ELECTRIC CHARGE AND ELECTRIC FIELD

Transcription

1 Electic Chage and Electic Field MODULE - 5 Electicity and 5 ELECTRIC CHARGE AND ELECTRIC FIELD So fa you have leant about mechanical, themal and optical systems and vaious phenomena exhibited by them. The impotance of electicity in ou daily life is too evident. The physical comfots we enjoy and the vaious devices used in daily life depend on the availability of electical enegy. An electical powe failue demonstates diectly ou dependence on electic and magnetic phenomena; the lights go off, the fans, cooles and ai-conditiones in summe and heates and gyses in winte stop woking. Similaly, adio, TV, computes, micowaves can not be opeated. Wate pumps stop unning and fields cannot be iigated. Even tain sevices ae affected by powe failue. Machines in industial units can not be opeated. In shot, life almost comes to a stand still, sometimes even evoking public ange. It is, theefoe, extemely impotant to study electic and magnetic phenomena. In this lesson, you will lean about two kinds of electic chages, thei behaviou in diffeent cicumstances, the foces that act between them, the behaviou of the suounding space etc. Boadly speaking, we wish to study that banch of physics which deals with electical chages at est. This banch is called electostatics. OBJECTIVES Afte studying this lesson, you should be able to : state the basic popeties of electic chages; explain the concepts of uantisation and consevation of chage; explain Coulomb s law of foce between electic chages; define electic field due to a chage at est and daw electic lines of foce; define electic dipole, dipole moment and the electic field due to a dipole;

2 MODULE - 5 Electicity and Electic Chage and Electic Field state Gauss theoem and deive expessions fo the electic field due to a point chage, a long chaged wie, a unifomly chaged spheical shell and a plane sheet of chage; and descibe how a van de Gaaff geneato functions. 5. FRICTIONAL ELECTRICITY The ancient Geeks obseved electic and magnetic phenomena as ealy as 6 B.C. They found that a piece of ambe, when ubbed, becomes electified and attacts small pieces of feathes. The wod electic comes fom Geek wod fo ambe meaning electon. You can pefom simple activities to demonstate the existence of chages and foces between them. If you un a comb though you dy hai, you will note that the comb begins to attact small pieces of pape. Do you know how does it happen? Let us pefom two simple expeiments to undestand the eason. ACTIVITY 5. Take a had ubbe od and ub it with fu o wool. Next you take a glass od and ub it with silk. Suspend them (ubbe od and a glass od) sepaately with the help of non-metallic theads, as shown in Fig. 5.. ubbe-od Attaction Glass-od Repulsion ubbe-od ubbe-od (a) (b) Fig. 5.: Foce of attaction/epulsion between chages: a) a chaged ubbe od epels anothe chaged ubbe od : like chages epel each othe; and b) a chaged glass od attacts a chaged ubbe od : unlike chages attact each othe.

3 Electic Chage and Electic Field Now bing ubbe od ubbed with wool nea these ods one by one. What do you obseve? You will obseve that when a chaged ubbe od is bought nea the chaged (suspended) ubbe od, they show epulsion [Fig. 5.(a)]; and when the chaged ubbe od is bought nea the (suspended) chaged glass od, they show attaction [Fig 5.(b)]. Simila esults will be obtained by binging a chaged glass od. On the basis of these obsevations, we can say that A chaged ubbe od attacts a chaged glass od but epels a chaged ubbe od. A chaged glass od epels a chaged glass od but attacts a chaged ubbe od. Fom these activities we can infe that the ubbe od has acuied one kind of electicity and the glass od has acuied anothe kind of electicity. Moeove, like chages epel and unlike chages attact each othe. Fanklin (Benjamin Fanklin, 76-79) suggested that the chage on glass od is to be called positive and that on the ubbe od is to be called negative. We follow this convention since then. Once a body is chaged by fiction, it can be used to chage othe conducting bodies by conduction, i.e., by touching the chaged body with an unchaged body; and induction, i.e., by binging the chaged body close to an unchaged conducto and eathing it. Subseuently, the chaged body and the eathing ae emoved simultaneously. MODULE - 5 Electicity and 5.. Consevation of Chage In Activity 5., you have seen that when a glass od is ubbed with silk, the od acuies positive chage and silk acuies negative chage. Since both mateials in the nomal state ae neutal (no chage), the positive chage on the glass od should be eual in magnitude to the negative chage on silk. This means that the total chage of the system (glass silk) is conseved. It is neithe ceated no destoyed. It is only tansfeed fom one body of the system to the othe. The tansfe of chages takes place due to incease in the themal enegy of the system when the glass od is ubbed; the less tightly bound electons fom the glass od ae tansfeed to silk. The glass od (deficient in electons) becomes positively chaged and silk, which now has excess electons, becomes negatively chaged. When ubbe is ubbed with fu, electons fom the fu ae tansfeed to ubbe. 3

