NATIONAL OPEN UNIVERSITY OF NIGERIA SCHOOL OF SCIENCE AND TECHNOLOGY COURSE CODE: PHY 204 COURSE TITLE: ELECTROMAGNETISM

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1 PHY4 MODULE NATIONAL OPEN UNIVERSITY OF NIGERIA SCHOOL OF SCIENCE AND TECHNOLOGY COURSE CODE: PHY 4 COURSE TITLE: ELECTROMAGNETISM

2 PHY4 ELECTROMAGNETISM Couse code: PHY 4 Couse Title: Electomagnetism Couse Adapted Fom: IGNOU Couse Wite: D. A. B. Adeloye Depatment of Physics Univesity of Lagos Akoka, Lagos Editoial Team: D. makanjuola Oki National Open Univesity of Nigeia Lagos D. A. B. Adeloye Depatment of Physics Univesity of Lagos Akoka, Lagos D. H. M. Olaitan Lagos State Univesity ojo Pogamme Leade: National Open Univesity of Nigeia Lagos NATIONAL OPEN UNVERSITY OF NIGERIA 4/6 Ahamadu Bello Way Victoia island Lagos.

3 PHY4 MODULE MODULE Unit Unit Unit 3 Unit 4 Unit 5 Macoscopic Popety of Dielectics Capacito Micoscopic Popeties of Dielectics Magnetism of Mateials I Magnetism of Mateials II UNIT MACROSCOPIC PROPERTIES OF DIELECTRICS CONTENTS. Intoduction. Objectives 3. Main Content 3. Simple Model of Dielectic Mateial 3. Behaviou of a Dielectic in an Electic Field 3.. Non-Pola and Pola Molecules 3.. Polaisation Vecto P 3.3 Gauss's Law in a Dielectic Medium 3.4 Displacement Vecto 3.5 Bounday Conditions on D and E 3.6 Dielectic Stength and Dielectic Beakdown 4. Conclusion 5. Summay 6. Tuto-Maked Assignment 7. Refeences/Futhe Reading. INTRODUCTION In the pevious couses in electomagnetism, you have leant the concepts of electic field, electostatic enegy and the natue of the electostatic foce. Howeve, fo easons of simplicity we confined, fo most pats, ou consideations of these concepts fo chages that ae placed in vacuum. Fo example, Coulomb's law of electostatic foce is the electic field due to a distibution of chages given in Unit 4; efe to the situation in which the suounding medium is vacuum. Of equal impotance is the situation in which the electical phenomenon occus in the pesence of a mateial medium. Hee we must distinguish between two diffeent situations, as the physics of these situations is completely diffeent. The fist situation is when the medium consists of insulating mateials i.e., those mateials which do not conduct electicity. The second situation coesponds to the case when the medium consists of conducting mateials, i.e. mateials like metals which ae conducto of 3

4 PHY4 ELECTROMAGNETISM electicity. The conducting mateials contain electons which ae fee to move within the mateial. These electons move unde the action of an electic field and constitute cuent. We shall study conducting mateials and electic fields in conducting mateials at a late stage. In the pesent unit, you will study the electic field in the pesence of an insulato. In these mateials thee ae pactically no fee electons o numbe of such electons is so small that the conduction is not possible. In 837, Faaday expeimentally found that when an insulating mateial also called dielectic (such as mica, glass o polystyene etc.) is intoduced between the two plates of a capacito, it is found that the capacitance is inceased by a facto which is geate than one. This facto is known as the dielectic constant (K) of the mateial. It was also found that this capacitance is independent of the shape and size of the mateial but it vaies fom mateial to mateial. In the case of glass, the value of the dielectic constant is 6, while fo wate it is 8. All the electons in these mateials ae bound to thei espective atoms o molecules. When a potential diffeence is applied to the insulatos no electic cuent flows; howeve, the study of thei behaviou in the pesence of an electic field gives us vey useful infomation. The choice of a pope dielectic in a capacito, the undestanding of double efaction in quatz o calcite cystals is based on such studies. Natual mateials, such as wood, cotton, natual ubbe, mica ae some popula examples of electic insulatos. A lage numbe of vaieties of plastics ae also good dielectics. Dielectic substances ae insulato (o non-conducting) substances as they do not allow conduction of electicity though them. In this unit fist, of all we will study a simple model of dielectic mateial and deduce a elationship between applied field E and the dipole moment p of a molecule/atom. You will lean about electic polaisation in a dielectic mateial and define polaisation vecto P. You must have studied Gauss's law in vacuum. You will now apply it to a dielectic medium. Hee we will also intoduce you to a new vecto known as the electic displacement vecto D. Afte that we will discuss the continuity of D and E at the inteface between two dielectics. In ecent yeas dielectic mateials have become impotant especially due to thei lage scale use in electic and electonic devices. Thee is high demand fo the impovement of opeating eliability of these devices. Reliability of these devices is measued to a geat extent by the quality of electical insulation. In the last section you will study dielectic stength and beak down in dielectics. 4

5 PHY4 MODULE In the next unit you will study the details of capacitos, especially the capacitance of a capacito, enegy stoed in a capacito, capacito with dielectic, diffeent foms of capacitos, etc.. OBJECTIVES At the end of this unit, you should be able to: explain the behaviou of a dielectic in an electic field deduce Gauss's law fo a dielectic medium define dielectic polaisation and classify dielectics as pola and nonpola explain Displacement Vecto (D) and elate it to the electic field stength (E) define dielectic constant state and deive the bounday conditions on E and D explain dielectic stength and dielectic beakdown. 3. MAIN CONTENT 3. Simple Model of the Dielectic Mateial You must be awae that: evey mateial is made up of a vey lage numbe of atoms/molecules, an atom consists of a positively chaged nucleus and negatively chaged paticles, with electons evolving aound it, the total positive chage of the nucleus is balanced by the total negative chage of the electons in the atom, so that the atom, as a whole, is electically neutal w..t. any point pesent outside the atom, a molecule may be constituted by atom of the same kind, o of diffeent kinds. To undestand the polaisation we shall conside a cude model of the atom. A simple cude model of an atom is shown in Fig... Fig.. Model of an Atom 5

6 PHY4 ELECTROMAGNETISM The nucleus is at the cente, and the vaious electons evolving aound it can be thought of as a spheically symmetic cloud of electons. Fo points outside the atom this cloud of electons can be egaded as concentated at the cente of the atom as a point chage. In most of the atoms and molecules, the centes of positive and negative chages coincide with each othe, wheeas, in some molecules the centes of the two chages ae located at diffeent points. Such molecules ae called pola molecules. Futhe, we note that in dielectics, all the electons ae fimly bound to thei espective atoms and ae unable to move about feely. In the absence of an electic field, the chages inside the molecules/atoms occupy thei equilibium positions. The aangement of the molecules in a dielectic mateial is shown in Fig... Fig.. The aangement of the atoms in a dielectic mateial The chage centes ae shown coincident at the cente of the sphee. Keeping this pictue of a dielectic in mind we shall poceed to study its behaviou in an electic field in the next section. 3. Behaviou of a Dielectic in an Electic Field You have seen in Section. that in a dielectic mateial, the centes of positive and negative chages of its atoms ae found to coincide at the cente of the sphee. It is shown in Fig..3. 6

7 PHY4 MODULE Fig..3 Atoms in which the centes of chages ae coincident with the cente of the sphees A chage expeiences a foce in the pesence of an electic field. Theefoe, when a dielectic mateial is placed in an electic field, the positive chage of each atom expeiences a foce along the diection of the field and the negative chage in a diection opposite to it. This esults in small displacement of chage centes of the atoms o molecules. This is also tue of molecules whose chage centes do not coincide in the absence of an electic field. The sepaation of the chage centes due to an applied field E is shown in Fig..4. Electic dipole moment pe unit volume is known as polaisation Fig..4 The sepaation of the chage centes due to an applied field E. This phenomenon is called polaisation. Thus when an electically neutal molecule is placed in an electic field, it gets polaised, with positive chages moving towads one end and negative chages towads the othe. The othewise neutal atom thus becomes a dipole with a dipole moment, which is popotional to electic field. Now we conside anothe kind of molecule in which the chage centes do not coincide as shown in Fig..5. 7

8 PHY4 ELECTROMAGNETISM Fig..5 A dielectic mateial in which chage centes do not coincide Due to this eason the molecule aleady possesses a dipole moment. Such mateials ae called pola mateials. Fo such mateials, let the initial oientation of the dipole axis be AOB as shown in Fig..6. Fig..6 Molecule possessing a dipole moment Now an electic field E is applied. This field pulls the chage centes along lines paallel to its diection. Thus the electic field exets a toque on the dipole causing it to eoient in the diection of the field. In the absence of an electic field these pola mateials do not have any esultant dipole moment, as the dipoles of the diffeent molecules ae oiented in andom diections due to themal agitation. When an electic field is applied, each of these molecules eoients itself in the diection of the field, and a net polaisation of the mateial esults. The eoientation o polaisation of the medium is not pefect again due to themal agitation. Thus polaisation depends both on field (linealy) and tempeatue. SELF ASSESSMENT EXERCISE What ae dielectics? In what espects do they diffe fom conducto? 8

9 PHY4 MODULE 3.. Non-Pola and Pola Molecules We have consideed two types of molecules. One in which the cente of positive chages coincide with the cente of negative chages. The molecule as a whole has no esultant chage. Molecules of this type ae called Non-pola. Examples of Non-pola molecules ae ai, hydogen, benzene, cabon, tetachloide, etc. The second type is the one in which the cente of positive chages and the cente of negative chages do not coincide. In this case the molecule possesses a pemanent dipole moment. This type of molecule is called a Pola Molecule. Examples of pola molecules ae wate, glass, etc. Thus we see that, a Non-pola molecule acquies a Dipole Moment only in the pesence of an electic field: wheeas in a Pola Molecule the aleady existing dipole moment oients itself in the diection of the extenal electic field. Even in pola molecules, thee is some induced dipole moment due to additional sepaation of chages, howeve this effect is compaatively much smalle than the eoientation effect and is thus ignoed fo pola molecules. 3.. Polaisation Vecto P Let us study the effect of an electic field on a dielectic mateial by keeping a dielectic slab between two paallel plates as shown in Fig..7. The electic field is set up by connecting the plates to a battey. We limit ou discussion to a homogeneous and isotopic dielectic. A homogeneous and isotopic dielectic is one in which the electical popeties ae the same at all points in all diections. The applied electic field displaces the chage centes of the constituent molecules of the dielectic. The sepaation of the chage centes is shown in Fig..7. We find that the negative chages of one molecule faces the positive chages of its neighbou. Thus within the dielectic body, the chages neutalise. Howeve, the chages appeaing on the suface of the dielectic ae not neutalised. These chages ae known as Polaisation Suface Chages. The entie effect of the polaisation can be accounted fo by the chages which appea on the ends of the specimen. The net suface chage, howeve, is bound and depends on the elative displacement of the chages. It is easonable to expect that the elative displacement of positive and negative chages is popotional to the aveage field E inside the specimen. 9

10 PHY4 ELECTROMAGNETISM Fig..7: Effect of an Electic field on a dielectic mateial by keeping a dielectic slab between two paallel plates Fom Fig..7, we find that these polaisation chages appea only on those sufaces of the dielectic which ae pependicula to the diection of the field. No suface chages appea on faces paallel to the field. Such a situation occus only in the special case of a ectangula block of dielectic kept between the plates of a paallel plate condense. It is shown late in this section that suface density of bound chages depends on the shape of the dielectic mateial. The polaisation of the mateial is quantitatively discussed in tems of dipole moment induced by the electic field. Recall that the moment of a dipole consisting of chages q and -q sepaated by a displacement d is given by P q d. It is known fom expeiments that the induced dipole moment (p) of the molecule inceases with the incease in the aveage field E. We can say that p is popotional to E o p α E (.) whee α is the constant of popotionality known as Molecula/Atomic Polaisability. Let us now define a new vecto quantity which we shall epesent by P and shall call it polaisation of the dielectic o just polaisation. Polaisation P is defined as the electic dipole moment pe unit volume of the dielectic. It is impotant to note that the tem polaisation is used in a geneal sense to descibe what happens in a dielectic when the dielectic is subjected to an extenal electic field. It is also used in this specific sense to denote the dipole moment pe unit volume. Let us fist conside a special case of n polaised molecules each with a dipole moment p pesent pe unit volume of a dielectic and let all the dipole moments be paallel to each othe. Then fom the definition of P P np

11 PHY4 MODULE Fom the above definition, units of P ae Units of P Coulombm m Coulomb m C m 3 In geneal, P is a point function depending upon the coodinates. In such cases, whee the ideal situation mentioned above is not satisfied, we would conside an infinitesimal volume V thoughout which all the p's can be expected to be paallel and wite the equation N Pi P lim ( N is the numbe of dipoles in volume V) (.la) V V i Hee V is lage compaed to the molecula volume but small compaed to odinay volumes. Thus, although p is a point function, it is a space aveage of p. The diection of p will, of couse, be paallel to the vecto sum of the dipole moment of the molecules within V. In such a case whee the p's ae not paallel, as in a dielectic that has pola molecules, Eq. (.la) still holds as the defining equation fo p. SELF ASSESSMENT EXERCISE Show that the dipole moment of a molecule p and the dipole moment pe unit volume ae elated by P np whee n is the numbe of molecules pe unit volume of the dielectic. To undestand the physical meaning of P, we conside the special case of a ectangula block of a dielectic mateial of length L and cosssectional aea A. Fig..8 epesents such a block. Fig..8 Suface polaisation chages on a ectangula block of dielectic

12 PHY4 ELECTROMAGNETISM Fig..8a Suface polaisation chages. Actual displacement of chage on ight is dx cosθ Let p be the suface density of polaisation chages, viz., the numbe of chages on a unit aea o chage/unit aea on the suface. The total numbe of polaisation chages appeaing on the suface A σ Induced dipole moment Volume of the slab AL By definition dipole moment pe unit volume P Aσ L (.) Induced dipole moment P AL (.3) Now we can compae the magnitudes of Eqs. (5.) and (5.3) to obtain the magnitude P of the polaisation vecto to be P σ p (.4) Thus, the suface density of chages appeaing on the faces pependicula to the field is a measue of P, the polaisation vecto. Eq. (.4) is tue fo a special geomety when the dielectic mateial is a ectangula block. Fo a block shown in Fig..8a the suface on the ight is not pependicula to P. The nomal unit vecto (n) to the suface makes an angle θ with P. If the chages ae displaced by a distance dx the effective displacement is dx cosθ fo the suface on the ight. If n is the numbe of chaged paticle and q is the chage on each paticle, then the suface chage density σ is given by σ n qdx cos θ P n (.5) p P n whee q is the positive chage on each atom/molecule and P n is the component of P nomal to the suface on the ight. This also shows why no chages appea on the sufaces paallel to the applied field (θ 9 ) and on the left side of the block the angle between P and n, the unit

13 PHY4 MODULE vecto nomal to the suface is 8 ; the suface chage density is negative. Fo an ideal, homogeneous and isotopic dielectic, the polaisation P is popotional to the aveage field E, i.e., P χε E (.6) Whee χ P / ε E and is known as electical susceptibility. This elation is elated to Eq. (.); Eq. (.) efes to one molecule, wheeas Eq. (.6) efes to the mateial. Thus the latte is a macoscopic vesion of Eq. (.). The constant ε is included fo the pupose of simplifying the late elationships. The elation (.6) equies that P be linealy elated to the aveage (micoscopic) field. This aveage field would be the extenal applied field as modified by the polaisation suface chages. The susceptibility is a chaacteistic of the mateial and gives the measue of the ease with which it can be polaised, it is simply elated to α fo the nonpola mateials. Fom SAE σ np p P np using Eq. (, 4), we get The dipole moment pe atom in this case p 3.3 Gauss' Law in a Dielectic q dxcosθ You have studied Gauss law in vacuum. Hee, we shall modify and genealise it fo dielectic mateial. Conside two metallic plates as shown in Fig..9. Let E be the electic field between these two plates. Now, we intoduce a dielectic mateial between the plates. When the dielectic is intoduced, thee is a eduction in the electic field, which implies a eduction in the chage pe unit aea. Since no chage has leaked off fom the plates, such a eduction can be only due to the induced chage appeaing on the two sufaces of the dielectic. Due to this eason, the dielectic suface adjacent to the positive plate must have an induced negative chage, and the suface adjacent to the negative plate must have an induced positive chage of equal magnitude. It is shown in Fig..9. 3

14 PHY4 ELECTROMAGNETISM Fig..9 Induced chages on the faces of a dielectic in an extenal field Fo the sake of simplicity, you conside the chage on the suface of dielectic mateial as shown in Fig..9a. Now we apply Gauss' flux theoem to a egion which is wholly within the dielectic such as the Gaussian volume at egion of Fig..9a. Fig..9a Gaussian volumes at and inside a dielectic. The displacement of chages at the sufaces pependicula to the applied field is shown The net chage inside this volume is zeo even though this mateial is polaised. The positive chages and negative chages ae equal. Fo this volume the flux of field though the suface is zeo. We can wite suface at E ds S ε χ P ds (.7) This shows that "lines" of P ae just like lines of E except fo a constant ( ε ). Instead of this Gaussian volume, suppose we take anothe one at egion. In this Gaussian volume one suface is inside the dielectic and the othe is outside it The cuved suface is paallel to the lines of field (E o P). Fo the suface of this Gaussian volume outside the mateial, P is nonexistent. Howeve, lines of P must teminate inside the Gaussian volume. Hence the net flux of P is finite and negative as shown in Fig..9a since the component of P nomal to the suface, i.e. P and n 4

15 PHY4 MODULE σ p the suface chage density ae equal to each othe in magnitude, the suface integal P ds P ds σ ds (.8) n q p p Whee q p is the chage inside the Gaussian volume. Thus, the flux of P is equal to the negative of the chage included in the Gaussian volume. Notice the diffeence in the flux of P and the flux of E. Now we can genealise Gauss flux theoem. Since the effects of polaised matte can be accounted fo by the polaisation suface chages, the electic field in any egion can be elated to the sum of both fee and polaisation chages. Thus in geneal closed suface E ds ( q f + q p ) (.9) ε whee q p epesents fee chages and q p the polaisation chages. SELF ASSESSMENT EXERCISE 3 Two paallel plates of aea of coss section of cm ae given equal 7 and opposite chage of. C. The space between the plates is filled with a dielectic mateial, and the electic field within the dielectic is V/m. What is the dielectic constant of the dielectic and the suface chage density on the plate? Using Gauss' theoem fo vectos this suface integal can be conveted into a volume integal. Thus the above equation becomes ( ) ( ρ f + V E dv ρ p )dv (.) ε V whee ρ f and ρ p ae espectively the fee and bound chage densities. As this is tue fo any volume, the integands can be equated. Thus ε E ρ f + ρ p (.) The flux of ρ though the closed suface is given by (See equation.8) P ds q p ρ dv p 5

