The chage Q, which is poducing the electic field, is called a souce chage and the chage q, which tests the effect of a souce chage, is called a test chage. Note that the souce chage Q must emain at its oiginal location. Howeve, if a chage q is bought at any point aound Q, Q itself is bound to expeience an electical foce due to q and will tend to move. A way out of this difficulty is to make q negligibly small. The foce F is then negligibly small but the atio F/q is finite and defines the electic field: F E = lim (1.9) q q A pactical way to get aound the poblem (of keeping Q undistubed in the pesence of q) is to hold Q to its location by unspecified foces! This may look stange but actually this is what happens in pactice. When we ae consideing the electic foce on a test chage q due to a chaged plana sheet (Section 1.15), the chages on the sheet ae held to thei locations by the foces due to the unspecified chaged constituents inside the sheet. (ii) Note that the electic field E due to Q, though defined opeationally in tems of some test chage q, is independent of q. This is because F is popotional to q, so the atio F/q does not depend on q. The foce F on the chage q due to the chage Q depends on the paticula location of chage q which may take any value in the space aound the chage Q. Thus, the electic field E due to Q is also dependent on the space coodinate. Fo diffeent positions of the chage q all ove the space, we get diffeent values of electic field E. The field exists at evey point in thee-dimensional space. (iii) Fo a positive chage, the electic field will be diected adially outwads fom the chage. On the othe hand, if the souce chage is negative, the electic field vecto, at each point, points adially inwads. (iv) Since the magnitude of the foce F on chage q due to chage Q depends only on the distance of the chage q fom chage Q, the magnitude of the electic field E will also depend only on the distance. Thus at equal distances fom the chage Q, the magnitude of its electic field E is same. The magnitude of electic field E due to a point chage is thus same on a sphee with the point chage at its cente; in othe wods, it has a spheical symmety. Electic Chages and Fields Electic field due to a system of chages Conside a system of chages q 1, q,..., q n with position vectos 1,,..., n elative to some oigin O. Like the electic field at a point in space due to a single chage, electic field at a point in space due to the system of chages is defined to be the foce expeienced by a unit test chage placed at that point, without distubing the oiginal positions of chages q 1, q,..., q n. We can use Coulomb s law and the supeposition pinciple to detemine this field at a point P denoted by position vecto. 19

20 Physics Electic field E 1 at due to q 1 at 1 is given by 1 q1 E 1 = ˆ 1P 4πε 1P whee ˆ 1P is a unit vecto in the diection fom q 1 to P, and 1P is the distance between q 1 and P. In the same manne, electic field E at due to q at is 1 q E = ˆ P 4πε whee ˆ P is a unit vecto in the diection fom q to P FIGURE 1.1 Electic field at a and P is the distance between q and P. Simila point due to a system of chages is expessions hold good fo fields E 3, E 4,..., E n due to the vecto sum of the electic fields chages q 3, q 4,..., q n. at the point due to individual By the supeposition pinciple, the electic field E at chages. due to the system of chages is (as shown in Fig. 1.1) E() = E 1 () + E () + + E n () 1 q1 1 q 1 qn = ˆ 1P + ˆ P ˆ np 4πε 4πε 4πε P 1P P np E() = 1 4 q n i ˆ ip (1.1) πε i = 1 i P E is a vecto quantity that vaies fom one point to anothe point in space and is detemined fom the positions of the souce chages Physical significance of electic field You may wonde why the notion of electic field has been intoduced hee at all. Afte all, fo any system of chages, the measuable quantity is the foce on a chage which can be diectly detemined using Coulomb s law and the supeposition pinciple [Eq. (1.5)]. Why then intoduce this intemediate quantity called the electic field? Fo electostatics, the concept of electic field is convenient, but not eally necessay. Electic field is an elegant way of chaacteising the electical envionment of a system of chages. Electic field at a point in the space aound a system of chages tells you the foce a unit positive test chage would expeience if placed at that point (without distubing the system). Electic field is a chaacteistic of the system of chages and is independent of the test chage that you place at a point to detemine the field. The tem field in physics geneally efes to a quantity that is defined at evey point in space and may vay fom point to point. Electic field is a vecto field, since foce is a vecto quantity. The tue physical significance of the concept of electic field, howeve, emeges only when we go beyond electostatics and deal with timedependent electomagnetic phenomena. Suppose we conside the foce between two distant chages q 1, q in acceleated motion. Now the geatest speed with which a signal o infomation can go fom one point to anothe is c, the speed of light. Thus, the effect of any motion of q 1 on q cannot