4 MODULE - 5 Electicity and Electic Chage and Electic Field That is, ubbe gains negative chage and fu gains an eual amount of positive chage. Any othe kind of chage (othe than positive and negative) has not been found till today. 5.. Quantisation of Chage In 99, Millikan (Robet Millikan, ) expeimentally poved that chage always occus as some integal multiple of a fundamental unit of chage, which is taken as the chage on an electon. This means that if Q is the chage on an object, it can be witten as Q = Ne, whee N is an intege and e is chage on an electon. Then we say that chage is uantised. It means that a chaged body cannot have.5e o 6.4e amount of chage. In units 4-6, you will lean that an electon has chage e and a poton has chage e. Neuton has no chage. Evey atom has eual numbe of electons and potons and that is why it is neutal. Fom this discussion, we can daw the following conclusions : Thee ae only two kinds of chages in natue; positive and negative. Chage is conseved. Chage is uantised. INTEXT QUESTIONS 5.. A glass od when ubbed with silk cloth acuies a chage = 3. 7 C. i) Is silk cloth also chaged? ii) What is the natue and magnitude of the chage on silk cloth?. Thee ae two identical metallic sphees A and B. A is given a chage Q. Both sphees ae then bought in contact and then sepaated. (i) Will thee be any chage on B? (ii) What will the magnitude of chage on B, if it gets chaged when in contact with A. 3. A chaged object has = C. How many units of fundamental chage ae thee on the object? (Take e =.6 9 C ). 5. COULOMB S LAW You have leant that two stationay chages eithe attact o epel each othe. The foce of attaction o epulsion between them depends on thei natue. Coulomb studied the natue of this foce and in 785 established a fundamental law govening 4

5 Electic Chage and Electic Field it. Fom expeimental obsevations, he showed that the electical foce between two static point chages and placed some distance apat is diectly popotional to thei poduct ; invesely popotional to the suae of the distance between them; diected along the line joining the two chaged paticles ; and epulsive fo same kind of chages and attactive fo opposite chages. The magnitude of foce F can then be expessed as F = k (5.) MODULE - 5 Electicity and Fo fee space, we wite F = 4πε (5.) whee constant of popotionality k = 4πε fo fee space (vacuum) and k = 4πε fo a mateial medium. ε is called pemittivity of fee space and ε is the pemittivity of the medium. It means that if the same system of chages is kept in a mateial medium, the magnitude of Coulomb foce will be diffeent fom that in fee space. The constant k has a value which depends on the units of the uantities involved. The unit of chage in SI system is coulomb (C). The coulomb is defined in tems of the unit of cuent, called ampee. (You will lean about it late.) In SI system of units, the value of k is k = since ε = 8.85 C² N m. 4πε Nm = 9 9 C (5.3) Thus in tems of foce, one coulomb chage can be defined as : If two eual chages sepaated by one mete expeience a foce of 9 9 N, each chage has a magnitude of one coulomb. The value of electonic chage e is.6 9 C. Note that Coulomb s law is also an invese suae law just like Newton s law of Gavitation, which you studied in lesson 6. Coulomb s law holds good fo point chages only. Coulomb s foce acts at a distance, unlike mechanical foce. 5

6 MODULE - 5 Electicity and How Big is One Coulomb? Electic Chage and Electic Field The unit of electical chage is coulomb. Have you eve thought : How big a coulomb is? To know this, let us calculate the magnitude of foce between two chages, each of one coulomb, placed at a distance of one mete fom one anothe: F = k = 9. 9 = 9. 9 N If the mass of a loaded passenge bus is 5 kg, its weight mg = (5 ) N (assume g m s ²) = 5 4 N. Let us assume that thee ae, such loaded buses in Delhi. The total weight of all these buses will be 5 4, = 5 8 N. If thee ae cities having same numbe of buses as those in Delhi, the total weight of all these loaded buses will be 5 9 N. It means that the foce between two chages, each of C and sepaated by on mete is euivalent to the weight of about two hunded thousand buses, each of mass 5 kg. Chales Augustin de Coulomb (736 86) A Fench physicist, Coulomb stated his caee as militay enginee in West Indies. He invented a tosional balance and used it to pefom expeiments to detemine the natue of inteaction foces between chages and magnets. He pesented the esults of these expeiments in the fom of Coulomb s law of electostatics and Coulomb s law of magnetostatics. The SI unit of chage has been named in his honou. You now know that the atio of foces between two point chages and sepaated by a distance, when kept in fee space (vacuum) and mateial medium, is eual to ε/ε : F (in vaccum) F (in medium) ε = =ε ε whee ε is known as elative pemittivity o dielectic constant. Its value is always geate than one. We will define dielectic constant in anothe fom late. 6

7 Electic Chage and Electic Field 5.. Vecto Fom of Coulomb s Law You know that foce is a vecto uantity. It means that foce between two chages should also be epesented as a vecto. That is, En. (5.) should be expessed in vecto fom. Let us lean to do so now. Let thee be two point chages and sepaated by a distance (Fig. 5.3). Suppose that F denotes the foce expeienced by due to the chage and F denotes the foce on due to chage. We denote the unit vecto pointing fom to by ˆ. Then fom Fig. 5.3 (a), it follows that MODULE - 5 Electicity and F = k ˆ (5.4) Similaly, fo chages shown in Fig. 5.3 (b), we can wite F = k ˆ (5.5) F F F F (a) (b) Fig. 5.3 : Two point chages and sepaated by a distance : a) the diection of foces of epulsion between two positive chages, and b) the diection of foces of attaction between a positive and a negative chage. The positive sign in En. (5.4) indicates that the foce is epulsive and the negative sign in En. (5.5) indicates that the foce is attactive. The Coulomb s law obeys the pinciple of action and eaction between two chages and. Theefoe, F = F (5.6) In geneal, we can wite the expession fo foce between two chages as F = k ˆ (5.7) 5.. Pinciple of Supeposition If thee ae moe than two chages, we can calculate the foce between any two chages using En. (5.7). Suppose now that thee ae seveal chages,, 3, 4, etc. The foce exeted on due to all othe chages is given by En. (5.7): 7