16 PHY4 ELECTROMAGNETISM which can be witten using Gauss' flux theoem ε E ρ f P ε E + P ρ f ( ε E + P) ρ f D ρ f (.) whee D ε E + P is known as the electic displacement vecto. (Note that. is aleady Gauss's Law.) SELF ASSESSMNENT EXERCISE 4 Show that Eq. (.) educes to Eq. (.) when P. The dimension of D is the same as that of P. The units of D ae C m. Fom Eqs. (.) and (.) we obseve that the souce of D is the fee chage density ρ, wheeas the souce of E is the total chage density ρ f + ρ p. f When we wite P ε E (see Eq..5), we have D ( + χ) ε E (.3) Whee ε ( + χ) is known as the elative pemittivity of the medium. Anothe usual fom of electic displacement vecto D is given by D ε E (.4) whee ε ε ε. Eq. (.4) povides the elation between Electic displacement D and electic field E. SELF ASSESSMENT EXERCISE 5 Conside two ectangula plates of aea of a coss section of m. Each is kept paallel to the othe. The sepaation between, 3 them is m and a voltage of V is applied acoss these plates. If a mateial of dielectic constant 6. is intoduced within the egion between the two plates, calculate: 6

17 PHY4 MODULE (i) (ii) (iii) (iv) Capacitance The magnitude of the chage stoed on each plate. The dielectic displacement D The polaisation 3.4 Displacement Vecto D It is one of the basic vectos fo an electic field that depends only on the magnitude of fee chage and its distibution. In Section.4, we intoduced a new vecto D and called it Displacement Vecto (o) Electic Displacement. We found (see Sec..4) that the electic displacement is defined by D ε E + P ; Gauss' law in dielectic is given by D ds q dv. Fo an isolated chage q, kept at the cente of a dielectic sphee of adius, we find that the Gauss' flux theoem gives (being a case of spheical symmety) Which gives (4π )( D) q f D q / 4π (.5) D ε E we get E q / 4π ε (.6) Fom (.6) it follows that the foce F, between two chages q and q, kept at a distance in a dielectic medium is given by qq (.7) 4π ε F and the expession fo the potential φ at a distance fom q is φ q / 4πε (.8) When we compae Eq..6 with the coesponding expession fo E in fee space, Eq..7 and.8 show simila expessions fo Coulomb foce and potentials. We may find that in all these expessions, ε has been eplaced by ε in a dielectic medium. 7

18 PHY4 ELECTROMAGNETISM SELF ASSESSMENT EXERCISE 6 Two lage metal plates each of aea sq. mete face each othe at a distance one mete apat. They cay equal and opposite chages on thei sufaces. If the electic intensity between the plates is 5 newton pe coulomb, calculate the chage on the plates. With this backgound, we may wongly conclude that D fo a dielectic medium is same as E fo fee space. It is theefoe impotant to clealy distinguish between these two vecto quantities: E is defined as the foce acting on unit chage, iespective of whethe a dielectic medium is pesent o not. It is to be calculated taking into account the fee o extenal chages as well as the induced chages of the medium. On the othe hand D is defined as D ε E + P, and it is a vecto like electic field, but is detemined only by fee o extenal chages. Note fom Eqs. (.5) and (.6) that the value of D does not depend upon the dielectic constant while the value of E as well as the foce between the chages involve ε. The quantity D d S is usually efeed to as the electic flux though the element of aea d S. Fo this eason D is also known as electic flux density. Fom the integal fom of Gauss' law in dielectics, we find that the total flux is q, though an aea suounding a chage q, and this flux is unalteed by the pesence of a dielectic medium. This is not tue in the case of total flux of electic intensity, since E ds S q ε Since D is a vecto, we may daw lines of displacement in the same way as we daw the lines of foce. The numbe of lines of displacement passing though a unit aea is popotional to (D). These lines of displacement begin and end only on fee chages, since the oigin of D is the conduction chages/chage density (see Section.4). Again by using Gauss' law it can be shown easily that the lines of displacement ae continuous in a space containing no fee chages. In othe wods, at the bounday of two dielectics, if thee ae no fee chages the lines of D ae continuous, while the lines of E ae not continuous because lines of electic foce can end on both fee and polaisation chages. This behaviou of D and E is dealt with in a geate 8

19 PHY4 MODULE detail in the next section. These ules ae contained in two Bounday conditions at the inteface between two dielectic media. 3.5 Bounday Conditions on D and E We wish to detemine the elationships that E and D must satisfy at the inteface between two dielectics. Hee, we will assume that thee ae only polaisation chages at the inteface i.e., since the dielectics ae ideal they have no fee electons, and thus thee is no conduction chage at the inteface. Late, these bounday conditions will be useful fo poving laws of eflection and efaction of electomagnetic waves. Now we will detemine the bounday condition fo vecto D. Bounday conditions give the way in which the basic vectos change when they ae incident on the suface of discontinuity in dielectic behaviou. Bounday Condition fo D We apply the Gauss' law fo dielectics to a small cylinde in the shape of a pill box which intesects the bounday between two dielectic media and whose axis is nomal to the bounday. Fig.. shows the cylinde. Let the height of the pill box be vey small compaed to its coss sectional aea. The contibution to D d S comes fom the components of D nomal to the bounday. That is, D d S DndS Dn ds (.9) Fig.. Bounday condition fo D between two dielectic media whee D n and D n ae the nomal components of D in media and espectively. D n is opposite to the diection of the nomal to ds in the medium ( ε ). Futhe D d S since thee ae no fee chages on the bounday suface. D D (.) n n 9

20 PHY4 ELECTROMAGNETISM Thus the nomal components of electical displacement vectos ae continuous acoss the bounday (having no fee chages). D ds D n ds whee n is the unit vecto along the outwad dawn nomal to the aea ds. This epesentation gives the bounday condition as n D n D which gives Eq. (.). Othewise the bounday conditions becomes D cosθ D cosθ whee θ and θ ae the angles between n and D and n and D espectively. Bounday condition fo E We shall make use of the consevative natue of the electic field in this case. To obtain the bounday condition fo E, we calculate the wok done in taking a unit chage aound a ectangula loop ABCDA, Fig.. shows such a loop. The sides BC and AC of the loop ae vey small. As the wok done in taking a unit chage ound a closed path is zeo (consevative foce) E dl (.) ABCDA ε D C ε A B Fig.. Bounday condition fo E between two dielectic media Let Etand E t be the tangential components of E in the media and espectively as shown in Fig... Then, ABCDA E dl E dl E dl (.) AB t CD whee l AB CD. t Using Eq.. in Eq.. we get E E (.3) t t

21 PHY4 MODULE Eq..3 states that the tangential component of the electic field is continuous along the bounday. Note that to calculate wok done, we need foce, which is elated to the electic field. The bounday condition contained in Eq. (.3) may be witten in the vecto fom as n E n E whee E, E ae the coesponding electic fields and n is the unit vecto nomal to the bounday. SELF ASSESSMENT EXERCISE 7 Pove Eq..3a using equation.3. Using the vecto identity. E dl E) nds ( n Suface Note on Eq..3a We wite Eq. (.3a) as ( E) ds E θ (.3b) sinθ E sin whee θ and θ ae the angles between n and E and n and E espectively in the media and. This is yet anothe fom of the bounday condition. We wite Eq..3b as o D sinθ ε D sinθ ε θ D sin D sinθ ε ε (.3c) Eq. (.3c) implies that the tangential component of D is not continuous acoss the bounday. SELF ASSESSMENT EXERCISE 8 Show that the nomal component of E is discontinuous acoss a dielectic bounday.

22 PHY4 ELECTROMAGNETISM 3.6 Dielectic Stength and Beakdown We have seen that unde the influence of an extenal electic field, polaisation esults due to the displacement of the chage centes. In ou discussion, we have teated the phenomenon as an elastic pocess. A question that aises in ou minds is, "what would happen if the applied field is inceased consideably? One thing that is cetain is that the chage centes will expeience a consideable pulling foce. If the pulling foce is less than the binding foce between the chage centes, the mateial will etain the dielectic popety and on emoving the field the chage centes will etun to thei equilibium positions. If the pulling foce just balances the binding foce, the chages will just be able to ovecome the stain of the sepaation and any slight imbalance will loosen the bonds between the electons and the nucleus. A futhe incease of the applied field will esult in the sepaation of the chages. Once this happens the electons will be acceleated. The fast moving electons will collide with the othe atoms and multiply in numbe. This will esult in the flow of conduction cuent. The minimum potential that causes the chage sepaation is known as the beakdown potential and the pocess is known as the dielectic beakdown. Beakdown potential vaies fom substance to substance. It also depends on the thickness of the dielectic (thickness measued along the diection of the field). The field stength at which the dielectic is about to beak down is known as the Dielectic Stength. It is measued in kilovolts pe mete. Knowledge of the beakdown potential is vey impotant fo pactical situations, as in the use of capacitos in electical cicuits. When a dielectic is subjected to a gadually inceasing electic potential, a stage will be eached when the electon of the constituent molecule is ton away fom the nucleus. Now the dielectic beaks down, viz., loses its dielectic popeties, and begins to conduct electicity. The beakdown voltage is the applied potential diffeence pe unit thickness of the dielectic when the dielectic just beakdown. 4. CONCLUSION In this unit we have examined the behaviou of dielectics and the deduction of Gauss s law. In addition, we have explained the tems, dielectic beakdown and dielectic stength as well as defined dielectic constant.

23 PHY4 MODULE 5. SUMMARY When an electic field is applied to an insulating mateial, it gets polaised. This means that a dipole moment is ceated in the mateial. This dipole moment is also exhibited as a suface chage density. Electic dipole moment pe unit volume is known as polaisation. At the atomic level, the polaisation of a medium takes place in two ways, as thee ae two kinds of molecules: pola and nonpola. In nonpola molecules the centes of positive and negative chages lie at the same point and thei inheent dipole moment is zeo. In pola molecules the positive and negative chage centes lie at diffeent points and consequently thee is an inheent dipole moment associated with the molecules, though the net chage of the molecule is zeo. Fo a dielectic medium, it is convenient to intoduce anothe vecto elated to E and P, This is called the displacement vecto D defined as D ε E + P Fo the analysis of dielectic behaviou, the elation between the polaisation vecto P and the total electic field E is impotant. Fo an ideal, homogeneous and isotopic dielectic, the elation is expessed as P ε χ ee The constant χ e is known as the electic susceptibility of the medium. The constant α, coesponding to the susceptibility χ e is known as the atomic (o molecula) polaisability when we conside the polaisation of a single atom (o molecule). In a polaised piece of a dielectic, the volume chage density ρ p ( div P ) and the suface chage density σ p ae given by P n o P n. The pesence of dielectic leads to the modification of the Gauss' law. It's modification is ε D n ds q whee q is the total unit fee o extenal chage o div D ρ 3

24 PHY4 ELECTROMAGNETISM whee D depends only on the magnitude of fee chage and distibution. The geneal elation between the vectos D, E and P can be used to define the dielectic constant K and pemittivity ε of dielectic medium. Using the pemittivityε, the elation between D, P and E can be expessed in the linea fom D ε E P ε k ) E ( ε ε ) E ( The vectos E and D satisfy cetain bounday conditions on the inteface between two dielectic media. These conditions ae: (i) (ii) The tangential component of E is the same on each side of the bounday, i.e., E t E and t The nomal component of D is same on each side of the bounday, i.e., D n Dn Dielectic stength is the applied potential diffeence pe unit thickness of the dielectic when the dielectic just beaks down. 6. TUTOR-MARKED ASSIGNMENT () Calculate the elative displacement of the nucleus of the molecule modelled in Fig.. (spheically symmetic molecule) when it is subjected to an extenal electic field and hence, its polaisability. () Suppose two metallic conducting plates ae kept as shown in Fig..3. Fig..: Model of atom. Fig..a The aea of coss section of each plate is. m and they ae apat. The potential diffeence between them in vacuum V is 3 volts, and 4

25 PHY4 MODULE it deceases to volts when a sheet of dielectic cm thick is inseted between the plates. Calculate the following: (a) The elative pemittivity K of the dielectic, (b) its pemittivity, ε, (c) its susceptibility χ, (d) the electic intensity between the plates in vacuum (hee it is given that Intensity Voltage acoss the plate/aea of Coss section, (e) the esultant electic intensity in the dielectic, (f) the electic intensity set up by the bounded chages. Fig..3 Two metallic conducting plates (a) and (b) with dielectic mateial. (3) Conside two isotopic dielectic medium and sepaated by a chage fee bounday as shown in Fig..4 Fig..4: Line of foce acoss the bounday between two dielectics Now, an electic vecto E goes fom medium and entes into medium. If i is the angle of incidence and is the angle of eflection, pove that tan i tan ε ε (4) Show that the polaisation chage density at the inteface between two dielectics is ε ε p' ε ε n E 5

26 PHY4 ELECTROMAGNETISM 7. REFERENCES/FURTHER READING IGNOU (5). Electicity and Magnetism Physics PHE-7, New Delhi, India. 6

27 PHY4 MODULE UNIT CAPACITOR CONTENTS. Intoduction. Objectives 3. Main Content 3. Capacitance 3. Paallel Plate Capacito o Condense 3.. Enegy Stoed in a Capacito 3.3 Paallel Plate Capacito with Dielectics 3.3. Voltage Rating of Capacito 3.4 Capacitance of Cylindical Capacito 3.5 Capacitos in Seies and Paallel 3.5. Combination of Capacito in Paallel 3.5. Combination of Capacito in Seies 3.6 Enegy Stoed in a Dielectic Medium 3.7 Pactical Capacitos 3.7. Fixed Capacito 3.7. Ceamic Capacito Electolytic Capacito Vaiable Ai Capacito/Gang Capacito Guad Ring Capacito 4. Conclusion 5. Summay 6. Tuto-Maked Assignment 7. Refeences/Futhe Reading. INTRODUCTION You have studied in you ealie classes that the potential of a conducto inceases as the chage placed on it is inceased. Mathematically we wite Q φ o Q Cφ (.) whee C is the popotionality constant. We call this constant C the capacity o capacitance. We also call any device that has capacitance a the capacito (condense). You ae aleady familia with this device. We change the capacitance in ou adio-tansisto while opeating the 'tuning' knob and get the adio station of ou choice. Capacitos ae used in many electical o electonic cicuits, they povide coupling between amplifie stages, smoothen the output of powe supplies. They ae used 7

28 PHY4 ELECTROMAGNETISM in motos, fans, in combination with inductances to poduce oscillations which when tansmitted become adio signals/tv signals etc. Besides these, capacitos have a vaiety of applications in electic powe tansmission. In the pesent unit, we shall lean about capacitance, capacitos of diffeent foms, enegy stoed in a capacito, and the woking pinciple of a capacito. We have studied the macoscopic popeties of dielectics in Unit. Hee we will study the effect on the capacitance of a capacito, when a dielectic is placed between the two plates of a capacito. Then we will intoduce some pactical capacitos. In next unit we will study the micoscopic popeties of the dielectics.. OBJECTIVES At the end of this unit, you should be able to: define capacitance of a capacito descibe capacitos of diffeent geometies and obtain mathematical expession fo thei capacitance able to calculate the enegy stoed in a capacito descibe the effect of intoducing a dielectic mateial in a capacito obtain expessions fo the effective capacitance of gouping a numbe of capacitos in seies and in paallel descibe pactical capacitos such as a guad condense and an electolytic capacito. 3. MAIN CONTENT 3. Capacitance A capacito o a condense is an electonic device fo stoing electical enegy by allowing chages to accumulate on metal plates. This electical enegy is ecoveed when these chages ae allowed to move away fom these plates into the cicuit of which the capacito foms a pat. Any device which can stoe chages is a capacito. Fo example, an insulated conducting spheical shell of adius R can stoe chages; hence it can be used as a condense. Let us see how it woks as a capacito. If a chage Q is placed on it, the oute suface of the shell becomes an equipotential suface. The potential of the oute suface of the shell is given by 8

29 PHY4 MODULE Q φ (.) 4πε R with infinity as zeo potential. Instead of infinity we can egad the gound (eath) as zeo potential. Then the capacitance of this shell (w..t. gound) is Q R φ 4πε Coulomb/Volt (.3) C The unit of capacitance C in SI system is the faad. Faad Coulomb/Volt (.4) If R cm in the above spheical shell its capacity in faads is (4 πε ). Faad Thus it is clea fom this that if a capacito is to be made with one unit (faad) capacity it has to have huge dimensions ( m in the above case). Pactical foms of condenses have small dimensions and smalle units such as picofaad ( 6 Faad) and micofaad ( Faad) ae moe commonly used. The symbolic epesentation of a capacito is o o. The above example of a spheical conducto as a capacito is given only to illustate the concept. Howeve, the most commonly used pactical fom of condenses always has a system of two metal sheets (cicula, cylindical o ectangula) kept close to each othe with an insulato sepaating the two sheets. This system has the ability to have lage capacity without having the coesponding lage dimensions. You will lean moe about this in detail in the next section. 3. The Paallel Plate Capacito o Condense This is the simplest and most commonly used fom of a condense. A paallel plate condense consists of two ectangula o cicula sheets (plates) of a metal aanged paallel to each othe, sepaated by a distance d. The value of d is usually vey small and an insulating mateial is nomally inseted between the two sheets. See Fig... A chage Q (positive) placed on the uppe plate distibutes equally on this plate to make it an equipotential suface. The lowe plate is shown gounded. The lowe plate is theefoe at gound potential (zeo potential). Because of electostatic induction an equal amount of negative chage appeas on the uppe side of the lowe plate. This 9

30 PHY4 ELECTROMAGNETISM induced negative chage pulls up almost all the positive chage placed on the uppe plate to the lowe side of the uppe plate. Thus the electic field now gets confined to the space between the two plates: the positive chage acting as souces and the negative chage as sink (the lines of foce oiginate on the positive chages and end on negative chages). The induced negative chage is equal lo the amount of positive chage because of the zeo field equiement inside the mateial of the conducting sheets. Besides, both the metal sheets ae equipotential sufaces. The lines of foce field lines ae nomal to these sheets except at edges. See Fig... Since all the field lines oiginate fom the uppe plate and end on the lowe plate, the value of the electic field, E is unifom in the space between the plates except at the edge. The edge effects ae negligible if the aea of the plates, A, is lage compaed to d. Since E is unifom the potential diffeence between the uppe and the lowe plates is given by Fig.. Paallel plate condense A and B ae the metal plates sepaated at a distance 'd'. φ φ E dl Ed whee φ, φ efe to the potentials of uppe and lowe plates espectively. As the lowe plate is eathed, φ ; φ Ed (.5) To evaluate E let us use Gauss's theom. Suppose we evaluate the electic flux fo a closed cylindical suface EFGH of base aea S with its axis nomal to the plate. See Fig.. 3