21 aise instantaneously. Thee will be some time delay between the effect (foce on q ) and the cause (motion of q 1 ). It is pecisely hee that the notion of electic field (stictly, electomagnetic field) is natual and vey useful. The field pictue is this: the acceleated motion of chage q 1 poduces electomagnetic waves, which then popagate with the speed c, each q and cause a foce on q. The notion of field elegantly accounts fo the time delay. Thus, even though electic and magnetic fields can be detected only by thei effects (foces) on chages, they ae egaded as physical entities, not meely mathematical constucts. They have an independent dynamics of thei own, i.e., they evolve accoding to laws of thei own. They can also tanspot enegy. Thus, a souce of timedependent electomagnetic fields, tuned on biefly and switched off, leaves behind popagating electomagnetic fields tanspoting enegy. The concept of field was fist intoduced by Faaday and is now among the cental concepts in physics. Example 1.8 An electon falls though a distance of 1.5 cm in a unifom electic field of magnitude. 1 4 N C 1 [Fig. 1.13(a)]. The diection of the field is evesed keeping its magnitude unchanged and a poton falls though the same distance [Fig. 1.13(b)]. Compute the time of fall in each case. Contast the situation with that of fee fall unde gavity. Electic Chages and Fields FIGURE 1.13 Solution In Fig. 1.13(a) the field is upwad, so the negatively chaged electon expeiences a downwad foce of magnitude ee whee E is the magnitude of the electic field. The acceleation of the electon is a e = ee/m e whee m e is the mass of the electon. Stating fom est, the time equied by the electon to fall though a distance h is given by t e h = = a e hm e E Fo e = C, m e = kg, E =. 1 4 N C 1, h = m, t e = s In Fig (b), the field is downwad, and the positively chaged poton expeiences a downwad foce of magnitude ee. The acceleation of the poton is a p = ee/m p whee m p is the mass of the poton; m p = kg. The time of fall fo the poton is e EXAMPLE 1.8 1

22 Physics h hmp t = p. a = ee = p s Thus, the heavie paticle (poton) takes a geate time to fall though the same distance. This is in basic contast to the situation of fee fall unde gavity whee the time of fall is independent of the mass of the body. Note that in this example we have ignoed the acceleation due to gavity in calculating the time of fall. To see if this is justified, let us calculate the acceleation of the poton in the given electic field: a p ee = m = p ( C) ( N C ) kg EXAMPLE 1.9 EXAMPLE 1.8 = ms 1 which is enomous compaed to the value of g (9.8 m s ), the acceleation due to gavity. The acceleation of the electon is even geate. Thus, the effect of acceleation due to gavity can be ignoed in this example. Example 1.9 Two point chages q 1 and q, of magnitude +1 8 C and 1 8 C, espectively, ae placed.1 m apat. Calculate the electic fields at points A, B and C shown in Fig FIGURE 1.14 Solution The electic field vecto E 1A at A due to the positive chage q 1 points towads the ight and has a magnitude E 9-8 (9 1 Nm C ) (1 C) 1A = = N C 1 (.5 m) The electic field vecto E A at A due to the negative chage q points towads the ight and has the same magnitude. Hence the magnitude of the total electic field E A at A is E A = E 1A + E A = N C 1 E A is diected towad the ight.