8 MODULE - 5 Electicity and F F 3 = k = k and F 4 = k ˆ 3 3 ˆ 4 4 ˆ 3 4 (5.8) The esultant of all these foces, i.e., the total foce F expeienced by is thei vecto sum: This is known as pinciple of supeposition. Electic Chage and Electic Field 4 F = F F 3 F 4 (5.9) 3 3 Fig. 5.4: Pinciple of supeposition 3 Example 5. : A chage = C is placed at a distance of 4. m fom anothe chage = 6C, as shown in the Fig Whee should a negative chage 3 be placed on the line joining and so that the chage 3 does not expeience any foce? Solution : Let 3 be placed between and at a distance of x mete fom. (It can be easily seen that on placing 3 on the left of o on the ight of o at any position othe than the one between the line joining and, the esultant foce can not be zeo.) The foce exeted on 3 by will be F 3 = k 3 3 ˆ 3 towads 3 F 3 = k x The magnitude of foce on 3 due to is given by F 3 = k (4 x) 4m 3 towads x 3 Fig. 5.5 : Thee point chages, and 3 placed in a staight line The esultant foce on 3 will be zeo when F 3 = F 3. Theefoe, on substituting the numeical values, we get 8

9 Electic Chage and Electic Field k x 3 = k 63 (4 x) Note that 6 3 k is common on both sides and cancels out. Theefoe, on simplification, we get = x (4 x) o (4 x)² = x² x² 6x 3 = On solving this, we get two values of x :.35 m and 3.65 m. The latte value is inadmissible because it goes beyond. Theefoe, the chage 3 should be placed at a distance of.35 m fom. It is a easonable solution ualitatively also. The chage is stonge than. Hence the distance between and 3 should be geate than that between and 3. The oots of a uadatic euation of the fom ax bx c = ae given by b± b 4ac x = a In this case, a =, b = 6 and c = 3. 6 ± x = =.35, 3.65 MODULE - 5 Electicity and Example 5. : Two chages, each of 6. C, ae sepaated by a distance of. m. Calculate the magnitude of Coulomb foce between them. Solution : We know that the magnitude of Coulomb foce between two chages is given by En. (5.) :. F = k Given, = = 6. C and =. m, Theefoe on putting these values, we get 9 (9 N m C ) (6. C) F = m = 4 = 8 N N INTEXT QUESTIONS 5.. Two chages = 6µC and = 9 µc ae sepaated by a distance m. Detemine the magnitude of the foce expeienced by due to and also the diection of this foce. What is the diection of the foce expeienced by due to? 9

10 MODULE - 5 Electic Chage and Electic Field Electicity and. Thee ae thee point chages of eual magnitude placed at the thee cones of a ight angle tiangle, as shown in Fig. 5.. AB = AC. What is the magnitude and diection of the foce exeted on? B 5.3 ELECTRIC FIELD To explain the inteaction between two chages placed at a distance, Faaday intoduced the concept of electic field. The electic field E at a point is defined as the electic foce F expeienced by a positive test chage placed at that point divided by the magnitude of the test chage. Mathematically, we wite A Fig. 5. : Thee chages placed at the thee cones of a ight angle tiangle. C E = F (5.) This is analogous to the definition of acceleation due to gavity, g = F/m, expeienced by mass m in the gavitational field F. The electic field E is a vecto uantity and has the same diection as the electic foce F. Note that the electic field is due to an extenal chage and not due to the test chage. The test chage should, theefoe, be so small in magnitude that it does not distub the field due to extenal chage. (In pactice, howeve, even the smallest test chage will distub the extenal field.) Stictly speaking, mathematical definition given below is moe accuate : E = lim F (5.) In SI system, the foce is in newton and the chage is in coulomb. Theefoe, accoding to En.(5.), the electic field has the unit newton pe coulomb. The diection of E is same as that of F. Note that the action of electic foce is mediated though electic field. Let us now examine why the test chage should be infinitesimally small. Refe to Fig It shows a unifomly chaged metallic sphee with chage and a test chage (< < ). It means that chage density pe unit aea is same aound points A, B, C and D. The test chage must measue the foce F without distubing the chage distibution on the sphee. Fig. 5.6 (b) shows the situation when ~. In this case, the pesence of the test chage modifies the suface

11 Electic Chage and Electic Field B B P P F C A C A D D >> (a) (b) F MODULE - 5 Electicity and Fig. 5.6 : a) unifomly chaged metallic sphee and a test chage, and b) edistibution of chage on the sphee when anothe chage is bought nea it. chage density. As a esult, the electical foce expeienced by the test chage will also change, say fom F to F. That is, the foce in the pesence of test chage is diffeent fom that in its absence. But without, the foce cannot be measued. If is infinitesimally small in compaison to, the chage distibution on the sphee will be minimally affected and the esults of measuement will have a value vey close to the tue value. That is, F will be vey nealy eual to F. We hope you now appeciate the point as to why the test chage should be infinitesimally small. Let thee be a point chage. A test chage is placed at a distance fom. The foce expeienced by the test chage is given by F = k ˆ (5.) The electic field is defined as the foce pe unit chage. Hence E = k ˆ (5.3) If is positive, the field E will be diected away fom it. If is negative, the field E will be diected towads it. This is shown in Fig Fig. 5.7 : Diection of electic field due to positive and negative chages The pinciple of supeposition applies to electic field also. If thee ae a numbe of chages,, 3,..., the coesponding fields at a point P accoding to En. (5.3) ae