31 PHY4 MODULE Fig.. Gaussian closed cylinde EFGH One of the hoizontal sufaces is inside the metal and the othe in the space between the plates; the cuved faces ae paallel to the field lines. Thee is no flux though EF as the field inside the conducting suface is zeo. Similaly, thee is zeo flux though EH and FG as the cuved sufaces of the Gaussian cylinde ae paallel to the field lines. Since the potential is defined as the wok done pe unit chage, the wok done in moving a small chage δ q against a chage potential φ will be wok done φδ q. But φ q / C. The total wok done in chaging a capacito to Q coulombs is given by Total wok done Q qdq C q C The flux though the suface HG of aea S is equal to ES. Since E is along the nomal to the aea, hence, we can apply Gauss' theoem. Accoding to Gauss theoem S ES, ε σ E (.6) ε whee, σ is the chage pe unit aea on the condense plate. The potential φ of the uppe plate is Ed fom Eq (.). The total chage Q is σ A Q A C ε φ d (.7) By keeping a small value fo d, the capacity C can be inceased. In the above deivation we have taken the medium between the plates to be vacuum. The above aangement has the advantage of the electic field being unaffected by the pesence of othe chages o conductos in the 3

32 PHY4 ELECTROMAGNETISM neighbouhood of the capacito. Moeove, if the aea A of the plates is much geate than d the coection fo the capacitance due to the nonunifom field at the edges is negligible. SELF ASSESSMENT EXERCISE Suppose we have the distance of sepaation between the plates, what happens to the capacitance? SELF ASSESSMENT EXERCISE Find the chage on a pf capacito when chaged to a voltage of 4 V. In the next subsection you will lean about the enegy stoed in a capacito. 3.. Enegy Stoed in a Capacito The wok done, W in assembling a chage Q by adding infinitesimal incements of chage is given by: W Qφ (.8) Whee φ is the final potential of the chaged body. In the case of a capacito of capacitance C, this wok, done in placing a chage Q on the capacito must also be given by simila expession, i.e., W Qφ (.9) This can be witten in tems of the capacitance C Q / φ as W Q Cφ Joules. C This wok is stoed up in the electic field as potential enegy. SELF ASSESSMENT EXERCISE 3 Show that in a paallel plate capacito of aea A and the sepaation of plates by a distance d in vacuum the enegy stoed in the (space) volume of the electic field between the plates is given by Qφ. 3

33 PHY4 MODULE 3.3 Paallel Plate Capacito with Dielectics When a dielectic slab is inseted between the paallel plates of a condense the capacity inceases. The polaised dielectic slab ABCD (see Fig..3) educes the electic field E inside the dielectic by a facto ( / ε ) whee ε is the elative pemittivity as discussed in the last unit: This can be poved by computing the electic field by using Gauss' law fo electic displacement, D inside the dielectic ABCD. Recall the Gaussian cylinde used in evaluating E in Section.. The flux of D is now given by (only fee chages contibute to the flux) Fig..3 Dielectic slab between capacito plates DS σ S (.) as the bound suface chages do not contibute to this flux and D ε E ε (.) fo an isotopic unifomly polaised dielectic. Thus the field σ E (.3) ε ε The potential diffeence between the plates is equal to Ed, whee d is now the thickness of the slab filling the entie space between the plates. The capacitance now becomes Q σ A ε ε A C (.4) φ Ed Ed The value of the capacitance C inceases by the facto ε, which is the elative pemittivity of the dielectic mateial. Fom Eq. (.4) we note that the capacitance of a paallel plate capacito inceases with the incease in suface aea (A) of the plates and also with the decease of the distance sepaating the plates. 33

34 PHY4 ELECTROMAGNETISM The effect of intoducing a dielectic in between the plates inceases the capacitance (Q ε > ). Thus the inclusion of a dielectic enables the capacito to hold moe chages at a given potential diffeence between the plates. We ewite Eq. (.4) as C ε A/( d / ε ) (.5) and compae it with Eq. (.7). We find that a dielectic of thickness d has an equivalent fee space thickness ( d / ε ). This obsevation will be useful late when we deal with the capacito in which the space in between the plates is only patially filled with a dielectic. SELF ASSESSMENT EXERCISE 4 Find the capacitance of the paallel plate capacito consisting of two paallel plates of aea.4 m 3 each and placed m apat in fee space. A capacito is shown in Fig..4 in which a dielectic slab of thickness t is inseted between the plates kept apat at a distance d. We wite the capacitance of this capacito, on the basis of the equivalent fee space thickness of the dielectic. We find the fee space thickness between the plates ( d t), whee t is the thickness of the dielectic mateial. This t is equivalent to t / ε in fee space. The capacito of Fig..4 is equivalent to a capacito with fee space between the plates, with the sepaation of ( d t + t / ε ). We wite the expession fo the capacitance as A C ε d t + t / ε (.6) Fig..4 34

35 PHY4 MODULE Fig..5 The equivalent capacito Now we will obtain Eq. (.6) with anothe simple method. Let the voltage acoss the capacito which is shown in Fig..4 be V. When a dielectic of thickness 't ' is intoduced between the two plates of the capacito, the distance between the positive plate of the capacito and the uppe suface of the dielectic is say d and fom lowe suface of dielectic to negative plate of the capacito is d. Now assume that the voltage between positive plate and uppe suface of the dielectic is V, the voltage between uppe and the lowe suface of the dielectic is V and the lowe plate of the dielectic to the negative plate of the capacito is V 3. The total voltage V acoss capacito is the sum of these thee voltages i.e., V V + + V V3 Let E be the field inside the dielectic. Then d E, V E t / ε and V 3 d E V V ( d + d ) E + E t / ε Fom the figue d d + d + t d + d ( d t) Fom the above equation we get V ( d t) E + E t / ε Using Eq. (.5), we get that in this case d [( d t) + t / ε ] 35

36 PHY4 ELECTROMAGNETISM Fom Eq. (.5), we get ε A C d ε A [( d t) + t / ε ] We can also find that the atio of the capacitance with dielectic between the plates to the capacitance with fee space between the plates is equal to the elative pemittivity, viz., ε Capacit. with dielectic between plates Capacit. with fee space between plates The elative pemittivities ( ε ) of some of the most common mateials ae given in Table.. Table.: Relative pemittivity ( ε ) of some common mateials Ai Casto Oil Mica Glass Bakelite Pape Pocelain Quatz Wate SELF ASSESSMENT EXERCISE 5 A dielectic of elative pemittivity 3 is filled into the space between the plates of a capacito. Find the facto by which the capacitance is inceased, if the dielectic is only sufficient to fill up 3/4 of the gap Voltage Rating of a Capacito Capacitos ae designed and manufactued to opeate at a cetain maximum voltage, which depends on the distance between the plates of the capacito. If the voltage is exceeded, the electons jump acoss the space between the plates and this can esult in pemanent damage to the capacito. The maximum safe voltage is called the woking voltage. The capacity and the woking voltage (WV) is maked on the capacito in the case of bigge capacitos and indicated by the colou code (simila to 36

37 PHY4 MODULE that of esistance) in the case of capacitos having low values of capacitance. 3.4 Capacitance of a Cylindical Capacito In Section.3, we have calculated the capacitance of a paallel plate capacito. Anothe impotant fom of capacito is a cylindical capacito. This is shown in Fig..6a. A section of this capacito is shown in Fig..6b. It is made up of two hollow coaxial cylindical conductos of adii a and b. The space between the cylindes is filled with a dielectic of elative pemittivity ε. Pactical foms of such capacitos ae: (i) a coaxial cable, in which the inne conducto is a wie and the oute conducto is nomally a mesh of conducting wie sepaated fom the inne conducto by an insulato (usually plastic). In Fig..6b, the diection of the field lines is adial, viz., nomal to the suface of the cylinde. Small lines in between the two cylindes, show the diection of fixed line. Fig..6: (a) Cylindical capacito {b) coss section of the cylindical capacito (ii) the submaine cable, in which a coppe conducto is coveed by polystyene (the oute conducto is sea wate). Since both the inne and oute cylindes ae conductos, they ae equipotential sufaces. The field is adial (nomal to the suface of the cylinde): Because of cylindical symmety we conclude that the capacitance is popotional to the length of the cylinde (as the length inceases, the aea of the plot inceases). We shall now find the capacitance pe unit length of the capacito. A B D Fig..7 Gaussian suface ABCD Let the chage pe unit length placed on the inne cylinde of the capacito be λ. The oute cylinde is gounded. An equal and C 37

38 PHY4 ELECTROMAGNETISM opposite amount of chage appeas on the inne side of the oute cylinde. This is because of the zeo field in the conducto. To evaluate the field let us conside a coaxial closed cylindical suface ABCD of unit length and of adius. See Fig..7. The electic field is nomal to the inne cylindical suface and is also confined to the space between the cylindes. The flux of electic displacement vecto, D, though the bottom and top sufaces of this Gaussian cylinde ABCD is zeo as D is paallel to these faces. The flux of D is only though the cuved suface of ABCD and as D is nomal to this at all points; the flux though this closed Gaussian suface is given by D d S ( π ) D δ l (.7) Now D ε ε E fo isotopic unifomly polaised dielectics. Using Gauss' law we get π D (π ) ε ε E λ (.8) whee λ is the fee chage enclosed by the Gaussian suface. Thus λ E (.9) π ε ε To find the capacitance, we equie the potential diffeence between the two cylindes. The expession fo potential diffeence is given by a φ Ed (.) b Fou ou case, Eq. (.) becomes a φa φ b b Ed b a λ d π ε ε λ b d πε ε a 38

39 PHY4 MODULE λ ln( b / a) πε ε (.) As the oute cylinde is gounded, φ b. Now, capacitance pe unit length, C is given by λ / φ a πε ε ln( b / a) (.3) Note: In the expession fo the capacitance pe unit length of a cylindical capacito, Eq. (.3), we find that the capacitance depends on the atio of the adii and on thei absolute values. SELF ASSESSMENT EXERCISE 6 Two cylindical capacitos ae of equal length and have the same dielectic. In one of them the adii of the inne and oute cylindes ae 8 and cm, espectively and in the othe they ae 4 and 5 cm. Find the atio of thei capacitances. 3.5 Capacitos in Seies and Paallel In Section.5, we have seen the method of finding the capacitance pe unit length of a cylindical capacito. We multiply the capacitance pe unit length by the length fo cylindical capacitos and get the capacitance. Now we can conside a cylindical capacito of length units as consisting of two cylindical capacitos of unit length joined end to end so that the inne cylindes ae connected togethe and the oute cylindes also get connected similaly. This is shown in Fig..8. Fig..7 A long cylindical capacito seen as a paticula combination of unit cylindical capacito We find immediately that in such a combination the chage on the capacito is doubled and so the capacitance is also doubled since the potential diffeence emains constant. Two capacitos connected in paallel (symbolic epesentation) ae shown in Fig..8a. 39

40 PHY4 ELECTROMAGNETISM Fig..8a Two capacitos connected la paallel In this combination, we find that the potential diffeence between the plates emains the same; the chage on each capacito adds up (moe aea is available fo stoing chages). We can find an equivalent capacito that holds the same chage when kept at the same potential diffeence as the combinations of the capacitos. The capacitance of that capacito is known as the Effective Capacitance of the combination. Befoe we poceed futhe, we note that capacitos can be gouped o combined in anothe way too. Hee altenate plates of the capacitos ae connected to the succeeding capacito so that they fom a seies. Fig..9 shows the combination; it is known as combination of capacitos in seies. Fig..9 Capacitos in Seies If a voltage souce is connected acoss the two end plates of the fist and last capacitos of the seies, equal chages will be induced in each capacito wheeas the potential diffeence acoss each capacito will depend upon its capacitance. We shall find the mathematical fomulas fo the equivalent capacitance of the combination of capacitances in paallel and in seies. 4

41 PHY4 MODULE 3.5. Combination of Capacitos in Paallel Fig.. shows the combination of thee capacitos in paallel. Fig.. Capacitos in paallel Hee, C, C and C 3 ae the capacitances of the individual capacitos, Q, Q and Q 3 ae espectively the chages on them and φ is the potential diffeence between the plates of each capacito. We take C to be the effective capacitance of the combination. The total chage Q of the paallel combination is equal to Q Q + + Q Q3 Since φ is same fo this equivalent C of the paallel combination, Q Q C φ + Q + φ Q3 Q Q Q3 + + φ φ φ C + C + (.4) C3 Thus the effective capacitance of the paallel combination of capacito is equal to the sum of the individual capacitances Combination of Capacitos in Seies Fig.. shows the combination of thee capacitos in seies. Fig.. Capacitos In seies and the equivalent capacitos 4

42 PHY4 ELECTROMAGNETISM Hee C, C and C 3 ae the capacitances of the individual capacitos. The application of a voltage will place a chage + Q on one plate which induces a chage Q on the othe plate. The intemediate plates acquie equal and opposite chages, because of electostatic induction. The potential dop acoss each will be invesely popotional to its capacitance. (Since C Q / φ gives φ Q / C, since Q is fixed φ / C ). Thus φ, φ and φ 3, the potential dops acoss the capacitos ae such that φ / C, φ / C and φ 3 / C3. Now we eplace the capacitos by a single capacito of capacitance C that holds the chage Q when subjected to a potential diffeence φ φ + φ + φ3. This capacitance C is known as the effective capacitance of the combination. We now wite C Q /φ o / C φ / Q. But φ φ + φ + φ3. Theefoe, i.e. φ + φ + φ3 C Q C + + (.5) C C C 3 Thus fo capacitos connected in seies the ecipocals of the capacitances add up to give the ecipocal of the effective capacitance. SELF ASSESSMENT EXERCISE 7 Detemine the equivalent capacitance of the netwok shows in Fig.. and the voltage dop acoss each of the capacito of the seies of capacitos. Fig.. 4

43 PHY4 MODULE SELF ASSESSMENT EXERCISE 8 Calculate the effective capacitance of thee capacitos aanged in such a way that two of them C and C ae in seies and the thid, C 3, is in paallel with this seies combination. 3.6 Stoed Enegy in a Dielectic Medium In Section.3., we have studied that the enegy stoed in a paallel plate capacito is given as U Cφ We know that and C φ ε d Ed A Putting these values in the above Eq. we get ε A d U E d ε ( Ad) E o U v ε E (since Ed v ) This is the enegy pe unit volume. When a dielectic of elative pemittivity ε fills the space between the plate of the capacito, then the effective capacitance is given by C die ε ε A d 43

44 PHY4 ELECTROMAGNETISM The enegy stoed in a capacito with a dielectic mateial is given by U Cφ ε ε A ( d E) d U u ε ε E ( A d) ε ε E ( A d u ) In the case of a paallel plate condense, the enegy stoed pe unit volume is ε E, which becomes ε ε E E D with the dielectic mateial. Whee D is the electic displacement in the dielectic. We have consideed hee the case of a linea dielectic whee E and D ae in the same diection. Howeve, thee ae dielectics in which E and D ae not in the same diection. Thus the enegy stoed pe unit volume in a dielectic medium is given by E D Joules/m (.6) 3.7 Pactical Capacitos We shall now study some of the capacitos that ae commonly in use. Capacitos may be boadly classified into two goups i.e., fixed and vaiable capacitos. They may be futhe classified accoding to thei constuction and use. The following ae the classifications of the capacito. Now, we will discuss each type of the capacito one by one. 44

45 PHY4 MODULE 3.7. Fixed Capacitos These have fixed capacitance. These ae essentially paallel plate capacitos, but compact enough to occupy less space. In thei make they consist of two vey thin layes of metal coated on the suface of mica o pape having a unifom coating of paaffin. The mica o pape foms the dielectic between the conductos. They ae shown in Fig..3. Fig..3 Fixed capacitos This aangement is olled up to the compact fom. Usually they ae piled up in paallel to give a lage capacitance. Though paaffin-waxed pape capacitos ae cheape, they absob a good amount of powe. Fo this eason these capacitos ae used in altenating cuent cicuits, adiosets, etc Ceamic Capacitos These ae low loss capacitos at all fequencies. Ceamic mateials can be made to have vey high elative pemittivity. Fo example, teflon has ε 8 but by the addition of titanim the value of ε becomes and on adding baium titanate the value of ε may be inceased to 5,. Each piece of such dielectic is coated with silve on the two sides to fom a capacito of lage capacitance. Yet anothe advantage with the ceamic dielectics is that they have negative tempeatue coefficient. Ceamic capacitos ae widely used in tansisto cicuits Electolytic Capacitos An electolytic capacito consists of two electodes of aluminium, called the positive and the negative plates. The positive plate is electolytically coaled with a thin laye of aluminium oxide. This coating seves as the dielectic. The two electodes ae in contact though the electolyte which is a solution of glyceine and sodium (o a paste of boates, fo example, ammonium boate). Thee ae two types of electolytic capacitos the wet type and the dy type. 45

46 PHY4 ELECTROMAGNETISM In the wet type the positive plate (A) is in the fom of a cylinde to pesent a lage suface aea. This is immesed in the electolyte (E) contained in a metal can (M). This can act as a negative plate. It is shown in Fig..4. In the dy type both plates ae in the fom of long stips of aluminium foils. Aluminium oxide is deposited electically on one (A) of the foils. This is kept sepaated fom the othe (B) by cotton gauze (C) soaked in the electolyte. It is then olled up to a cylindical fom. The oxide films on aluminium offe a low esistance to cuent in one diection and a vey high esistance in the othe diection. Hence an electolytic capacito must be placed in a DC cicuit such that the potential of the oxide plate is always positive elative to the othe plate. It is shown in Fig..5 Fig..4 Wet type capacito (electolytic) Fig..5 Dy type electolytic capacito Vaiable Ai Capacito/Gang Capacito A vey common capacito whose capacitance can be vaied continuously is used fo tuning in a adio station. The capacity of this capacito can be unifomly vaied by otating a knob (diffeent foms of such a type of capacito ae shown in Fig. (.6)). 46

47 PHY4 MODULE Fig..6 Vaiable ai capacito The capacito consists of two sets of semicicula aluminium plates. One set of plates is fixed and the othe set of plates can be otated with the knob. As it is otated, the moving set of plates gadually gets into (o comes out of) the intespace between the fixed set. The aea of ovelap between the two sets of plates can thus be unifomly vaied. This changes the capacitance of the capacito. The ai between the plates acts as the dielectic. Usually it consists of two condenses attached to the same knob (ganged). When the knob is otated the vaiation of C in both the plates takes place simultaneously. This is widely used in wieless sets and electonic cicuits. See Table. fo a compaative ange of voltages fo diffeent types of condenses. SELF ASSESSMENT EXERCISE 9 What is a vaiable capacito? Give an example of a vaiable capacito with a solid dielectic Guad Ring Capacito In Section. we calculated the capacitance of a paallel plate condense. We neglected the nonunifomity of electic field at the edges. It is possible to get ove the poblem of edge effects by using a guad ing capacito. In this capacito a ing R is used aound the uppe plates of the paallel plate capacito. The inne diamete of the ing is slightly lage than the diamete of the capacito plate. The diamete of the othe capacito plate is equal to the oute diamete of the ing. Now the edge effects ae absent as fa as the plates and ae concened. In estimating the capacitance of the guad ing capacito, we take the effective aea of the plates as equal to the sum of the aea of the plate A and half the aea of the gap between A and R. In Table., the capacity ange, max. Rating voltage and use of diffeent types of capacitos ae shown. 47

48 PHY4 ELECTROMAGNETISM 4. CONCLUSION In unit, we have defined and descibed capacitos and expessions fo thei capacitance have been obtained. The effects of adding dielectic mateials on thei popeties have been highlighted while thei pactical uses have been descibed. 5. SUMMARY Any device which can stoe chage is a capacito. The capacity of capacito is given by, Q C φ ε d A Whee the symbols have thei usual meaning. Type of Dielectic Capacitance Range Max. Rating Voltage Remaks Pape 5 pf- µ F 5KV Cheap, used in cicuits whee losses ae not impotant. Mica 5 pf-.5 µ F KV High quality, used in low cicuit Ceamic.5 pf-. µ F 5 KV High quality used in low loss pecision cicuit whee miniatuisation is impotant. Electolytic µ F- µ F 6 V at small Used whee lage (Aluminium Oxide) capacitance capacitance is needed. The enegy stoed in a capacito is given by W Cφ Q C Joules The symbols have thei usual meanings. If you intoduce an insulato of thickness t between the two plates of a capacito, then the esultant capacity is given by C ε A /( d t + t / ε ) The maximum safe voltage is called ating voltage of a capacito. 48

49 PHY4 MODULE The capacitance of a cylindical capacito, pe unit length is given πε ε by ln( b / a) If two capacitos C and C ae connected in seies, then the CC esultant capacity is given by C C + C The esultant capacity of two capacitos C and C, when connected in paallel is given by C C + C The enegy stoed in a dielectic medium is given by E D Pactical capacitos ae made in diffeent ways, to suit the paticula application. Layes of conducting foil and pape olled up give a cheap fom of capacito, mica and metal foil stands high electic field but ae moe expensive. Electolytic capacitos, in which the dielectic is a vey thin oxide film deposited electolytically, give vey lage capacitance. Ceamic capacitos ae useful in tansisto cicuits whee voltages ae low but small size and compactness ae vey desiable. 6. TUTOR-MARKED ASSIGNMENT () A capacito has n simila plates at equal spacing, with the altenate plates connected togethe. Show that its capacitance is equal to ( n ) ε ε A/ D. () What potential would be necessay between the paallel plates of a capacito sepaated by a distance of.5cm in ode that the gavitational foce on a poton would be balanced by the electic 7 field? Mass of poton.67 kg. (3) A capacito is made of two hollow concentic metal sphees of adii a and b (b>a). The oute sphee is eathed. See Fig..8. Find the capacity. 49

50 PHY4 ELECTROMAGNETISM Fig..8 (4) In the aangement shown in Fig..9, find the values of the capacitances such that when a voltage is applied between the teminals A and B no voltage diffeence is set up between teminals C and D. Fig..9 (5) Two capacitos one chaged and the othe unchaged ae joined in paallel. Show that the final enegy is less than the initial enegy and deive the fomula fo the loss of enegy in tems of the initial chages and the capacitances of the two capacitos. 7. REFERENCES/FURTHER READING IGNOU (5) Electicity and Magnetism. Physics PHE-7, New Delhi, India. 5

51 PHY4 MODULE UNIT 3 MICROSCOPIC PROPERTIES OF DIELECTRICS CONTENTS. Intoduction. Objectives 3. Main Content 3. Micoscopic Pictue of a Dielectic in a Unifom Electic Field Review 3.. Definition of Local Field 3. Detemination of Local Field: Electic Field in Cavities of a Dielectic 3.3 Clausius-Mossotti Fomula 3.3. Polaisation in a Gas 3.3. Relation between Polaisability and Relative Pemittivity 3.4 Relation between the Polaisability and Refactive Index 3.5 Behaviou of Dielectic in Changing o Altenating Fields 3.6 Role of Dielectic in Pactical Life 4. Conclusion 5. Summay 6. Tuto-Maked Assignment 7. Refeences/Futhe Reading. INTRODUCTION In Unit, we have studied the macoscopic (aveage) behaviou of a dielectic in an electic field. We also found that the field is alteed within the body of the dielectic. This can be accounted fo by the chages appeaing on the suface of the dielectic in the case of an isotopic mateial. In Unit, the macoscopic study of the dielectic behaviou was used to study the incease of capacitance in a condense when a dielectic is placed between the plates of the condense. In the pesent unit, we will descibe the micoscopic pictue of a dielectic in which we will define the local field ( E loc ), and the aveage macoscopic field inside the dielectic ( E i ). Futhe, we will deive the elationship between the local field and the macoscopic field. We will also study the effects of polaisation in nonpola and pola molecules and deive the famous Clausius-Mossotti fomula fo polaisation of these molecules. Then we will deive Clausius-Mossotti equation fo a gas. We will also study the elationship between polaisability and elative pemittivity. Afte that, we will deive the elationship between polaisability and efactive index. As you know that capacitos ae used in altenating fields, so we will also study the effect of an altenating 5

52 PHY4 ELECTROMAGNETISM field on a dielectic. In the last section of this unit we will study the ole of dielectics in ou daily life.. OBJECTIVES At the end of this unit, you should be able to: define the local field and elate it with polaisation, find the macoscopic field within the dielectic and elate it to polaisation. elate the macoscopic electic field, the local field and the micoscopic field within the dielectic, wite Clausius-Mossotti equation fo a liquid and a gas, establish a elationship between polaisability and Refactive index, discuss the ole of dielectics in daily life. 3. MAIN CONTENT 3. micoscopic pictue of a dielectic in a unifom electic field-eview In Unit you have studied the aveage (macoscopic) behaviou of dielectics. In this section, we will study the micoscopic pictue of a dielectic in a unifom electic field. Let us conside a dielectic in a unifom electic field as shown in Fig. 3.. Fig. 3. A molecule in a dielectic medium In an electic field, the electons and atomic nuclei of the dielectic mateial expeience foces in opposite diections. We know that the electons in a dielectic cannot move feely as in a conducto. Hence each atom becomes a tiny dipole with the positive and negative chage centes slightly sepaated. Taking the chage sepaation as a, the chage as q the dipole moment p in the diection of field associated with the atom o molecule 5

53 PHY4 MODULE p qa (3.) Eq. (3.) gives the dipole moment induced in the atom/molecule by the field. Hence we call it the induced dipole moment. If thee ae n such dipoles in an element of volume V of the mateial, we can define the polaisation vecto P as the (dielectic) dipole moment pe unit volume as npv p V Within the dielectic the chages neutalise each othe, the negative chage of one atom/molecule is neutalised by the positive chage of its neighbou. Thus within the bulk of the mateial, the electic field poduces no chage density but only a dipole moment density. Howeve, at the suface this chage cancellation is not complete, and polaisation chage densities of opposite signs appeas at the two sufaces pependicula to the field. Now what is the consequence of the appeaance of polaisation chages? The consequence of this is that the electic field inside the dielectic is less than the electic field causing the polaisation. The polaisation chages give ise to an electic field in the opposite diection. This field opposes the electic field causing polaisation. It is shown in Fig. 3.. Fig. 3. Field Inside a dielectic Hence we conclude that inside the dielectic, the aveage electic field is less than the electic field causing polaisation. Howeve, the macoscopic o aveage field is not a satisfactoy measue of the local field esponsible fo the polaisation of each atom. Let us denote the field at the site o location of the atom o molecule as the local field. In the next section, we will calculate the local field inside a dielectic. 53

54 PHY4 ELECTROMAGNETISM 3.. Definition of Local Field In this section we will define the local field in a dielectic mateial. This is the field on a unit positive chage kept at a location o site fom which an atom o molecule has been emoved povided the othe chages emain unaffected. Fig.3.3 shows a site in a unifomly polaised medium fom which a molecule/atom is emoved when all othe chages ae kept intact at thei positions. Fig. 3.3 A site in a unifomly polaised medium The extent of the chage sepaation depends on the magnitude of the local field. Hence we conclude that the induced dipole moment, p, is diectly popotional to the local field, E. Thus we have, P α E loc (3.3) Whee α is the constant of popotionality and is known as atomic/molecula polaisability and E the local field. To use Eq. (3.3) we equie the value of loc loc E loc. 3. Detemination of local field: electic fields in cavities of a Dielectic The polaisation of dense mateials such as liquids and many solids changes the electic field inside the mateial. The field expeienced by an individual atom/molecule depends on the polaisation of atoms in its immediate vicinity. The actual value of the field vaies apidly fom point to point. Vey close to the nucleus it is vey high and it is 54

55 PHY4 MODULE elatively small in between the atoms/molecules. By taking the mean of the fields ove a space containing a vey lage numbe of atoms one gets the aveage value of the field. SELF ASSESSMENT EXERCISE Show that the field at the cente of a spheical cavity (filled with ai) is zeo. The field expeienced by an individual atom/molecule may be called the local field, which is diffeent fom the aveage field. The local field is the one which causes the polaisation of the atom. The aveage field can be expessed as V / d whee V is the potential diffeence between two points of a dielectic, distant d apat. (as one obtains the field between the plates of a paallel plate condense). The estimation of local field is not so easy. Let us conside thee diffeent cavities to find the local field in a dense dielectic, which has been unifomly polaised. See Fig Fig. 3.4 The field in a slot cut in a dielectic depends on the shape and oientation of the slot E shown Is the aveage field The diections of electic (aveage) field E and P ae shown in Fig Suppose we cut a ectangula slot ABCDEFGH as in (a) of Fig. (3.4). The field E and the polaisation P ae paallel to the faces ABCD, EFGH. The field inside this slot can be found out by evaluating the line integal of E aound the cuve C shown in Fig. 3.4(b). Since E d l has to be zeo fo the closed cuve C the field inside this slot has to be the same as the field outside the slot. Theefoe the field inside a thin slot cut paallel to the field is equal to the aveage field E. Now conside a thin ectangula slot with faces pependicula to the aveage field E cut fom the dielectic as shown in (c) A'B'C'D'E'F'G'H' of Fig To find the field inside this slot we use the Gauss' flux theoem on a suface S with one face outside the slot and one face inside 55

56 PHY4 ELECTROMAGNETISM the slot. See Fig. 3.4(d). The flux of E though faces paallel to E is zeo. Instead of the flux of E, let us conside the flux of electic displacement D. Let E loc be the field inside the slot, then D inside the slot is ε Eloc. The D vecto outside the slot is ε E + P. Now, as the flux of D though the closed suface S has to be zeo (no fee o extenal chages inside the Gaussian suface), we must have ε loc ε E E + P E loc P / ε E + (3.4) The field inside the slot in this case is diffeent fom the field outside by P / ε because of the suface polaisation chages appeaing on A'B'C'D' shown in Fig. 3.4(c). Anothe possible slot is a spheical hole, which is the most likely way an atom finds itself in most liquids and solids. We would expect that an atom finds itself, on the aveage, suounded by othe atoms in what would be a good appoximation to a spheical hole. What is the local field in a spheical hole? Suppose we cut a spheical hole afte "feezing" the state of polaisation fom a unifomly polaised mateial. If we call E loc the field inside the spheical hole at its cente and E the field poduced by the unifomly polaised dielectic spheical plug at its cente, then by adding E loc and E p, we should get the aveage field E inside the dielectic. See Fig This should be tue because of the supeposition pinciple. Thus E E loc + E p (3.5) p Fig. 3.5: The field at any point A in a dielectic can be consideed as a sum of the field in a spheical hole plus the field due to the spheical plugand the equied field E E (3.6) loc E p 56

57 PHY4 MODULE One can calculate E p (the field poduced by the unifom polaised dielectic) as follows: The field E p aises fom bound chages of density σ n P cosθ. Hence the field due to the chages ove an aea ds is given as: (cosθ ) ds de p 4πε whee, is the unit vecto fom the suface to the cente of the sphee whee the field is to be calculated. Resolving de p into components paallel and pependicula to P, it is clea fom the symmety of the situation that only the components paallel to the diection of P will contibute to the total field E. Thus E p de p cosθ p It should be noted that the diection of then have E p is paallel to that of P. We E cos θ p ds 4πε P Now, ds sinθ dθ dφ and the limits of θ ae fom to π and that of φ fom to π. Hence E p P d cos 4πε θ sinθ dθ P cos θ d 4πε 3 P 4πε P 3ε (3.7) 57

58 PHY4 ELECTROMAGNETISM Then the field expeienced by an atom in a spheical hole is E loc E P 3ε To detemine the field E p at an abitay point inside the dielectic sphee, we conside the polaised sphee as a supeposition of slightly displaced sphees of positive and negative chages. See Fig Futhe note that the field at point is entiely detemined by the chage contained in the sphee of adius, inteio to point. Fig. 3.6 Supeposition of slightly displaced sphee of positive and negative chages The sphee of positive chage can be egaded as a point chage at its cente and if P is the volume chage density then the positive chaged 4π 3 sphee is equivalent to chage at its cente equal to. Similaly the 3 negative chaged sphee is equivalent to a point chage at its cente. The 4π 3 magnitude of this point chage is same as. If 'a' is the sepaation of 3 the positive and negative chages in an atom, then the unifomly 4π 3 polaised dielectic is equivalent to a dipole of moment a. If thee 3 ae n dipoles pe unit volume, q is the chage on each dipole then σ qn. [The numbe of positive o negative chages pe unit volume is p also equal to n in the sphees consideed above]. Then the dipole moment of the sphee is given by 4π 3 3 4π nqa 3 3 P 58

59 PHY4 MODULE and the polaised sphee is equivalent to a dipole of moment 4π 3 3 P kept at its cente. The potential due to this dipole at the point on the suface is given by 3 4π P cosθ 3 4πε P cosθ 3ε whee p,, θ ae as shown in the Fig Fig. 3.7 Field outside a unifomly polaised sphee The polaisation is in the diection of E and if we take this to be the z - diection with the oigin at the cente then the potential at T is, PZ φ 3ε This shows that the potential at a point depends only on its z coodinate. Hence the electic field is along the z diection and is given by: E p φ P Z 3ε This shows that the electic field inside the dielectic sphee is unifom and in the diection of the polaisation vecto. Hence the field expeienced by an atom in a spheical hole is, P E loc E + (3.8) 3ε The field in a spheical hole is geate than the aveage field by P / 3ε. p 59

60 PHY4 ELECTROMAGNETISM SELF ASSESSMENT EXERCISE Show that the field inside a unifom spheically symmetic chage distibution with chage density is equal to whee is the position 3ε vecto of the point with oigin at the cente. 3.3 The Clausius-Massotti Equation In a liquid we would expect an individual atom to be polaised by a field obtained in a spheical cavity athe than by the aveage (macoscopic) field. Thus using Eq. 3.8 and Eq. 3.3 we have, P nα E loc P P n α E + (3.9) 3ε This can be ewitten as: nα P E (3.) nα 3ε The susceptibility χ was defined in Unit by the equation, Hence, P ε χe nα / ε nα / 3ε (3.) Eq. 3. gives the elation between susceptibility and atomic/molecula polaisability. This is one fom of the Clausius-Mossotti Equation Polaisation in a Gas Unlike the atoms/molecules of a liquid o solid it is possible to conside the atoms/ molecules of a gas as fa apat and independent. We can neglect the field due to the dipoles on the immediate neighbouhood of an individual molecule. Hence the local field causing polaisation is the aveage o macoscopic field E. Theefoe we can wite, P χ E np ε 6

61 PHY4 MODULE whee, n is the numbe of molecules pe unit volume. If we conside only an individual atom/molecule and wite the dipole moment p as: P ε α E (3.) whee, α is known as the atomic polaisability. Theefoe α has the dimensions of volume and oughly equals the volume of an atom. We can elate α o χ to the natual fequency of oscillation of electons in the atom/molecule. If the atom is placed in an oscillating field E the cente of chage of electons obeys the equation d x m + mω x qe dt whee m is the mass of electon of chage q, mω x is the estoing foce tem and qe the foce fom outside field - this equation is the same as the equation of foced oscillation. If the electic field vaies with angula fequency ω then, qe x m ( ω ω ) Fo ou puposes in the electostatic case ω which means that qe x mω and the dipole moment P is P q E q x mω Fom Eq. (3.) we can wite the atomic polaisability as q α ε mω (3.3) and P ε εχ ( ε ) ε nα E 6

62 PHY4 ELECTROMAGNETISM ε nα nq ε mω Fo hydogen gas we can get a ough estimate of ω. The enegy needed hω to ionise the hydogen atom is equal to 3.6 ev. Equating this to π whee h is the Planck's constant, we get π ω.65 5 Substituting this in the equation 3.3 (a) we get nq ε +. ε m The expeimentally obseved value is ε Relation between Polaisability and Relative Pemittivity In Unit, you have noted that one can wite P as: P ε ( ε )E whee ε is the elative pemittivity. Using Eq. 3.4 in Eq. 3.8 we get P E E + ( ε + ) E / 3 3ε p (3.5) Using Eqs. 3.4 and 3.5 one can ewite Eq. 3.9 as: ( ε + ) P ε ( ε ) E nα E 3 which yields, α 3ε n ( ε ) ( ε + ) (3.6) 6