12 MODULE - 5 Electic Chage and Electic Field Electicity and E = k ˆ, E = k ˆ and E = k ˆ 3 The total field at point P due to all chages is the vecto sum of all fields. Thus, E= E E E 3 o E = N ˆ k (5.5) i= i i i whee i is the distance between P and chage i and ˆ i is the unit vecto diected fom to P. The foce on a chage in an electic field E is î F = E (5.6) Example 5.3 : The electic foce at some point due to a point chage = 3.5µC is N. Calculate the stength of electic field at that point. Solution : Fom E. (5.6) we can wite E = F 4 = 8.5 N C =.43 NC Example 5.4 : Thee eual positive point chages ae placed at the thee cones of an euilateal tiangle, as shown in Fig Calculate the electic field at the centoid P of the tiangle. Solution : Suppose that a test chage has been placed at the centoid P of the tiangle. The test chage will expeienced foce in thee diections making same angle between any two of them. The esultant of these foces at P will be zeo. Hence the field at P is zeo. A C Fig. 5.8 : Electic field at the centoid of an euilateal tiangle due to eual chages at its thee cones is zeo. P B INTEXT QUESTIONS 5.3. A chage Q is placed at the oigin of co-odinate system. Detemine the diection of the field at a point P located on a) x-axis b) y-axis c) x = 4 units and y = 4 units

13 Electic Chage and Electic Field. The Δ ABC is defined by AB = AC = 4 cm. And angle at A is 3. Two chages, each of magnitude 6 C but opposite in sign, ae placed at B and C, as shown in Fig Calculate the magnitude and diection of the field at A. 3. A negative chage is located in space and the electic field is diected towads the eath. What is the diection of the foce on this chage? 4. Two identical chages ae placed on a plane suface sepaated by a distance d between them. Whee will the esultant field be zeo? B A Fig. 5.9 C MODULE - 5 Electicity and 5.3. Electic Field due to a Dipole l Fig. 5.: Two unlike chages of eual magnitude sepaated by a small distance fom a dipole. Its SI unit is coulomb-mete. If two eual and opposite chages ae sepaated by a small distance, the system is said to fom a dipole. The most familia example is H O. Fig 5. shows chages and sepaated by a small distance l. The poduct of the magnitude of chage and sepaation between the chages is called dipole moment, p : p = l (5.7) The dipole moment is a vecto uantity. En. (5.7) gives its magnitude and its diection is fom negative chage to positive chage along the line joining the two chages (axis of the dipole). Having defined a dipole and dipole moment, we ae now in a position to calculate the electic field due to a dipole. The calculations ae paticulaly simple in the following cases. CASE I : Electic field due to a dipole at an axial point : End on position To deive an expession fo the electic field of a dipole at a point P which lies on the axis of the dipole, efe to Fig. 5..This is known as end-on position. The point chages and at points A and A B B ae sepaated by a distance l. The point O P E O is at the middle of AB. Suppose that E l point P is at a distance fom the mid point O. Then electic field at P due to Fig. 5. : Field at point P on the dipole axis at B is given by E = k ( l) in the diection AP 3

14 MODULE - 5 Electicity and Electic Chage and Electic Field Similaly, the electic field E at P due to is given by E = k in the diection PA ( l) The esultant field E at P will be in the diection of E, since E is geate than E [as ( l) is less than ( l)]. Hence k E = ( l) k ( l) = k ( ) ( ) l l = k ( l) ( l) ( l ) (a b) (a b) = 4ab (a b) (a b) = a b 4l = k ( l ) ( l) = k ( l ) p k ( l ) = whee dipole moment p = l. Since k = /4πε, we can ewite it as E = p 4 4πε ( l / ) If >> l, l²/² will be vey small compaed to. It can even be neglected and the expession fo electic field then simplifies to p E = 3 4πε (5.8) It shows that electic field is in the diection of p and its magnitude is invesely popotional to the thid powe of distance of the obsevation point fom the cente of the dipole. CASE II : Electic field due to a dipole at a point on the pependicula bisecto : Boad-on position Suppose that point P lies on the pependicula bisecto of the line joining the chages shown in Fig. 5.. Note that AB = l, OP =, and AO = OB = l. 4

15 Electic Chage and Electic Field MODULE - 5 E E E sin Electicity and A E O (a) P B E cos E cos Fig. 5. : a) Field at point P on the pependicula bisecto of the line joining the chages, and b) esolution of field in ectangula components. E (b) P E sin The angle θ is shown in Fig. 5.(a). Fom ight angled Δs PAO and PBO, we can wite AP = BP = l The field at P due to chage at B in the diection of BP can be witten as E = k l Similaly, the field at P due to chage at A in the diection of PA is given as E = k l Note that the magnitudes of E and E ae eual. Let us esolve the fields E and E paallel and pependicula to AB. The components paallel to AB ae E cos θ and E cos θ, and both point in the same diection. The components nomal to AB ae E sin θ and E sin θ and point in opposite diections. (Fig. 5.b) Since these component ae eual in magnitude but opposite in diection, they cancel each othe. Hence, the magnitude of esultant electic field at P is given by E = E cos θ E cos θ = k l cosθ k l cos θ l But cos θ =. Using this expession in the above esult, the electic ( l ) field at P is given by k E = ( l ) l ( l ) 5