63 PHY4 MODULE Eq. 3.6 gives us the elation between atomic/molecula polaisability and the elative pemittivity. Eq. (3.6) is anothe fom of the Clausius- Mossoti equation. SELF ASSESSMENT EXERCISE 3 Obtain Eq. 3.5 fom Eq Relation between the Polaisability and Refactive Index Fo a dielectic, the efactive index µ defined as the atio of the speed of light in vacuum to the speed in the dielectic medium, can be shown to be equal to ε. µ ε Using Eq. 3.5 in Eq, 3.4 we get 3ε α n ( µ ) ( µ + ) (3.7) Eq. 3.7 gives the elation between polaisability and efactive index. This elation is known as the Loentz-Loenz fomula. In all the equations discussed above, n epesents the numbe density of atoms o molecules which is equal to N A d / W whee N A is the Avogado numbe, d the mass density and W the molecula weight. Fo gases, we have the gas equation elating pessue, P, volume, V and absolute tempeatue T given by P ' V RT N A KT whee q is the mole numbe and P' qn A KT / V nkt Theefoe, n p' / KT Thus if we detemine ε at diffeent pessues fo a gas, we can calculate the atomic/ molecula polaisability of a gas. Fo this we wite Eq. 3.6 as α 3ε KT ( ε ) p' ( ε + ) 63

64 PHY4 ELECTROMAGNETISM o ( ε ) α ( p' / T) ( ε + ) 3ε k (3.8) Eq. 3.8 epesents the linea elation between ( ε ) /( ε + ) and ( p '/ T ). If now a gaph is dawn with ( ε ) /( ε + ) on the y-axis and ( p '/ T ) on the x -axis, we get a staight line, the slope of which gives ( α / 3ε k ). 3.6 Behaviou of Dielectic in Changing o Altenating Fields So fa we have consideed only electostatic fields in matte. Now we would like to look at the effects of electic fields that vay with time, like the field in the dielectic of a capacito used in an altenating cuent cicuit. Will the changes in polaisation keep up with the changes in the field? Will the polaisability, the atio of P to E, at any instant be the same as in a static electic field? Fo vey slow changes o small fequencies we do not expect any diffeence. Howeve, fo high fequencies o faste pocess we have to look at the esponse time fo the polaisation. We have to sepaately conside two polaisation pocesses viz., induced polaisation and the oientation of pemanent dipoles. We know that the induced polaisation occus by the distotion of the electonic stuctue. In the distotion mass involved is that of electon and the distotion is vey small, which means the stuctue is vey stiff. Fom ou knowledge of oscillatoy motion (see the couse on oscillations and waves), its natual fequencies of vibation ae extemely high. Altenatively, the motions of electons in atoms and molecules ae chaacteised by peiods of the ode of the peiod of a 6 visible light wave ( seconds). Thus the eadjustment of the electonic stuctue i.e. the polaisation esponse is vey apid, occuing 4 at the time scale of sec. Fo this eason we find that nonpola substances behave the same way fom dc up to fequencies close to those of visible light. We shall examine the situation in the light of Eq. 3.5, whee we have expessed the Clausius-Mossotti fomula in tems of the efactive index. We know that the efactive index is dependent on the wavelength o fequency. Thus, in a way 3.3 implies the vaiation of the polaisability with fequency. 64

65 PHY4 MODULE Expeimentally, d.c. values of ε can be found. The efactive index of the same substance can be detemined by optical methods, using a spectomete. A faily good ageement is found between the efactive index and ε values fo non-pola substances. Howeve fo pola substances, ε vaies with fequency; it deceases with incease in fequency. The dop in the value of ε at high fequencies is due to the fact that the pemanent dipoles ae not able to follow the apid altenation of the field. In othe wods the polaisation esponse of pola molecules is much slowe. Howeve, in the fequency ange of visible light the efactive index and ε values show a faily good ageement as indicated by nonpola substances. 3.7 The Role of Dielectic Capacito in Ou Pactical Life Dielectics have seveal applications. Dielectics ae used vey widely in capacitos. Although the actual equiements vay depending on the application, thee ae cetain chaacteistics which ae desiable fo thei use in capacitos. A capacito should be small, have high esistance, be capable of being used at high tempeatues and have long life. Fom a commecial point of view it should also be cheap. Specially pepaed thin kaft pape, fee fom holes and conducting paticles, is used in powe capacitos whee withstanding high voltage stesses is moe impotant than incuing dielectic losses. In addition, the kaft pape is impegnated with a suitable liquid such as chloinated diphenyl. This inceases the dielectic constant and thus educes the size of the capacito. In addition the beakdown stength is inceased. In addition to pape capacitos fo geneal pupose, othe types of capacitos ae used. In the film capacitos, thin film of teflon, myla o polythene ae used. These not only educe the size of the capacito but also have high esistivity. Teflon is used at high fequencies as it has low loss. In electic capacitos, an electolyte is deposited on the impegnating pape. The size of such a capacito is small as the film is vey thin. Polaity and the maximum opeating voltage ae impotant specifications fo these capacitos. Some ceamics can be used as tempeatue compensatos in electonic cicuits. High dielectic constant mateials, whee small vaiations in dielectic constant with tempeatue can be toleated, help miniatuise capacitos. Baium titanate and its modifications ae the best examples of such mateials. 65

66 PHY4 ELECTROMAGNETISM 4. CONCLUSION In this unit, we have leant about diffeent types of fields elative to thei polaisation. The Claussius-Massoti equation fo liquids and gases has been deived and the oles of dielectics in daily life have been highted. 5. SUMMARY Inside a dielectic the aveage electic field is less than the electic field which causes the polaisation. In a dielectic mateial, the induced dipole moment p, is diectly popotional to the local field and mathematically given by: P α E loc whee the symbols have thei usual meanings. The field inside a spheical hole is given by: E E + P / 3ε loc which shows that the field in a spheical hole is geate than the aveage field. The elation between susceptibility and atomic/molecula polaisability is given by: nα / ε nα / 3ε 6. TUTOR-MARKED ASSIGNMENT () A sphee of linea dielectic mateial is placed in a unifom electic field E (see Fig. TQ). Find the field inside the sphee and polaisation in tems of extenal field () The electic field inside a polaised sphee is unifom and equal to P / 3ε. Pove this by supeposing the intenal fields of two sphees of chage whose centes ae sepaated. 66

67 PHY4 MODULE Fig. TQ A linea dielectic mateial placed in a unifom magnetic field (3) Show that ε times the foce on a unit chage placed in a disc shaped cavity will measue the electic displacement (D) in a solid dielectic. (4) A dielectic consists of a cubical aay of atoms (o molecules) with spacing d between each atom along the ( x, y, z ) axis. It is influenced by a field E loc applied along the diection of z -axis. Evaluate the aveage field poduced by all the dipoles. 7. REFERENCE/FURTHER READING IGNOU (5). Electicity and Magnetism, Physics PHE-7. New Delhi, India. 67

68 PHY4 ELECTROMAGNETISM UNIT 4 MAGNETISM OF MATERIALS - I CONTENTS. Intoduction. Objectives 3. Main Content 3. Response of Vaious Substances to a Magnetic Field 3. Magnetic Moment and Angula Momentum of an Atom 3.3 Diamagnetism and Paa magnetism 3.3. Diagmagnetism Effect of Magnetic Field on Atomic Obits 3.3. Paamagnetism Toque on Magnetic Dipoles 3.4 The Inteaction of an Atom with Magnetic Field Lamo Pecession 3.5 Magnetisation of Paamagnets 4. Conclusion 5. Summay 6. Tuto Maked Assignments 7. Refeences/Futhe Reading. INTRODUCTION In Units and, we lean how the magnetic field affects mateials and how some mateials poduce magnetic field. You must have leant in you school Physics Couse that in equipment such as geneato and moto, ion o ion alloy is used in thei stuctue fo the pupose of enhancing the magnetic flux and fo confining it to a desied egion. Theefoe, we will study the magnetic popeties of ion and a few othe mateials called feomagnets, which have simila popeties as ion. We shall also lean that all the mateials ae affected by the magnetic field to some extent, though the effect in some cases is weak. When we speak of magnetism in eveyday convesation, we almost cetainly have in mind an image of a ba magnet. You may have obseved that a magnet can be used to lift nails, tacks, safety pins, and needles (Fig. 4.a) while, on the othe hand, you cannot use a magnet to pickup a piece of wood o pape (Fig. 4.b). 68

69 PHY4 MODULE Fig. 4. (a) Mateials that ae attacted to a magnet ae called magnetic mateials, (b) Mateials that do not eact to a magnet ae called nonmagnetic mateials Mateials such as nails, needles etc., which ae influenced by a magnet ae called magnetic mateials wheeas othe mateials, like wood o pape, ae called non-magnetic mateials. Howeve, this does not mean that thee is no effect of magnetic field on non-magnetic mateials. The diffeence between the behaviou of such mateials and ion like magnetic mateials is that the effect of magnetic field on non-magnetic mateial is vey weak. Thee ae two types of non-magnetic mateials: diamagnetic and paamagnetic. Unit 4 deals with diamagnetic and paamagnetic effects. The ideas, concepts and vaious tems that you become familia with in this Unit would help you in the study of feomagnetism in the next Unit. In this unit, we pesent a simple classical account of magnetism, based on notion of classical physics. But you must keep in mind that it is not possible to undestand the magnetic effects of mateials fom the point of view of classical physics. The magnetic effects ae a completely quantum mechanical phenomena. Only moden quantum physics is capable of giving a detailed explanation of the magnetic popeties of matte because the study equies the intoduction and utilization of quantum mechanical popeties of atoms. Fo a complete explanation, one must take ecouse to quantum mechanics; howeve, a lot of, though incomplete, infomation about matte can be extacted by combining classical and quantum concepts. Basically, in this unit, we will ty to undestand, in a geneal way, the atomic oigin of the vaious magnetic effects. The next unit is an extension of this unit. Thee, we will ty to develop a teatment of magnetised matte based on some obseved elations between the magnetic field and the paametes which chaacteise the mateial. Finally, we conside the analysis of the magnetic cicuit, which is of paticula impotance in the design of the electomagnets. 69

70 PHY4 ELECTROMAGNETISM. OBJECTIVES Afte studying this unit you should be able to: undestand and explain: gyo-magnetic atio, paamagnetism, diamagnetism, Lamo fequency elate the magnetic dipole moment of an atomic magnet with its angula momentum explain the phenomenon of diamagnetism in tems of Faaday induction and Lenz's pinciple explain paamagnetism in tems of the toque on magnetic dipoles find the pecessional fequency of an atomic dipole in a magnetic field appeciate that a lot of infomation about magnetism of matte can be obtained fom the classical ideas of atomic magnetism. 3. MAIN CONTENT 3. Response of Vaious Substances to a Magnetic Field To show how the magnetic mateials espond to a magnetic field, conside a stong electomagnet, which has one shaply pointed pole piece and one flat pole piece as shown in Fig. 4.. Fig. 4. A small cylinde of bismuth is weakly epelled by the shap pole) a piece of aluminium is attacted The magnetic field is much stonge in the egion nea the pointed pole wheeas nea the flat pole the field is weake. This is because the lines must concentate on the pointed pole. When the cuent is passed though the electomagnet (i.e., when the magnet is tuned on), the hanging mateial is slightly displaced due to the small foce acting on it. Some mateials get displaced in the diection of inceasing field, i.e., towads the pointed pole. Such mateials ae paamagnetic mateials. Examples of such mateial ae aluminium and liquid oxygen. On the 7

71 PHY4 MODULE othe hand, thee ae mateials like bismuth, which ae attacted in the diection of the deceasing field, i.e., it gets epelled fom the pointed pole. Such mateials ae called diamagnetic. Finally, thee is a small class of mateials which feel a consideable stonge foce ( 3-5 times) towads the pointed pole. Such substances ae called feomagnetic mateials. Examples ae ion and magnetite. How does a substance expeience a foce in a magnetic field? And why does the foce act in a paticula diection fo some substance while in opposite diection fo othe substance? If we can answe these questions, we will undestand the mechanisms of paamagnetism, diamagnetism and feomagnetism. Magnetic fields ae due to electic chages in motion. In fact, if you could examine a piece of mateial on an atomic scale, you would visualize tiny cuent loops due to (i) electons obiting aound nuclei and (ii) electons spinning on thei axes. Fo macoscopic puposes, these cuent loops ae so small that they ae egaded as the magnetic dipoles having magnetic moment. It is this magnetic moment, via which the atoms at a substance inteact with the extenal field, and give ise to diamagnetic and paamagnetic effects. In this unit, you will undestand the oigin of paamagnetism and diamagnetism. Feomagnetism has been left to be explained in the next unit. Let us fist find out the value of the magnetic moment and see how it is elated to the angula momentum of the atom. 3. Magnetic Moment and Angula Momentum of an Atom Electons in an atom ae in constant motion aound the nucleus. To descibe thei motion, one needs quantum mechanics, howeve, in this unit we shall use only classical aguments to obtain ou esults, though we epeat hee that ou desciption of the physical wold is incomplete as we shall be leaving out quantum mechanics. We conside an electon in the atom to be moving, fo simplicity, in a cicula obit aound the nucleus unde the influence of a cental foce, known as the electostatic foce, as shown in Fig. 4.3(a). As a esult of this motion, the electon will have an angula momentum L about the nucleus. 7

72 PHY4 ELECTROMAGNETISM Fig.4.3 (a) Classical model of an atom in which an electon moves at speed v in a cicula obit (b) The aveage electic cuent is the same as if the chage - e wee divided into small bits, foming a otating ing of chage, (c) The obital angula momentum vecto and the magnetic moment vecto both point in opposite diections. The magnitude of this angula momentum is given by the poduct of the mass m of the electon, its speed v and the adius of the cicula path (see Fig. 4.3), i.e., L mv (4.l) Its diection is pependicula to the plane of the obit. The fact that obital motion of the electon constitutes an electic cuent will immediately stike you mind. The aveage electic cuent is the same as if the chage on the electon wee distibuted in small bits, foming a otating ing of chage, as shown in Fig. 4.3(b). The magnitude of this cuent is the chage times the fequency as this would equal to the chage pe unit time passing though any point on its obit. The fequency of otation is the ecipocal of the peiod of otation, π / v, hence the fequency of otation has the value, v / π. The cuent is then I ev (4.) π The magnetic moment due to this cuent is the poduct of the cuent and the aea of which the electon path is the bounday, that is, µ Iπ. Hence we have ev µ (4.3) It is also diected pependicula to the plane of the obit. Using Eq. (4.) in Eq. (4.3) we get as follows: e µ L (4-4) m 7

73 PHY4 MODULE The negative sign above indicates that µ and L ae in opposite diections, as shown in Fig. 4.3(c). Note that L is the obital angula momentum of the electon. The atio of the magnetic moment and the angula momentum is called the gyo-magnetic atio. It is independent of the velocity and the adius of the obit. Accoding to quantum mechanics, L h l( l +) whee l is a positive h intege and h, h being Planck's constant. Howeve, in some π physical cases the applicability of classical models is close to eality, theefoe, we will go ahead with the classical ideas. Futhe, the ealy wok on the natue of magnetic mateials was based on classical ideas, which gave intelligent guesses at the behaviou of these mateials. SELF ASSESSMENT EXERCISE () Show that the magnetic dipole moment can be expessed in units of JT (Joule pe Tesla). () In the Boh hydogen atom, the obital angula momentum of the 34 electon is quantized in units of h, whee h 6.66 Js is Planck's constant. Calculate the smallest allowed magnitude of the atomic dipole moment in JT. (This quantity is known as 3 Boh magneton.) The mass of the electon is 9.9 kg. In addition to its obital motion, you know that, the electon in an atom behaves as if it wee otating aound an axis of its own as shown in Fig Fig. 4.4 The spin and the associated magnetic moment of the electon This popety is called spin. Though stictly it is not possible to visualise the spin of a point paticle like electon, fo many puposes it helps to egad the electon as a ball of negative chage spinning aound its axis. Then you can say that it is a cuent loop. Spin is entiely a quantum mechanical idea. Nevetheless, the spin of the electon has 73

74 PHY4 ELECTROMAGNETISM associated with it an angula momentum and a magnetic moment. Fo puely quantum mechanical easons with no classical explanation, we have e µ S (4.5) m whee S is the spin angula momentum and µ is the spin magnetic moment. The gyomagnetic atio in this case is twice that in the obital case. In geneal, an atom has seveal electons. The obital and spin angula momenta of these electons can be combined in a cetain way, the ules of which ae given by quantum mechanics, to give the total angula momentum J and a esulting total magnetic moment. It so happens that the diection of the magnetic moment is opposite to that of the angula momentum in this case as well, so that we have e µ g J (4.6) m whee g is a numeical facto known as Lande g-facto which is a chaacteistic of the state of the atom. The ules of quantum mechanics enable us to calculate the g-facto fo any paticula atomic state, g fo the pue obital case and g fo the pue spin case. The atom o molecules inteacts with the extenal magnetic field due to its magnetic moment. But thee is anothe way in which atomic cuents and hence moments ae affected by the field. In this case the magnetic moment is induced by the field. This effect leads to diamagnetism which we study in the next section. But befoe moving to the next section, ty the following SAQ. SELF ASSESSMENT EXERCISE () Compae Eq. (4.6) with (4.4) and (4.5), to find the value of g fo (i) pue obital case and fo (ii) pue spin case. () The expeimentally measued electon spin magnetic moment is Am. Show that this value is consistent with the fomula given by Eq h h (Hint: Accoding to Boh's theoy S. Hee h, h being Planck's π constant.) 74

75 PHY4 MODULE 3.3 Diamagnetism and Paamagnetism In many substances, atoms have no pemanent magnetic dipole moments because the magnetic moments of vaious electons in the atoms of these substances tend to cancel out, leaving no net magnetic moment in the atom. The obital and spin magnetic moments exactly balance out. These mateials exhibit diamagnetism. If a mateial of this type is placed in a magnetic field, little exta cuents ae induced in thei atoms, accoding to the laws of electomagnetic induction, in such a diection as to oppose the magnetic field aleady pesent. Hence, in such a substance, the magnetic moments (on account of induced cuents) ae induced in a diection opposite to that of the extenal magnetic field. This effect is diamagnetism. It is a weake effect. Howeve, this effect is univesal. Thee ae othe substances of which the atoms have pemanent magnetic dipole moments. This is due to the fact that the magnetic moments due to obital motion and spins of thei electons do not cancel out, but have a net value. When such a substance is placed in a magnetic field, besides possessing diamagnetism, which is always pesent, the dipoles of such a mateial tend to line up along the diection of the magnetic field. This is paamagnetism and the mateial is called paamagnetic. In a paamagnetic substance, the paamagnetism usually masks the eve pesent popety of diamagnetism in evey substance. Diamagnetism involves a change in the magnitude of the magnetic moment of an atom wheeas paamagnetism involves change in the oientation of the magnetic moment of an atom. Let us see how Diamagnetism Effect of Magnetic Field on Atomic Obits We conside an atom, which has no intinsic magnetic dipole moment, and imagine that a magnetic field is slowly tuned on in the space occupied by the atom. The act of switching the magnetic field intoduces change in the magnetic field which, in tun, geneates an electic field given by Faaday's law of induction. It states that the line integal of E aound any closed path equals the ate of change of the magnetic flux Φ though the suface enclosed by the path. Fo simplicity, we choose a cicula path along which the electon in the atom is moving (see Fig. 4.5). The electic field aound this path is given by Faaday's law as: dφ E d l dt 75

76 PHY4 ELECTROMAGNETISM o d E π ( B π ) (4.7) dt Fig. 4.5 An electon moving in cicula obit m a unifom magnetic Held that is nomal to the Obit whee, is the adius of the cicula path pependicula to B. The above equation gives the ciculating electic field whose stength is, db E (4.8) dt This electic field exets a toque τ ee on the obiting electon dl which must be equal to the ate of change of its angula momentum, dt that is, dl dt ee o o dl dt dl dt db dt e db e (4.9) dt The change in angula momentum, L, due to tuning on the field is obtained by integating Eq. (4.9) with espect to time fom zeo field as follows: e L B (4.) 76

77 PHY4 MODULE Thus Eq. (4.) shows that a build up of a magnetic field B causes a change in the angula momentum of the electon, L, and hence a change in the magnetic moment govened by Eq. (4.4) as follows: e µ L m e µ B (4.) 4m The diection of the induced magnetic moment is opposite to that of B, which poduces it as can be seen fom the negative sign in the Eq. (4.). In this equation, we have the tem which is the squae of the adius of the paticula electon obit whose axis is along B. If B is along the z -axis, we put x + y. Thus, the aveage < > would be < x >, since < x >< y >< z >, due to spheical symmety. Futhe < x > < y > < z > < x + y + z > < > 3 3 gives < >. 3 Hence the Eq. (4.), which we shall wite as becomes, e µ < 4m > B e µ < > B (4.) 6m We find that the induced magnetic moment in a diamagnetic atom is popotional to B and opposes it. This is diamagnetism of matte. If each molecule has n electons, each with an obit of adius, then the change in the magnetic moment of the atom is e µ 6m < >B all electons Thee is an altenative way of undestanding the oigin of diamagnetism which is based on the fact that an electon eithe speeds up o slows down depending on the oientation of the magnetic field. Let us see how. As shown in Fig. 4.6, in the absence of the magnetic field, the mv centipetal foce is balanced by the electical foce as follows: 77

78 PHY4 ELECTROMAGNETISM 4πε e mv (4.3) Let us find out what happens to one of the obits when an extenal magnetic field is applied as shown in Fig Fig. 4.6 Thee is no extenal magnetic field. Centipetal foce is balanced by the electical foce Fig. 4.7 Magnetic field is pependicula to the plane of the obit In the pesence of the magnetic field thee is an additional tem e( v B) and unde these conditions speed of the electon changes. Suppose the new speed is v, then evb + 4π ε e mv o m m evb ( v v ) ( v + v)( v v) If we assume that the change v v v is small, we get o m ev B ( v ) v e B v (4.4) m 78

79 PHY4 MODULE A change in obital speed means a change in the dipole moment given by Eq. (4.3) as follows: e µ e( v) B (4.5) 4m This shows that change in µ is opposite to the diection of B. In the absence of an extenal magnetic field, the electon obits ae andomly oiented and the obital dipole moments cancel out. But in the pesence of a magnetic field, the dipole moment of each atom changes and all get aligned antipaallel to the extenal field. This is the mechanism esponsible fo diamagnetism. This popety of magnetic mateial is obseved in all atoms. But as it is much weake than paamagnetism it is obseved only in those mateials whee paamagentism is absent Paamagnetism Toque on Magnetic Dipoles Paamagnetism is exhibited by those atoms which do not have magnetic dipole moment. The magnetic moment of an atom is due to the moment poduced by the obital cuents of electons and thei "unpaied spins". A cuent loop having µ as its magnetic dipole moment when placed in a unifom field expeiences a toque T which is given by τ µ B The toque tends to align the dipoles so that the magnetic moment is lined up paallel to the field (in the way the pemanent dipoles of dielectic ae lined up with electic field). It is this toque which accounts fo paamagnetism. You might expect evey mateial to be paamagnetic since evey spinning electon constitutes a magnetic dipole. But it is not so, as vaious electons of the atom ae found in pais with opposing spins. The magnetic moment of such a pai of electons is cancelled out. Thus paamagnetism is exhibited by those atoms o molecules in which the spin magnetic moment is not cancelled. That is why the wod "unpaied spins" is witten above. Paamagnetism is geneally weak because the linings up foces ae elatively small compaed with the foces fom the themal motion which ty to destoy the ode. At low tempeatues, thee is moe lining up and hence stonge the effect of paamagnetism. SELF ASSESSMENT EXERCISE 3 A. Of the following mateials, which would you expect to be paamagnetic and which diamagnetic? Coppe, Bismuth, Aluminium, Sodium, Silve. 79

80 PHY4 ELECTROMAGNETISM B. Would it be possible to pepae an alloy of, say, a diamagnetic mateial like coppe and a paamagnetic mateial like aluminium so that the alloy will neithe be paamagnetic no diamagnetic? 3.4 The Inteaction of an Atom with Magnetic Field-Lamo Pecession In the last subsection, while explaining paamagnetism we, consideed an atom as a magnet with the magnetic moment µ. When placed in a unifom magnetic field B, it is acted upon by a toque τ µ B, which tends to line it up along the diection of the magnetic field. But it is not so fo the atomic magnet, because it has an angula momentum J like a spinning top. We aleady know that a apidly spinning top o a gyoscope in the gavitational field is acted upon by a toque, the esult of which is that it pecesses about the diection of the field. Similaly, instead of lining up with the diection of the magnetic field, the atomic magnet pecesses about the field diection. The angula momentum and with it the magnetic moment pecess about the magnetic field, as shown in Fig. 4.8a. Due to the pesence of the magnetic field, the atom will feel a toque T whose magnitude is given by: τ µ B sinθ (4.6) whee θ is the angle which µ makes with B. The diection of the toque is pependicula to the diection of magnetic field and also of µ, as shown in Fig. 4.8b. Fig. 4.8 (a) The angula momentum associated with atomic magnet pocesses about magnetic field (b) The pesence of magnetic field esults in the toque T. It is at ight angles to the angula momentum; (c) The toque changes the diection of the angula momentum vecto, causing pecession 8

81 PHY4 MODULE Notice that the toque is pependicula to the vecto J. Now accoding to Newton's second law τ dj (4.7) dt Fo small changes, we can wite it as J τ t (4.8) In othe wods, the toque will poduce a change in the angula momentum with time. Suppose that J is the change in the angula momentum in an inteval of time t. This J will be in the diection of τ. This will esult in the tip of J moving in a cicle about B as the axis. This is, in fact, a pecession of J (so also of µ) about the diection of B. The magnitude of J can be witten by using Eq. (4.6) in Eq. (4.8) as follows: J τ t ( µ B sinθ ) t (4.9) Although the toque τ, being at ight angles to J, cannot change the magnitude of J, it can change its diection. Fig. 4.8c shows how the vecto J adds vectoially onto the vecto J to bing this about. If ω is the angula velocity of the pecession and time t, then φ is angle of pecession in ω p φ (4.) t Fom Fig. 4.8c we see that p φ J ( µ Bsinθ ) t J sinθ J sinθ Dividing above by t, appoaching the diffeential limit and putting dφ ω p, we get dt µ B ω p (4.) J Substituting fo µ / J fom the Eq. (4.6), we get e ω p g B (4.) m 8

82 PHY4 ELECTROMAGNETISM as the angula speed of pecession of an atomic magnet about the diection of B. If in Eq. (4.) g, then ω is called the Lamo fequency, and is popotional to B. It should be bone in mind that this is the classical pictue. Now you may wonde if the atomic magnets (dipoles) pecess about magnetic field, how many of these dipoles get aligned along the diection of magnetic field. We know that the potential enegy of a dipole in the applied field is given by µ B µ B cosθ. Theefoe, an unaligned dipole has a geate potential enegy than an aligned one. If the enegy of the dipole is conseved then it cannot change its diection with espect to the field, i.e. the value of angle θ emains constant. So it keeps pecessing about the field. Howeve, by losing enegy the atomic dipole gets aligned with the field. In a solid, the dipole can lose enegy in vaious ways as its enegy is tansfeed to othe degees of feedom and so it gets aligned with the field depending upon the tempeatue of the solid. To change the oientation of the dipole, the maximum enegy 3 equied is µ B. If µ is about Am 3 and a lage field, say, 5T is applied then the potential enegy will be of the ode of joules. This is compaable to the themal enegy kt at oom tempeatue. Thus only a small faction of the dipoles will be aligned paallel to B. In the next section it will be shown, using statistical mechanics, what faction of dipoles is aligned along B. In the pesence of the magnetic field, when the tiny magnetic dipoles pesent in the mateial get aligned along a paticula diection we say that mateial becomes magnetized o magnetically polaized. The state of magnetic polaization of a mateial is descibed by the vecto quantity called magnetisation, denoted by M. It is defined as the magnetic dipole moment pe unit volume. It plays a ole analogous to the polaization P in electostatics. In the next section we will also find the expession of magnetisation fo paamagnets. But befoe poceeding do the following SAE. SELF ASSESSMENT EXERCISE 4 Wate has all the electon spins exactly balanced so that thei net magnetic moment is zeo, but the wate molecules still have a tiny magnetic moment of the hydogen nuclei. In the magnetic field of. Wb m potons (in the fom of H- nuclei of wate) have the pecession fequency of 4 MHz. Calculate the g - facto of the poton. p 8

83 PHY4 MODULE Accoding to Boltzmann's law the pobability of finding molecules in a given state vaies exponentially with the negative of the potential enegy of that state divided by kt. In this case the enegy E depends upon the angle θ that the moment makes with the magnetic field. So pobability is popotional to exp( U ( θ ) / kt ). 3.6 Magnetisation of Paamagnets In the pesence of an extenal magnetic field, the magnetic moment tends to align along the diection of the magnetic field. But the themal enegy of the molecules in a macoscopic piece of magnetic mateial tends to andomise the diection of molecula dipole moments. Theefoe, the degee of alignment depends both on the stength of the field and on the tempeatue. Let us deive the degee of alignment of the molecula dipoles, quantitatively, using statistical methods. Suppose thee ae N magnetic molecules pe unit volume, each of magnetic moment µ, at a tempeatue T. Classically, the magnetic dipole can make any abitay angle with the field diection (Fig. 4.9). In the absence of an extenal field, the pobability that the dipoles will be between angles θ and θ + dθ is popotional to π sinθ dθ, which is the solid angle d Ω subtended by this ange of angle. This pobability leads to a zeo aveage of the dipoles. When a magnetic field B is applied in the z - diection, the pobability becomes also popotional to the U / kt Boltzmann distibution, e. Hee U µ B µ B cosθ is the magnetic enegy of the dipole when it is making an angle θ with the magnetic field, k is the Boltzmann constant and T is the absolute tempeatue. Fig. 4.9 Calculation of the paamagnetic popeties of mateials in an extenal magnetic field 83

84 PHY4 ELECTROMAGNETISM Hence, the numbe of atoms (o molecules) dn pe unit volume fo which µ makes angles between θ and θ + dθ with B, is given by dn + µ B cosθ / kt πke sinθ dθ (4.3) whee K is a constant. Calling µ B / kt as a, the total numbe of dipoles pe unit volume of the specimen is N dn π + a cosθ πke sinθ d θ Putting cos θ x, we have N + + ax π K e dx π K a ( e e a ) a (4.4) The magnetic dipole, making an angle θ with B, makes a contibution µ cos θ to the intensity of magnetization M of the specimen. Hence, the magnetization of the specimen obtained by summing the contibutions of all the dipoles in the unit volume is given by: M dn µ cosθ + µ B cosθ / kt πke µ cosθ sinθ dθ + + ax π K e xdx whee, again, we have put above integal, we obtain cos θ x and µ B / kt a. Evaluating the a a M π Kµ ( e + e ) + ( e a a Substituting fo a + e a ) π K fom the Eq. (4.4), we get 84

85 PHY4 MODULE M M ( e µ N ( e + e e a a a M s a ) ) a coth a (4.5) a whee M s µ N is the satuation magnetization of the specimen when all the dipoles align with the magnetic field. The expession coth a is called the Langevin function which is denoted by L (a). µ B We now conside two cases: (i) when is vey lage. This would kt happen if the tempeatue wee vey low and/o B vey lage. Fo this case, a Hence, a a + ( ) e + e e L a coth a a a a e e a e M M s. These would be satuation. a a a (ii) When µ B kt small. In this case is small which means that T is lage and / o B is a coth a and M M s ( µ B / 3kT ) µ NB / 3kT. a 3 The complete dependence of M on B is shown in Fig. 4.. Fo you compaison, the dependence of M on B based on quantum mechanical calculation is also shown. Fig 4. The Magnetisation of paamagnetic mateial placed in a µ B magnetic field B as a function of a (i) is based on classical kt calculation with no estiction on the diection of dipole (ii) is based 85

86 PHY4 ELECTROMAGNETISM on quantum mechanical calculation with estiction on the diection of dipole SELF ASSESSMENT EXERCISE 5 + Evaluate the integal e ax xdx SELF ASSESSMENT EXERCISE 6 Show that when a µ B / kt is small, M M coth a a s M s 3 a Let us now sum up what we have leant in this unit. 4. CONCLUSION In unit 4, we have explained gyomagnetic atio, paamagnet-ism, diamagnetism and Lamo fequency. In addition, we have explained how to obtain infomation about magnetism of matte fom the classical ideas of atomic magnetism. 5. SUMMARY All mateials ae, in some sense, magnetic and espond to the pesence of a magnetic field. Mateials can be classified into mainly thee goups: diamagnetic, paamagnetic and feomagnetic. Diamagnetism is displayed by those mateials in which the atoms have no pemanent magnetic dipole moments. Paamagnetism and feomagnetism occus in those mateials in which the atoms have pemanent magnetic dipoles. The obital motion of the electon is associated with a magnetic moment n, which is popotional to its obital angula momentum J. We wite this as e µ g J m whee e is the chage on electon, m the mass of electon and g is Lande g - facto which has a value fo obital case and fo spin case. The atio of the magnetic dipole moment to the angula momentum is called the gyomagnetic atio. The magnetic dipoles in the magnetic mateials ae due to atomic cuents of electons in thei obits and due to thei intinsic spins. 86

87 PHY4 MODULE Change in the magnitude of the magnetic moment of atoms is esponsible fo diamagnetism wheeas change in the oientation of the magnetic moment accounts fo paamagnet ism. Because the magnetic moment is associated with angula momentum, in the pesence of a magnetic field, the atom does not simply tun along the magnetic field but pecesses aound it with a fequency ω g( e / m B. This is called the Lamo pecession. p ) When a diamagnetic atom is placed in an extenal magnetic field nomal to its obit, the field induces a magnetic moment opposing the field itself (Lenz's law) as µ e B 4m whee and m ae the adius of the obit and mass of the electon. When atoms of magnetic moment µ ae placed in a magnetic field B, then the Magnetisation M is given by M M s (coth a / a) µ B whee, a and M s µ N is the satuation magnetisation kt when all the dipoles ae aligned in the diection of field. 6. TUTOR-MARKED ASSIGNMENT. A unifomly chaged disc having the chage q and adius is otating with constant angula velocity of magnitude ω. Show that the magnetic dipole moment has the magnitude, ( qω ). 4 (Hint: Divide the sphee into naow ings of otating chage; find the cuent to which each ing is equivalent, its dipole moment and then integate ove all ings.). Compae the pecession fequency and the cycloton fequency of the poton fo the same value of the magnetic field B. 7. REFERENCES/FURTHER READING IGNOU (5). Electicity and Magnetism, Physics PHE-7, New Delhi, India. 87

88 PHY4 ELECTROMAGNETISM UNIT 5 MAGNETISM OF MATERIALS - II CONTENTS. Intoduction. Objectives 3. Main Content 3. Feomagnetism 3. Magnetic Field Due to a Magnetised Mateial 3.3 The Auxiliay Field H (Magnetic Intensity) 3.4 Relationship between B and H fo Magnetic Mateial 3.5 Magnetic Cicuits 4. Conclusions 5. Summay 6. Tuto Maked Assignments 7. Refeences/Futhe Reading. INTRODUCTION Ealie in this couse you have studied the behaviou of dielectic mateials in esponse to the extenal electic fields. This was done by investigating thei popeties in tems of electic dipoles, both natual and induced, pesent in these mateials and thei lining up in the electic field. The macoscopic popeties of these mateials wee studied using the so-called polaization vecto P, the electic dipole moment pe unit volume. The magnetic popeties of mateials have a simila kind of explanation, albeit in a moe complicated fom, due to the absence of fee magnetic monopoles. The magnetic dipoles in these mateials ae undestood in tems of the so-called Ampeian cuent loops, fist intoduced by Ampee. All mateials ae, in some sense, magnetic and exhibit magnetic popeties of diffeent kinds and of vaying intensities. As you know, all mateials, can be divided into thee main categoies: (i) Diamagnetic; (ii) Paamagnetic and (3) Feomagnetic mateials. In this unit, we shall study the macoscopic behaviou of these mateials. We undestood the macoscopic popeties of the dielectic mateials using the fact that the atoms and molecules of these substances contain electons, which ae mobile and ae esponsible fo the electic dipoles, natual and induced, in these substances. The polaisation of these substances is the goss effect of the alignment of these dipoles. Similaly we descibe the magnetic popeties of vaious mateials in tems of the magnetic dipoles in these mateials. 88