16 MODULE - 5 Electicity and l = k 3/ ( l ) l = k 3 3/ ( l / ) Electic Chage and Electic Field But p = l. If >> l, the facto l / can be neglected in compaison to unity. Hence p E = 3 (5.9) 4πε Note that electic field due to a dipole at a point in boad-on position is invesely popotional to the thid powe of the pependicula distance between P and the line joining the chages. If we compae Ens. (5.8) and (5.9), we note that the electic field in both cases is popotional to / 3. But thee ae diffeences in details: The magnitude of electic field in end-on-position is twice the field in the boad-on position. The diection of the field in the end-on position is along the diection of dipole moment, wheeas in the boad-on position, they ae oppositely diected Electic Dipole in a Unifom Field A unifom electic field has constant magnitude and fixed diection. Such a field is poduced between the plates of a chaged paallel plate capacito. Pictoially, it is epesented by euidistant paallel lines. Let us now examine the behaviou of an electic dipole when it is placed in a unifom electic field (Fig 5.3). Let us choose x-axis such that the electic field points along it. Suppose that the dipole axis makes an angle θ with the field diection. A foce E acts on chage along the x diection and an eual foce acts on chage in the x diection. Two eual, unlike and paallel foces fom a couple and tend to otate the dipole in clockwise diection. This couple tends to align the dipole in the diection of the extenal electic field E. The magnitude of toue τ is given by E l Fig. 5.3 : A dipole in a unifom electic field. The foces on the dipole fom a couple and tend to otate it. τ = Foce am of the couple = E y = E l sin θ = pe sin θ E y E 6

17 Electic Chage and Electic Field In vecto fom, we can expess this esult to τ = p E (5.) We note that when θ =, the toue is zeo, and fo θ = 9, the toue on the dipole is maximum, eual to pe. So we may conclude that the electic field tends to otate the dipole and align it along its own diection. Example 5.5 : Two chages and, each of magnitude 6. 6 C, fom a dipole. The sepaation between the chages is 4 m. Calculate the dipole moment. If this dipole is placed in a unifom electic field E = 3. NC at an angle 3 with the field, calculate the value of toue on the dipole. Solution : The dipole moment p = d Since toue τ = pe sin θ, we can wite = (6. 6 C ) (4. m) = 4 6 Cm. τ = (4 6 cm) 3. NC ) sin 3º MODULE - 5 Electicity and = 7 4 Nm = 36 4 Nm If a dipole is placed in a non-unifom electic field, the foces on the chages and will be uneual. Such as electic field will not only tend to otate but also displace the dipole in the diection of the field Electic Lines of Foce (Field Lines) A vey convenient method fo depicting the electic field (o foce) is to daw lines of foce pointing in the diection of the field. The sketch of the electic field lines gives us an idea of the magnitude and diection of the electic field. The numbe of field lines passing though a unit aea of a plane placed pependicula the diection of the field is popotional to the stength of the field. A tangent at any point on the field lines gives the diection of the field at that point. Note that the electic field lines ae only fictitious constuction to depict the field. No such lines eally exist. But the behaviou of chages in the field and the inteaction between chages can be effectively explained in tems of field lines. Some illustative examples of electic field lines due to point chages ae shown in Fig 5.4. The field lines of a stationay positive chage point adially in outwad 7

18 MODULE - 5 Electicity and Electic Chage and Electic Field diection. But fo stationay negative chage, the lines stat fom infinity and teminate at the point chage in adially inwad diection (towads the point chage). You must undestand that the electic field lines in both cases ae in all diections in the space. Only those which ae in the plane containing the chage ae shown hee. (a) Fig. 5.4 : Electical field lines of single point chages : a) The field lines of positive chage, and b) the field lines of negative chage. Fig 5.5(a) shows a sketch of electic field lines of two eual and simila positive chages placed close to each othe. The lines ae almost adial at points vey close to the positive chages and epel each othe, bending outwads. Thee is a point P midway between the chages whee no lines ae pesent. The fields of the two chages at this point cancel each othe and the esultant field at this point is zeo. Fig. 5.5(b) depicts the field lines due to a dipole. The numbe of lines leaving the positive chage is eual to the numbe of lines teminating on the negative chage. (b) P (a) (b) Fig. 5.5 : Electic field lines due to a system of two point chages : a) Two positive chages at est, and b) The field lines due to a dipole stat fom the positive chage and teminate on the negative chage. You must emembe the following popeties of the electic field lines : The field lines stat fom a positive chage adially outwad in all diections and teminate at infinity. The field lines stat fom infinity and teminate adially on a negative chage. 8

19 Electic Chage and Electic Field Fo a dipole, field lines stat fom the positive chage and teminate on the negative chage. A tangent at any point on field line gives the diection of electic field at that point. The numbe of field lines passing though unit aea of a suface dawn pependicula to the field lines is popotional to the field stength on this suface. Two field lines neve coss each othe. MODULE - 5 Electicity and 5.4 ELECTRIC FLUX AND GAUSS LAW Let us conside a sphee of adius having chage located at its cente. The magnitude of electic field at evey point on the suface of this sphee is given by E = k The diection of the electic field is nomal to the suface and points outwad. Let us conside a small element of aea Δs on the spheical suface. Δs is a vecto whose magnitude is eual to the element of aea Δs and its diection is pependicula to this element (Fig.5.6). The electic flux Δφ is defined as the scala poduct of Δs and E : Δφ = E. Δs The total flux ove the entie spheical suface is obtained by summing all such contibutions: φ E = E Δs i. Δs i (5.) i Since the angle between E and Δs is zeo, the total flux though the spheical suface is given by φ E = k Σ Δs The sum of all elements of aea ove the spheical suface is 4π. Hence the net flux though the spheical suface is s E φ E = k 4π = 4 π k Fig