89 PHY4 MODULE In Unit 4, we have aleady explained diamagnetism and paamagnetism in tems of magnetic dipoles. In this unit, fist, we will mention the oigin of feomagnetism. Late, we will develop a desciption of the macoscopic popeties of magnetic mateials. OBJECTIVES Afte studying this unit you should be able to: undestand and explain the tems: feomagnetism, ampeian cuent, magnetisation, magnetic intensity H, magnetic susceptibility, magnetic pemeability, elative pemeability elate magnetisation M (which is expeimentally measuable) and the atomic cuents (which is not measuable) within the mateial deive and undestand the diffeential and integal equations fo M and H and apply these to calculate fields fo simple situations inteelate B, H, M, no, H and y_< elate B & H fo vaious magnetic and non-magnetic mateials deive an equation in analogy with Ohm's law fo a magnetic cicuit. Fig. 5.: Domain Conside two electons on atoms that ae close to each othe. If the electon spins ae paallel, they stay away fom each othe due to Pauli s pinciple, theeby educing thei coulomb enegy of epulsion. On the othe hand, if these spins ae anti-paallel, the electons can come close to each othe and thei coulomb enegy is highe. Thus, by making thei spins paallel, the electons can educe thei enegy. 3. MAIN CONTENT 3. Feomagnetism Feomagnetic mateials ae those mateials, which espond vey stongly to the pesence of magnetic fields. In such mateials, the magnetic dipole moment of the atoms aises due to the spins of unpaied 89

90 PHY4 ELECTROMAGNETISM electons. These tend to line up paallel to each othe. Such a line-up does not occu ove the whole mateial, but it occus ove a small volume, known as 'domain\ as shown in Fig... Howeve, these volumes ae lage compaed to the atomic o molecula dimensions. Such line-ups take place even in the absence of an extenal magnetic field. You must be wondeing about the natue of foces that cause the spin magnetic moments of diffeent atoms to line up paallel to each othe. This can be explained only by using quantum mechanical idea of "exchange foces". We will not go into the details of exchange foces. About this, you will study in othe couses of physics, but we ae giving you some idea of exchange foces in the magin emak. Fig. 5. The domains in an unmagnetised ba of ion. The aows show the alignment diection of the magnetic moment in each domain In an unmagnetized feomagnetic mateial, the magnetic moments of diffeent domains ae andomly oiented, and the esulting magnetic moment of the mateial, as a whole is zeo, as shown in Fig. 5.. Howeve, in the pesence of an extenal magnetic field, the magnetic moments of the domains line-up in such a manne as to give a net magnetic moment to the mateial in the diection of the field. The mechanism by which this happens is that the domains with the magnetic moments in the favoued diections incease in size at the expense of the othe domains, as shown in Fig. 5.3a. Fig. 5.3 In a feomagnetic mateial domain changes, esulting in a net magnetic moment, occu though (a) domain gowth and (b) domain ealignment 9

91 PHY4 MODULE In addition, the magnetic moments of the entie domains can otate, as shown in Fig. 5.3b. The mateial is thus magnetised. If, afte this, the extenal magnetic field is educed to zeo, thee still emains a consideable amount of magnetization in the mateial. The mateial gets pemanently magnetized. The behaviou of feomagnetic mateials, unde the action of changing magnetic fields, is quite complicated and exhibits the phenomenon of hysteesis which liteally means 'lagging behind'. You will study moe about this in Sec Above a cetain tempeatue, called the 'Cuie Tempeatue', because the foces of themal agitation dominate 'exchange' foces, the domains lose thei dipole moments. The feomagnetic mateial begins to behave like a paamagnetic mateial. When cooled, it ecoves its feomagnetic popeties. Finally, we biefly mention two othe types of magnetism, which ae closely elated to feomagnetism. These ae anti-feomagnetism and feimagnetism (also called feites). In this couse, we will not study the physics of antifeo- and feimagnetism. The main eason fo mentioning these mateials ae that they ae of technological impotance, being used in magnetic ecoding tapes, antenna and in compute memoy. In antifeomagnetic substances, the 'exchange' foces, as we mentioned ealie, play the ole of setting the adjacent atoms into antipaallel alignment of thei equal magnetic moments, that is, adjacent magnetic moments ae set in opposite diections, as shown in Fig. 5.4 a. Such substances exhibit little o no evidence of magnetism pesent in the body. Howeve, if these substances ae heated above the tempeatue known as Neel tempeatue, the exchange foce ceases to act and the substance behaves like any othe paamagnetic mateial. In feimagnetic substances, known geneally as feites, the exchange coupling locks the magnetic moments of the atoms in the mateial into a patten, as shown in Fig. 5.4b. The extenal effects of such an alignment is intemediate between feomagnetism and antifeomagnetism. Again, hee the exchange coupling disappeas above a cetain tempeatue. Fig. 5.4 Relative oientation of electon spins in (a) antifeomagnetic mateial and (b) feite. 9

92 PHY4 ELECTROMAGNETISM Thus, we find that the magnetization of the mateials is due to pemanent (and induced) magnetic dipoles in these mateials. The magnetic dipole moments in these mateials ae due to the ciculating electic cuents, known as ampeian cuents at the atomic and molecula levels. You ae expected to undestand the coect elationship between magnetization in a mateial and the ampeian cuents, togethe with the basic diffeence (and sometimes similaities) between the behaviou of the magnetic mateials in magnetic fields, and dielectics (and conductos) in electic fields. Though the physics of paamagnetic and feomagnetic mateials have analogues in the electic case, diamagnetism is peculia to magnetism. The student is advised to ead the matte in this unit and find the analogies and appeciate the diffeences, if any, by efeing back to the units on dielectics. In the next section, we will find out the elationship between the macoscopic quantity M, which is expeimentally measuable and the atomic cuents (a micoscopic quantity) within the mateial which is not measuable. With the help of this elationship, we can find out the magnetic field that magnetised matte itself poduces. 3. Magnetic Field Due to a Magnetised Mateial In Unit, we have descibed the macoscopic popeties of dielectic mateials in tems of the polaization vecto P, the oigin of which is in the dipole moments of its natual o induced electic dipoles. We shall adopt a simila pocedue in the study of magnetic mateials. You would be tempted to say that we should cay ove all the equations in the study of dielectics to magnetic mateials. One way of doing this would be to eplace the electic field vecto E by B, then eplace P by an analogous quantity which we shall call magnetization vecto M which is the magnetic dipole moment pe unit volume. Futhe, we eplace the polaization chage density ρ by magnetic 'chage' density ρ p whateve that means, by witing M ρ m just as we had P ρ p. In fact, people did something like this, and they believed that magnetic chages o monopoles exist. They have built a whole theoy of electomagnetism on this assumption. Howeve, we know that magnetic 'chages' o monopoles have not yet been detected in any expeiment so fa, despite a long seach fo them. Now, we know that the magnetization of matte is due to ciculating cuents within the atoms of the mateials. This was oiginally suggested by Ampee, and we call these ciculating cuents as 'ampeian' cuent loops. These cuents aise due to eithe the obital motion of electons in the atoms o thei spins. These cuents, obviously, do not involve lage scale chage tanspot in the magnetic mateials as in the case of conduction cuents. m 9

93 PHY4 MODULE These cuents ae also known as magnetization cuents, and we shall elate these cuents to the magnetization vecto M. Let us conside a slab of unifomly magnetised mateial, as shown in Fig. 5.5a. It contains a lage numbe of atomic magnetic dipoles (evenly distibuted thoughout its volume) all pointing in the same diection. If µ is the magnetic moment of each dipole, then the magnetisation M will be the poduct of µ and the numbe of oiented dipoles pe unit volume. You know that the dipoles can be indicated by tiny cuent loops. Suppose the slab consists of many tiny loops, as shown in Fig. 5.5b. Let us conside any tiny loop of aea a, as shown in Fig. 5.5c. In tems of magnetisation M, the magnitude of dipole moment µ is witten as follows: µ Madz (5.) whee dz is the thickness of the slab. Fig. 5.5: (a) A thin slab of unifomly magnetized mateial, with the dipoles indicated by (b) and (c) tiny cuent loops is equivalent to (d) a ibbon of cuent/ flowing aound the bounday 93

94 PHY4 ELECTROMAGNETISM If the tiny loop has a ciculating cuent I, then the dipole moment of the tiny loop is given by µ Ia (5.) Equating (5.) and (5.) we get I M o I Mdz (5.3) dz Hee we have assumed that the cuent loops coesponding to magnetic dipoles ae lage enough so that magnetisation does not vay appeciably fom one loop to the next, so Eq. 5.3 shows that the cuent is the same in all cuent loops of Fig. 5.5b. Notice that within the slab, cuents flowing in the vaious loops cancel, because evey time if thee is one going in one paticula diection, then a continuous one is going in the exactly opposite diection. At the bounday of the slab, thee is no adjacent loop to do the cancelling. Hence the whole thing is equivalent to the single loop of cuent I flowing aound the bounday, as shown in Fig. 5.5d. Theefoe, the thin slab of magnetised mateial is equivalent to a single loop caying the cuent Mdz, Hence, the magnetic field at any point extenal to the slab, is the same as that of the cuent Mdz. In case thee is non-unifom magnetization in the mateial, the atomic cuents in the (ampeian) ciculating cuent loops do not have the same magnitude at all points inside the mateial and, obviously, they do not cancel each othe out inside such a mateial. Still we will find that magnetised matte is equivalent to a cuent distibution J cul M. Let us see how we have aived at this elation. In the non-unifomly magnetised mateial conside two little blocks of the volume x y z, cubical in shape adjacent to each othe along the y- axis (see Fig. 5.6a). Let us call these blocks ' ' and ' espectively. Let the z -component of M in these blocks be Mz ( y) and M z ( y + y) espectively. Let the ampeian cuents ciculating ound the block ' ' be I ( ) and ound the block '' be I ().Using Eq. (5.3) and efeing to Fig. 5.6a we wite, and I ( ) M ( y) z x z I ( ) M ( y + y) z x z 94

95 PHY4 MODULE Fig. 5.6: Two adjacent chunks of magnetised mateial, with a lage aow on the one to the ight in (a) and above in (b), suggesting geate magnetisation at that point. On the suface whee they join thee is a net cuent in the x-diection. At the inteface of the two blocks, thee will be two contibutions to the total cuent: I () flowing in the negative x -diection, poduced due to block, and I () flowing in the positive x-diection poduced due to Block. The total cuent in the positive x -diection is the sum: o I ( ) I () [ M ( y + y) M ( y)] z x x z M z I x + y z (5.4) y Eq. (5.4) gives the net magnetization cuent in the mateial at a point in the x -diection in tems of the z-component of M. The cuent pe unit aea, i.e., cuent density J m flowing in the x -diection is given as follows: z ( J m ) x I x y z whee y z is the aea of coss-section of one such block fo the cuent I x. Hence M z ( J m ) x + (5.5) y In these equations, we have put suffixes x to the cuents to indicate that, at the inteface of the blocks, the cuent is along the x -axis. 95

96 PHY4 ELECTROMAGNETISM If the magnetisation in the fist block is M M ( x, y, z) + y + highe ode tems. y The z -component of magnetisation of the fist block in tems of I () is witten as M z (). x z I x Similaly, the z -component of magnetisation of the second block neglecting high-ode tems which vanish in the limit whee each block becomes vey small, is given by M + y z M z z I x () Thee is anothe way of obtaining the cuent flowing in x-diection by consideing these two tiny blocks, one above the othe, along the z-axis, as shown in Fig. 5.6b. We obtain the elation as M z ( J m ) x (5.6) y By supeimposition of these two situations, we get M M ( J m ) x M) y z z y ( x (5.7) Eq. (5.7) is obviously the x -component of a vecto equation elating J m and the cul of M. Combining this with y and z components, we obtain J m M (5.8) Eq. (5.8) is a moe geneal expession epesenting the elationship between the magnetisation and the equivalent cuent. We see fom Eq. (5.8) that inside a unifomly magnetized mateial in which case M constant; we have J m. This is tue. See Eq. (5.8), the cuent is only at the suface of the mateial whee the magnetization has a discontinuity (dopping fom a finite M to zeo). Inside a non-unifomly magnetized mateial J m is nonzeo. We shall see in the next section that J m, which is intoduced to explain the oigin of magnetisation in a mateial, is made to make its exit fom 96

97 PHY4 MODULE the equation, and only the conduction cuent density indicating the actual chage tanspot and which is expeimentally measuable emains. 3.3 The Auxiliay Field H (Magnetic Intensity) So fa we have been consideing that magnetisation is due to cuent associated with atomic magnetic moments and spin of the electon. Such cuents ae known as bound cuents o magnetisation ampeian cuent. The cuent density J m in Eq. (5.8) is the bound cuent set up within the mateial. Suppose you have a piece of magnetised mateial. What field does this object poduce? The answe is that the field poduced by this object is just the field poduced by the bound cuents established in it. Suppose we wind a coil aound this magnetic mateial and send though this coil a cetain cuent, I Then the field poduced will be the sum of the field due to bound cuents and the field due to cuent, I. The cuent I is known as the fee cuent because it is flowing though the coil and we can measue it by connecting an ammete in seies with the coil. (In case the magnetic mateial happens to be a conducto, the fee cuent will be the cuent flowing though the mateial itself.) Remembe that fee cuents ae those caused by extenal voltage souces, while the intenal cuents aise due to the motion of the electons in the atoms. The cuent is fee, because someone has plugged a wie into a battey and it can be stated and stopped with a switch. Theefoe, the total cuent density J can be witten as: J J + (5.9) f J m whee, J f epesents the fee cuent density. Let us use Ampee's law to find the field. In diffeential fom, it is witten as: B µ J (*) Using Eq. (5.9), Ampee's law would then take the fom as follows: B µ ( J f + J m ) As mentioned ealie, we have no way to measue J m expeimentally, but we have a way to expess it in tems of a measuable quantity, the magnetization vecto M though the Eq. (5.8). We then have 97

98 PHY4 ELECTROMAGNETISM B µ µ ( J f + M ) o B µ M J f (5.) B Eq. (5.) is the diffeential equation fo the field M in tems of µ its souce J, the fee cuent density. This vecto is given a new symbol H, i.e., f B µ M H (5.) The vecto H is called the magnetic 'intensity' vecto, a name that ightly belongs to B, but, fo histoical easons, has been given to H. Using Eq. (5.), Eq. (5.) becomes, H J f (5.) In othe wods, H is elated to the fee cuent in the way B is elated to the total cuent, bound plus fee. This suely has made you think ove the pupose of intoducing the new vecto field H. Fo pactical easons the vecto H is vey useful as it can be calculated fom the knowledge of extenal cuent only, wheeas B is elated to the total cuent, which is not known. Eq. (5.) can also be witten in the integal fom as H d l I f (5.3) whee I f is the conduction cuent though the suface bounded by the path of the line integal on the left. Hee the line integal of H is aound the closed path, which may o may not pass though the mateial. This equation can be used to calculate H, even in the pesence of the magnetic mateial. SELF ASSESSMENT EXERCISE Fig. 5.7 shows a piece of ion wound by a coil caying a cuent of 5A. Find the value of H d l I f aound the path (), () and (3). Also state fo which path(s) B H and B H. 98

99 PHY4 MODULE Fom Eq. (5.3), we see that the unit in which M is measued is ampees pe mete. Eq. (5.) shows that the vecto H has the units as M, hence H is also measued in ampees pe mete. The electical enginees woking with electomagnets, tansfomes, etc., call the unit of H ampee tuns pe mete. Since 'tuns', which is supposed to imply the numbe of tuns in the coil caying a cuent, is dimensionless, it need not confuse you. Magnetic Popeties of Substance In paamagnetic and diamagnetic mateials, the magnetisation is maintained by the field. When the field is emoved, M disappeas. In fact, it is found that M is popotional to B, povided that the field is not too stong. Thus M B (5.4) It is conventional to expess Eq. (5.4) in tems of H instead of B. Thus we have M χ m H (5.5) The constant of popotionality χ m is called the magnetic susceptibility of the mateial. It is a dimensionless quantity, which vaies fom one substance to anothe. We can chaacteise the magnetic popeties of a substance by χ m. It is negative fo diamagnetic substances and positive fo paamagnetic mateials. Its magnitude is vey small compaed to unity, that is χ m < <. Fo vacuum χ m is zeo, since M can only exist in magnetised matte. We give below a shot table giving the values of χ fo diamagnetic and paamagnetic substances at oom tempeatue. m 99

100 PHY4 ELECTROMAGNETISM Paamagnetic Paamagnetic Paamagnetic Paamagnetic Diamagnetic Diamagnetic Diamagnetic Diamagnetic Mateial Aluminium Sodium Tungsten Oxygen Bismuth Coppe Silve Gold χ m We have not given a table fo the susceptibilities of feomagnetic substances as χ m depends not only on H but also on the pevious mangetic histoy of the mateial. Using Eq. (5.) in the fom we have, B µ ( H + M ) B µ ( + χ )H (5.6) m µ K m H B µ H (5.7) whee µ µ ( K m µ + χ m ) µ K (5.8) m µ µ is called the pemeability of the medium and K m is called the elative pemeability. We see that µ has the same dimensions as µ and K m is dimensionless. In vacuum χ m and µ µ. Relative pemeability K m diffes fom unity by a vey small amount as K m fo paa- and feomagnetic mateials ae geate than unity and fo diamagnetic mateial it is less than unity. The magnetic popeties of a mateial ae completely specified if any one of the thee quantities, magnetic susceptibility, χ m, elative pemeability K m o pemeability µ is known.

101 PHY4 MODULE Example A tooid of aluminium of, length m, is closely wound by tuns of wie caying a steady cuent of A. The magnetic field B in the tooid 4 is found to be.567 Wb m. Find (i) H, (ii) χ m, and K m (iii) M in the tooid and (iv) equivalent suface magnetization cuent I. m Solution (i) Accoding to Eq. (5.3) H d l I f To evaluate H poduced by the cuent, we conside a cicula integation path along the tooid. H is constant eveywhee along this path of length m. The numbe of cuent tuns theading this integation path is A. Since H is eveywhee paallel to the cicula integation path, we get o H m A H A A/m m (ii) Fom Eq. (5.6) o and B µ K m H 7 B 56.7 K m.5 7 µ H 4π ( + χ ) m K m χ K.5 m m 5 5 (iii) Fom Eq. (5.5) M χ H m

102 PHY4 ELECTROMAGNETISM 5 5 A m 5 3 A m (iv) I m ML 5 3 A m m 5 ma In this solution, we have assumed B, H and M to be unifom ove the coss-section of the tooid and along the axis of the tooid. Ty to do the following SAE SELF ASSESSMENT EXERCISE An ai-coe solenoid wound with tuns pe centimete caies a cuent of.8 A. Find H and B at the cente of the solenoid. If an ion 3 coe of absolute pemeability 6 H m is inseted in the solenoid, find the value of H and B? 3.4 Relationship between B and H fo Magnetic Mateial The specific dependence of M on B will be taken up in this section. The elationship between M and B o equivalently a elationship between B and H depend on the natue of the magnetic mateial, and ae usually obtained fom expeiment. A convenient expeimental aangement is a tooid with any magnetic mateial in its inteio. Aound the tooid, two coils (pimay and seconday) ae wound, as shown in Fig If we conside the adius of the coss-section of the tooidal windings to be small in compaison with the adius of the tooid itself, the magnetic field within the tooid can be consideed to be appoximately unifom. A cuent passing though the pimay coil establishes H. The establishment of the cuent in the pimay coil induces an electomotive foce (emf). By measuing the induced voltage, we can detemine changes in flux Φ and hence, in B inside the magnetic mateial. If we take H as the independent vaiable, and if we keep the tack of the changes in B stating fom B, we can always know what B is fo a paticula value of H. In this way, we can obtain a B-H cuve fo diffeent types of magnetic mateial.