20 MODULE - 5 Electic Chage and Electic Field Electicity and On substituting fo k = /4πε, we get φ E = 4πε 4π = /ε (5.) The spheical suface of the sphee is efeed to as Gaussian suface. En. (5.) is known as Gauss law. It states that the net electic flux though a closed gaussian suface is eual to the total chage inside the suface divided by ε. Gauss law is a useful tool fo detemining the electic field. You must also note that gaussian suface is an imaginay mathematical suface. It may not necessaily coincide with any eal suface. Cal Fiedich Gauss ( ) Geman genius in the field of physics and mathematics, Gauss has been one of the most influential mathematicians. He contibuted in such divese fields as optics, electicity and magnetism, astonomy, numbe theoy, diffeential geomety, and mathematical analysis. As child podgy, Gauss coected an eo in his fathe s accounts when he was only thee yea old. In pimay school, he stunned his teache by adding the integes to within a second. Though he shun inteactions with scientific community and disliked teaching, many of his students ose to become top class mathematicians Richad Dedekind, Behad Riemann, Fiedich Bessel and Sophie Gemain ae a few among them. Gemany issued thee postal stamps and a mak bank note in his honou. A cate on moon called Gauss cate, and asteoid called Gaussia have been named afte him Electic Field due to a Point Chage Let us apply Gauss law to calculate electic field due to a point chage. Daw a spheical suface of adius with a point chage at the cente of the sphee, as shown in Fig s The electic field E is along the adial diection pointing away fom the cente and nomal to the suface of the Fig. 5.7 : Electic field on a spheical suface due to a chage at its cente

21 Electic Chage and Electic Field sphee at evey point. The nomal to the element of aea Δs is paallel to E. Accoding to Gauss law, we can wite MODULE - 5 Electicity and φ E = i E i.δs i = /ε Since cos θ = and E is same on all points on the suface, we can wite o φ E = E 4π² /ε = E 4π² E = 4πε (5.3) If thee is a second chage placed at a point on the suface of the sphee, the magnitude of foce on this chage would be so that F = E F = 4πε (5.4) Do you ecogmise this esult? It is expession fo Coulomb s foce between two static point chages Electic Field due to a Long Line Chage A line chage is in the fom of a thin chaged wie of infinite length with a unifom linea chage density σ l (chage pe unit length). Let thee be a chage on the wie. We have to calculate the electic field at a point P at a distance. Daw a ight cicula cylinde of adius with the long wie as the axis of the cylinde. The cylinde is closed at both ends. The suface of this cylinde is the gaussian suface and shown in Fig The magnitude of the electic field E is same at evey point on the cuved suface of the cylinde because all points ae at the same distance fom the chaged wie. The electic field diection and the nomal to aea element Δs ae paallel. Gaussian suface Fig. 5.8 : Electic field due to an infinite line of chages having unifom linea chage density. The gaussian suface is a ight cicula cylinde. s E

22 MODULE - 5 Electicity and Electic Chage and Electic Field Let the length of the gaussian cylinde be l. The total chage enclosed in the cylinde is = σ l l. The aea of the cuved suface of the cylinde is πl. Fo the flat sufaces at the top and bottom of the cylinde, the nomals to these aeas ae pependicula to the electic field (cos 9 = ). These sufaces, theefoe, do not contibute to the total flux. Hence φ E = Σ E. Δs E = E πl Accoding to Gauss law, φ E = /ε. Hence E E πl = /ε = σ l l/ε o E = σ l πε (5.5) Fig. 5.9 : Vaiation of E with fo a line chage This shows that electic field vaies invesely with distance. This is illustated in Fig Electostatic Filte You must have seen black smoke and dit paticles coming out of a chimney of a themal powe station o bick klin. The smoke consists of not only gases but lage uantities of small dust (coal) paticles. The smoke along with the dit is dischaged into the atmosphee. The dust paticles settle down on eath and pollute the soil. The gases contibute to global waming. These ae extemely injuious to living systems (health). It is theefoe essential that the dit is emoved fom smoke befoe it is dischaged into the atmosphee. A vey impotant application of electical dischage in gases by application of high electic field is the constuction of a device called Electostatic Filte o Pecipitato. dity gases The basic diagam of the device is shown hee. The cental wie inside a metallic containe is maintained at a vey high negative potential (about kv). The wall of the containe is connected to the positive teminal of a high volt battey and is eathed. A weight W keeps the wie staight in the cental pat. The electic field thus ceated is fom the wall towads the wie. The dit and gases ae passed dust exist Clean ai

23 Electic Chage and Electic Field though the containe. An electical dischage takes place because of the high field nea the wie. Positive and negative ions and electons ae geneated. These negatively chaged paticles ae acceleated towads the wall. They collide with dust paticles and chage them. Most of the dust paticles become negatively chaged because they captue electons o negative ions. They ae attacted towads the wall of the containe. The containe is peiodically shaken so that the paticles leave the suface and fall down at the bottom of the containe. These ae taken out though the exit pipe. The undesiable dust paticles ae thus emoved fom the gases and the clean ai goes out in the atmosphee. Most efficient systems of this kind ae able to emove about 98% of the ash and dust fom the smoke. MODULE - 5 Electicity and Electic Field due to a Unifomly Chaged Spheical Shell A spheical shell, by definition, is a hollow sphee having infinitesimal small thickness. Conside a spheical shell of adius R caying a total chage Q which is unifomly distibuted on its suface. We shall calculate the electic field due to the spheical chage distibution at points extenal as well as intenal to the shell. (a) Field at an extenal point Let P be an extenal point distant fom the cente Q O of the shell. Daw a spheical suface (called s Gaussian suface) passing though P and concentic O P E with the chage distibution. By symmety, the electic field is adial, being diected outwad as shown in Fig. 5.. The electic field Fig. 5. E is nomal to the suface element eveywhee. Its magnitude at all points on the Gaussian suface has the same value E. Accoding to Gauss law, Q ΣEΔ scos = ε o 4 Q ΔE π = ε o Q E = 4 πε 3

24 MODULE - 5 Electicity and Electic Chage and Electic Field Fom the esult we can conclude that fo a point extenal to the spheical shell, the entie chage on the shell can be teated as though located at its cente. The electic field deceases with distance. Instead of a spheical shell if we had taken a chaged solid conducting sphee, we would have obtained the same esult. This is because the chage of a conducto always esides on its oute suface. (b) Field at an Intenal Point Let P be an intenal point distant fom the cente of the shell. Daw a concentic sphee passing though the point P. Applying Gauss Law, o Q ΣEΔ scos = ε E 4π Q = ε E = as Q = the electic field at an intenal point of the shell is zeo. The same esult is applicable to a chaged solid conducting sphee. The vaiation of the electic field with the adial distance has been shown in Fig Electic Field due to a Plane Sheet of Chage Conside an infinite plane sheet of chage ABCD, chaged unifomly with suface chage density σ. E P A D B C E R E= 4 o Q R =R Fig. 5. P Fig. 5. S Q O E P 4 Fig. 5.3

25 Electic Chage and Electic Field Fo symmety easons, the electic field will be pependicula to the sheet, diected away fom it, if σ >. Let P be the point in font of the sheet whee we want to find the electic field. Daw a Gaussian suface in the fom of a cylinde with its axis paallel to the field and one of its cicula caps passing though P. The othe cicula cap of the cylinde lies symmetically opposite at P, on the othe side of the sheet, being situated at the same distance as P. The electic flux though both the cicula caps is MODULE - 5 Electicity and E Δ s E Δ s = EΔ s EΔs = EΔs The electic flux though the cuved suface of the Gaussian suface is E Δ s = EΔ s cos 9 =. Hence, the total electic flux though the Gaussian cylinde is φ E = E Δ s = EΔs As the chage enclosed by the Gaussian cylinde is σδs, using Gauss Law we have o EΔ s = σδs ε σ E = ε Please note that the electic field is independent of the distance fom the sheet. 5.5 VAN DE GRAAFF GENERATOR Van de Gaaff Geneato is an electostatic device that can poduce potential diffeences of the ode of a few million volts. It was named afte its designe Robet J. van de Gaaff. It consists of a lage hollow metallic sphee S mounted on an insulating stand. A long naow belt, made of an insulating mateial, like ubbe o silk, is wound aound two pulleys P and P as shown in Fig The pulley P is mounted at the cente of the sphee S while the pulley P is mounted nea the bottom. The belt is made to otate continuously by diving the pulley P by an electic moto M. Two comb-shaped conductos C and C, having a numbe of shap points in the shape of metallic needles, ae mounted nea the pulleys. 5

26 MODULE - 5 Electic Chage and Electic Field Electicity and S C P T Insulating Stand Powe Supply To Eath C P M Fig. 5.4 The needles point towads the belt. The comb-shaped conducto C is maintained at a high positive potential (~ 4 V) elative to the gound with the help of a powe supply. The uppe comb C is connected to the inne suface of the metallic sphee S. Nea the shap points of the comb-shaped conducto C, the chage density and electostatic field ae vey high. Lage electostatic field nea thei pointed ends causes dielectic beakdown of the ai, poducing ions (both positive and negative) in the pocess. This phenomenon is known as coona dischage. The negative chages fom the ai move towads the needles and the positive chage towads the belt. The negative chages neutalize some of the positive chages on the comb C. Howeve, by supplying moe positive chages to C, the powe supply maintains its positive potential. As the belt caying the positive chages moves towads C, the ai nea it becomes conducting due to coona dischage. The negative chages of the ai move towads the belt neutalizing its positive chages while the positive chages of the ai move towads the needles of the comb C. These positive chages ae then tansfeed to the conducting sphee S which uickly moves them to its oute suface. The pocess continues and positive chages keep on accumulating on the sphee S and it acuies a high potential. As the suounding ai is at odinay pessue, the leakage of chage fom the sphee takes place. In ode to pevent the leakage, the machine is suounded by an eathed metallic chambe T whose inne space is filled with ai at high pessue. 6

27 Electic Chage and Electic Field By using Van de Gaaff geneato, voltage upto 5 million volts (MV) have been achieved. Some geneatos have even gone up to ceation of such high voltages as MV. MODULE - 5 Electicity and Van de Gaaff geneato is used to acceleate the ion beams to vey high enegies which ae used to study nuclea eactions. INTEXT QUESTIONS 5.4. If the electic flux though a gaussian suface is zeo, does it necessaily mean that (a) the total chage inside the suface is zeo? (b) the electic field is zeo at evey point on the suface? (c) the electic field lines enteing into the suface is eual to the numbe going out of the suface?. If the electic field exceeds the value 3. 6 NC, thee will be spaking in ai. What is the highest value of chage that a metallic sphee can hold without spaking in the suounding ai, if the adius of the sphee is 5. cm? 3. What is the magnitude and diection of the net foce and net toue on a dipole placed along a a) unifom electic field, and b) non-unifom field. WHAT YOU HAVE LEARNT Electic chage is poduced when glass od is ubbed with silk o ubbe is ubbed with fu. By convention, the chage on glass od is taken positive and that on ubbe is taken negative. Like chages epel and unlike chages attact each othe. Coulomb s law gives the magnitude and diection of foce between two point chages : whee k = 4πε F = k = 9. 9 Nm²C ². The smallest unit of chage in natue is the chage on an electon : e =.6 9 C ˆ 7

28 MODULE - 5 Electicity and Electic Chage and Electic Field Chage is conseved and uantised in tems of electonic chage. The electic field E due to a chage at a point in space is defined as the foce expeienced by a unit test chage : E= F/ = k Supeposition pinciple can be used to obtain the foce expeienced by a chage due to a goup of chages. It is also applicable to electic field at a point due to a goup of chages. Electic dipole is a system of two eual and unlike chages sepaated by a small distance. It has a dipole moment p = ; the diection of p is fom negative chage to positive chage along the line joining the two chages. The electic field due to a dipole in end-on position and boad-on position is espectively given by ˆ and E = p E = 3 4πε 4 p πε 3. Electic field lines (line of foce) ae only a pictoial way of depicting field. Electic flux is the total numbe of electic lines of foce passing though an aea and is defined as φ E =E.A. Gauss s law states that the total flux passing though a closed aea is times the total chage enclosed by it. The electic field due to a line chage is given by σl E =. πε ε TERMINAL EXERCISE. A μ C chage is at x = cm and a 8 μ C ( ) chage is at x = 9 cm on the x-axis. Calculate the magnitude and diection of the foce on a chage of 8μC. What is the diection of foce on μc chage?. Two point Chages and sepaated by a distance of 3. m expeience a mutual foce of 6 5 N. Calculate the magnitude of foce when = =. What will be the magnitude of foce if sepaation distance is changed to 6. m? 8

29 Electic Chage and Electic Field 3. Thee ae two points A and B sepaated by a distance x. If two point chages each ae on the points A and B, the foce between them is F. The point chages ae now eplaced by two identical metallic sphees having the same chage on each. The distance between thei centes is again x only. Will the foce between them change? Give easons to suppot you answe. 4. The foce of epulsion between two point chages placed 6 cm apat in vaccum is 7.5 N. What will be foce between them, if they ae placed in a medium of dielectic constant k =.5? MODULE - 5 Electicity and 5. Compae the electical foce with the gavitational foce between two potons sepaated by a distance x. Take chage on poton as.6 9 C, mass of poton as.67 7 kg and Gavitational constant G = 6.67 Nm kg. 6. Fou identical point chages each ae placed at the fou cones (one at one cone) of a suae of side. Find the foce expeienced by a test chage placed at the cente of the suae. 7. When ae the electic field lines paallel to each othe? 8. How many electons should be emoved fom a metallic sphee to give it a positive chage = C. 9. Conside an electic dipole of = 3. 6 C and l = 4 m. Calculate the magnitude of dipole moment. Calculate electic field at a point = 6 6 m on the euatoial plane.. A Chage = 5-6 C is placed on a metallic sphee of adius R=3. mm. Calculate the magnitude and diection of the electic field at a point =5 cm fom the cente of the sphee. What will be the magnitude and diection of the field at the same point if 3. mm sphee is eplaced by 9. mm sphee having the same Chage.. A chage of 5μC is located at the cente of a sphee of adius cm. Calculate the electic flux though the suface of the sphee.. A poton is placed in a unifom electic field E = 8. 4 NC. Calculate the acceleation of the poton. 3. Two point chages and ae 3. cm apat and ( ) = μc. If the foce of epulsion between them is 75N, calculate and. 5. ANSWERS TO INTEXT QUESTIONS. (i)yes (ii) Chage = 3. 7 C. 9

30 MODULE - 5 Electicity and Electic Chage and Electic Field. A has chage Q. When A and B ae bought in contact, chage will get distibuted eually. (i) Yes., (ii) Q/ 3. = Since Ne =, we get N = = 3. 3 chages 5.. Q = 6μC, Q = µc and = m Since F = = 4πε (9 Nm C )(6 C) ( C) = 9 3 N 44m (i) diection fom to (ii) diection fom to. The foce at A due to chage at B, F = k whee AB = a a Since AB = AC, the foce at A due to chage at B is 5.3 F = k a R = F F = F R = F at 45. (a) E along the x axis. (b) along the y axis. (c) at 45 with the x axis. AB = AC = 4 cm k E = = E = Nm C ( C) (.4m) =.5 NC 5 3

31 Electic Chage and Electic Field The esultant of E and E will be paallel to BC. Hence R = E E E E cos 5 = E E cos (8-3) 3 = E E cos 3 = E = 4.73 N C. Diection will be paallel to BC in the diection B C. MODULE - 5 Electicity and E = 75 A 3 E 3. E is diected towads the eath. The foce on ve chage will be vetically upwads. 4. The field will be zeo at the mid point between the chage (i) Yes (ii) not necessaily (iii) Yes. Q. E = 4 πε Q = 4πε E = (3 6 NC ) 9 (9 Nm C ) (5 4 m ) = C 3. (a) F =, τ = (b) F τ = B C 3

32 MODULE - 5 Electicity and Answes to Poblems in Teminal Execise Electic Chage and Electic Field. 4 N towads negative x diection foce on µc chage is towads x diection.. = 4 3 C 4. 3 N 5. Electic foce is 36 times the gavitational foce. 6. zeo electons 9. 6 Cm..5 5 o Nc. 6 6 NC towads the cente, same field μm. 7.6 ms 3. 5 µc and 5 µc. 3