103 PHY4 MODULE Fig. 5.8 Aangement fo investigating the elation between B and M, o B and H, in a magnetic mateial The expeiment descibed above can be caied out fo diamagnetic and paamagnetic mateials by commencing with I and slowly inceasing the value of I to obtain a seies of values of B and H. A plot of B against H fo these substances is shown in the Fig. 5.9(a). We see that the gaph is a staight line as expected fom the elation B µ ( + )H m (5.6) χ Fig. 5.9 Intenal magnetic field (B) vesus applied magnetic field (H) fo diffeent types of magnetic mateials, (a) In diamagnetic and paamagnetic mateials, the elationship is linea, (b) In feomagnetic mateials, the elationship depends on the stength of the applied field and on the past histoy of the mateial, in (b), the field stengths along the vetical am ae much geate than along the hoizontal axis. Aows indicate the diection in which the fields ae changed. whee µ and χ m ae constants. The slope of the gaph is given by can be detemined using the following elation: χ m slope µ χ m 3

104 PHY4 ELECTROMAGNETISM Fo diamagnetic substances, slope < µ making χ m <. Fo paamagnetic mateials slope > µ, so that χ m >. If in the expeiment given above we use feomagnetic mateials like ion, we obtain a typical B H cuve as shown in Fig. 5.9(b). (i) (ii) (iii) (iv) At I, i.e., when H, B is zeo. When I is inceased. B and H ae detemined fo inceasing values of I. At fist, B inceases with H along the cuve 'a'. At some high value of H, the cuve (shown by the dashed line in the figue) becomes linea, indicating that M ceases to incease, as the mateial has eached satuation with all the domain dipole moments in the same diection. If, afte eaching satuation, we decease the cuent in the coil to bing H back to zeo, the E-H cuve falls along the cuve 'b'. When I eaches zeo, thee is still some B left, implying that even when I, thee is still some magnetization of M left in the specimen. The mateial is pemanently magnetized. The value of B fo H is called emanence. If the cuent is evesed in the pimay coil and made to incease its value, the B-H cuve uns along the cuve 'b' until B becomes zeo at a cetain value of H. This value of H is called the coecive foce. If we continue to incease the value of the cuent in the negative diection, the cuve continues along 'b' until satuation is eached again. The cuent is now deceased until it becomes zeo once again. This coesponds to H, but B is not zeo and have magnetization in the opposite diection. Hee we evese the cuent again, so that the cuent in the coil is once moe along the positive diection. With the inceasing cuent in this diection, the cuve continues along the cuve 'c' to meet the cuve 'b' at satuation. If we altenate the cuent between lage positive and negative values, the B-H cuve goes back and foth along 'b ' and 'c ' in a cycle. This cycle cuve is called hysteesis cuve. It shows that B is not a single valued function of H, but depends on the pevious teatment of the mateial. The shape of the hysteesis loop vaies vey widely fom one substance to anothe. Those substances, like steel, alnico, etc., fom which pemanent magnets ae made, have a vey wide hysteesis loop with a lage value of the coecive foce (see Fig. 5.). Howeve, those 4

105 PHY4 MODULE substances, like soft ion, pemalloy, etc., fom which electomagnets (tempoay magnet) ae made, should have lage emanence but vey small coecive foce. Those feomagnetic mateials, which ae used in the coes of tansfomes, like ion-silicon (.8-4.8%) alloys, have vey naow hysteesis loop. Fig. 5. The hysteesis cuves fo a few mateials. Cuves (a) and (b) ae espectively fo specimen of soft - ion and steel mateials 3.5 Magnetic Cicuits A magnetic cicuit is the closed path taken by the magnetic flux set up in an electic machine o appaatus by a magnetising foce. (The magnetising foce may be due to a cuent coil o a pemanent magnet.) In ode to study the esemblance between a magnetic cicuit and an electic cicuit, we will develop a elation coesponding to Ohm's law, fo a magnetic cicuit. Let us conside the case of an ion ing (Fig. 5.) magnetised by a cuent flowing though a coil wound closely ove it. Suppose: I cuent flowing in the coil N numbe of tuns in the coil l length of the magnetic cicuit (mean cicumfeence of the ing) A aea of coss-section of the ing µ pemeability of ion. In this case, all the magnetic flux poduced is confined to the ion ing with vey little leakage (we shall see the eason fo this late). We have seen ealie that H inside the ing is given by 5

106 PHY4 ELECTROMAGNETISM Fig. 5. Magnetic cicuit H d l NI (fom Ampee's law) whee, the path of integation is along the axis of the ing. As the line integal of electic field E ove a cicuital path is the electomotive foce (e.m.f), by analogy, the line integal of H is temed as magnetomotive foce (M.M.F.) M.M.F. H d l NI At evey point along this path in the ing, we wite B H µ Futhe, if Φ is the magnetic flux given by Φ BA, then hence H Φ / µ A, M.M.F. dl H NI (5.9) µ A d l Φ whee we have taken Φ outside the integal as it is constant at all cosssections of the ing. Eq. (5.9) eminds us of a simila equation fo an electic cicuit containing a souce of E.M.F., namely, ρ dl e.m.f. cuent esistance I A (5.) The Eqs. (5.9) and (5.) suggest that: (i) The magnetomotive foce ( H d l ) is analogous with e.m.f. ( E d l ). 6

107 PHY4 MODULE (ii) The magnetic flux Φ is analogous with cuent I in Ohm's law, dl (iii) The magnetic esistance known as eluctance µ A ρ dl analogous with electic esistance A is M.M.F. flux eluctance o Total flux Φ M.M.F eluctance NI dl µ A (5.) If we take µ to be constant thoughout the ing then dl L eluctance R µ A µ A (5.) whee L is the length of the ing. Howeve, we must ecognise the significant diffeence between an electic cicuit and a magnetic cicuit: (i) (ii) (iii) Enegy is continuously being dissipated in the esistance of the electic cicuit, wheeas no enegy is lost in the eluctance of the magnetic cicuit. The electic cuent is a tue flow of the electons but thee is no flow of such paticle in a magnetic flux. At a given tempeatue, the esistivity ρ is independent of cuent, while the coesponding quantity µ in eluctance vaies with magnetic flux Φ. Reluctances in Seies Let us assume that the tooid is made of moe than one feomagnetic mateial, each of which is of the same coss-sectional aea A, but with diffeent pemeabilities µ, µ, 7

108 PHY4 ELECTROMAGNETISM Fig. 5. (a) A magnetic cicuit composed of seveal mateials: Reluctances in seies, (b) Magnetic cicuit consisting of two loops: Reluctances in paallel. Then, (see Fig. 5.a) as befoe, we have NI H d l H d l + H dl +... whee the integals on the ight ae taken ove axial paths in the mateials,,... Theefoe, Φ Φ M.M.F. dl dl µ A µ A dl dl Φ µ A µ A L L Φ µ A µ A Φ R + R +...) ΦR ( so that the total eluctance of the given magnetic cicuit is given by R R + R... (5.3) + Reluctances in Paallel We shall next illustate the case of a magnetic cicuit in which the eluctances ae in paallel. Fig. 5.b shows such a magnetic cicuit. The cuent caying coils have N tuns each, caying a cuent I ampees. 8

109 PHY4 MODULE The magnetic flux Φ theading the coil splits into two paths with fluxes Φ and Φ as shown in the figue. Obviously, Φ Φ + Φ. We assume that the aea of coss-section A is constant eveywhee in the cicuit. Let the lengths of the paths ABCD, DA and DEFA shown in the figue be L, L, and L espectively. Fo the path ABCDA, we have Φ Φ NI dl + dl µ A µ A ABCD DA Φ Φ L + L µ A µ A (5.4) Similaly fo the closed path ADEFA, we have Φ Φ dl + dl (5.5) µ A µ A AD DEFA Notice that we have used µ and µ fo the paths AD and DEFA. As Φ s being diffeent fo these paths, H would be diffeent. This makes µ s diffeent in these paths. Using Φ Φ + Φ and Eq. (5.5), we wite Φ Φ + Φ µ µ L L µ Φ + L µ L Substituting the value of Φ fom the above equation in the Eq. (5.4), we have L L Φ NI µ A µ A L + Φ µ A L L µ A µ A o R R NI Φ R + (5.6) R R 9

110 PHY4 ELECTROMAGNETISM This shows that the eluctances of the paths DA and DEFA ae in paallel as the magnetic flux Φ splits into Φ and Φ along these paths espectively. The combined eluctance R of these paths is given, in tems of the eluctances R and Rof these paths, as follows R R + R (5.7) Notice that the Eq. (5.4), (5.5) and Φ Φ + Φ ae the statements of Kichhoff's laws fo magnetic cicuits. Now we see why the magnetic flux does not leak though the ai. Ai foms a paallel path fo the flux, fo ai, µ µ and fo a feomagnetic 4 mateial µ ; hence the ai path is a vey high eluctance path compaed to that though the feomagnetic mateial. The magnetic flux will follow the path of least eluctance, a situation simila to that in the electic cicuit. The magnetic cicuit fomulae ae used by the electical enginees in calculations elating to electomagnets, motos and dynamos. The poblem is usually to find the numbe of tuns and the cuent in the winding of a coil, which is equied to poduce a cetain flux density in the ai gap of an electomagnet. Knowing the eluctance of the cicuit, M.M.F. is calculated fom the elation: M.M.F. flux eluctance Since M.M.F. is also NI (see Eq. (5.9)), the magnitude of ampee tuns can be calculated. Let us illustate it by studying the magnetic cicuit of an electomagnet. Magnetic Cicuit of an Electomagnet The magnetic cicuit of an electomagnet consists of the yoke which foms the base of the magnet, the limbs on which the coil is wound, the pole pieces and the ai gap. See Fig Let l be the effective length and a the aea of coss-section of the yoke. If µ is the pemeability of l its ion, then is the eluctance of the yoke. Similaly the eluctance µ a l of each limb is µ a

111 PHY4 MODULE and the eluctance of each pole piece is l4 the ai gap is µ a4 magnetic cicuit is l3 µ a 3 3, while the eluctance of (because µ ). Hence the total eluctance of the ai l µ a 3 l + µ a l + µ a 3 3 l4 + µ a 4 Fig. 5.3 Magnetic cicuit of an electomagnet. If the magnetic cicuit caies one and the same flux Φ acoss all its pats, then accoding to Eq. (5.9), the numbe of ampee tuns is: l Φ µ a l + µ a + 3 l 3 µ a 3 + l 4 µ a 4 (5.8) Let us take anothe example of calculating the magnetic field B in the ai gap of a tooid of Fig Hee the tooid is of a feomagnetic mateial (soft ion) with a small ai gap of width 'd ' which is small compaed to the length L of the tooid. Fo this case, we have ( L d ) d NI Φ +, Φ being the flux though this µ A µ A magnetic cicuit. o B [ µ ( L d) + µ d] µ µ B NIµ µ µ L + ( µ µ ) d (5.9)

112 PHY4 ELECTROMAGNETISM This is the value of the magnetic field in the ai gap. Read the following example which shows how the ai gap effectively inceases the length of the tooid. Fig. 5.4: Magnetic field in the ai gap Example Compae the examples of a complete tooid of length L wound with a coil of N tuns each caying a cuent I ampees and of a tooid of length ( L d ) with an ai gap of length d ( d << L). Show that the ai gap effectively inceases the length of the tooid by ( K m ) d, whee K m is the elative pemeability. Solution In the case of a complete tooid without the ai gap, we have L B NI /. In the event of an ai gap of length d, we have fom Eq. µ (5.9): B NIµ µ µ L + ( µ µ ) d Dividing both the numeato and the denominato by µ µ, we get B NI L + d µ µ µ NI d ( L d) + µ µ NI NI d ( L d) + [( L d) + K µ µ µ m d]

113 PHY4 MODULE so that B NI [ L + ( K µ m ) d] If we compae this fomula with that fo the complete tooid, we see that L is effectively inceased by ( K m ) d. Befoe ending this unit solve the following SAQ. SELF ASSESSMENT EXERCISE 3 A soft ion ing with a. cm ai gap is wound with a coil of 5 tuns and caies a cuent of A. The mean length of ion ing is 5 cm, its coss-section is 6cm, its pemeability is 5 µ. Calculate the magnetic induction in the ai gap. Find also B and H in the ion ing. Let us now sum up what we have leant in this unit. 4. CONCLUSION In this unit, we have explained the tems, feomagnetism, ampeian cuent, Magnetisation, M, Magnetic intensity, H, susceptibility, pemeability etc. An analogy has been deived fo magnetic cicuit fom Ohm s law. In addition, we have leant about the inte-elationship between, M, H and othe quantities. 5. SUMMARY The behaviou of the feomagnetic mateials is complicated on account of the pemanent magnetization and the phenomenon of hysteesis. This behaviou is explained by the pesence of the domains in these mateials. In each domain the dipole moments ae locked to emain paallel due to 'exchange' foce. Howeve, in the unmagnetised state, the magnetisation diections of diffeent domains ae andom, esulting in a zeo net magnetisation. Thee also exist two othe kinds of magnetic mateials: antifeomagnetic and feimagnetic. Fo non-unifom magnetisation, magnetised matte is equivalent to a cuent distibution J M, whee M is magnetisation o magnetic moment pe unit volume. The magnetic field poduced by the magnetised mateial is obtained by Ampee's law as follows: B J f + J m 3

114 PHY4 ELECTROMAGNETISM whee J f is the fee cuent density which flows though the mateial and J m is the bound cuent density which is associated with magnetisation. This gives B M J f µ whee B H M µ is a new field vecto. Fo paamagnetism and diamagnetism B, M and H ae linealy elated to each othe, but fo feomagnetic mateials which exhibit hysteesis, a non-linea behaviou. The study of the electomagnets, motos and dynamos involves the poblem of cuent caying coils containing feomagnetic mateials, i.e., it involves the study of magnetic cicuits. We speak of the magnetic cicuits when all the magnetic flux pesent is confined to a athe welldefined path o paths. The magnetic cicuit fomula is: magnetomotive foce (M.M.F.) flux eluctance M.M.F. is also equal to NI whee N is the numbe of tuns of the coil wound ove the magnetic mateial and I the cuent flowing though each coil. Reluctance l R µ a whee l, a and µ ae the length, aea of coss-section and pemeability of the mateial. Additions of eluctances obey the same ules as additions of esistances. 6. TUTOR-MARKED ASSIGNMENT. Find the magnetizing field H and the magnetic flux density B at (a) a point 5 mm fom a long staight wie caying a cuent of 5 A and (b) the cente of a -tun solenoid which is.4m 7 long and beas a cuent of.6 A. ( µ 4π H/m).. A tooid of mean cicumfeence.5 m has 5 tuns, each caying a cuent of.5 A. (a) Find H and B if the tooid has an 4

115 PHY4 MODULE ai coe, (b) Find B and the magnetization M if the coe is filled with ion of elative pemeability A tooid with 5 tuns is wound on an ion ing 36 mm in coss-sectional aea, of.75-m mean cicumfeence and of 5 elative pemeability. If the windings cay.4a, find (a) the magnetizing field H (b) the m.m.f., (c) the magnetic induction B, (d) the magnetic flux, and (e) the eluctance of the cicuit. 7. REFERENCES/FURTHER READING IGNOU (5). Electicity and Magnetism, Physics PHE-7, New Delhi, India. SOLUTIONS AND ANSWERS UNIT SELF ASSESSMNET EXERCISES () Please see text. () The dipole moment pe molecule p The numbe of molecules pe unit volume n The dipole moment pe unit volume n p By definition, the dipole moment pe unit volume Polaisation P P n p (3) The dielectic constant K is given by K E E without the dielectic the electic field would be q E ε A ( C C N m )( 4 m ) 5

116 PHY4 ELECTROMAGNETISM 6.3 Vm 6.3 Vm Dielectic constant Vm The suface chage density on the plate is q A (4) Fom Eq. (.) C Cm m D ρ f (i) we know that D ε E + P Putting the value of D in Eq. (i) D ε E + P ρ f When P, the above equation becomes ε E ρ f (ii) E ρ f ε Eq. (.) is ε ε E ρ f + ρ p when ρ p, the above equation educes to o ε ε E ρ f E ρ f ε (iii) Eq. (ii) and Eq. (iii) ae the same. Hence pove the esult. 6

117 PHY4 MODULE (5) (i) The capacitance is A C ε l Befoe, calculating the capacitance, we will calculate the pemittivity of the dielectic as follows: ε ε ε (6.)(8.85) 5.3 faad/m faad/m Thus, A l C ε (5.3 faad/m).7 faad (ii) (iii) We know that Q CV (.7 faad) ( V).7 C The dielectic displacement is calculated as follows: V (5.3 faad / m)(v ) ε E ε l m D Cm (iv) The polaisation is V P D ε E D ε l.66 7 Cm (8.85 faad / m)(v ) 3 m (6) Let σ be the chage density on the suface of the plates. Consideing each plate as an infinite plane sheet chage, the intensity at a point between them due to positively chaged plate σ / ε. 7

118 PHY4 ELECTROMAGNETISM The intensity at the point due to the negatively chaged plate is also σ / ε acting in the same diection. Hence the esultant intensity at the point is σ σ σ + ε ε ε Since E 5 N/C and ε σ 5 ε σ ε faad/m 44.7 Cm Total chage on each plate ε E Aea of each plate C (7) Fom Eq. (.) we have E d l Fom vecto analysis we have E dl ( E) n ds ( n E) ds Suface Fo ( n E) ds to be zeo, the integand ( n E) has to be to zeo. Again, in as much as ( n E) epesents a space deivative opeation we can set n E to be eithe a constant o zeo. If we set n E then a tivial esult follows. So it is bette to choose n E a constant Applying this to Fig.., we get 8

119 PHY4 MODULE n E n E which is Eq. (.3a). (8) The integal fom of Gauss' law in dielectics is D ds total fee chage enclosed Suface (Refe to the Fig..) ( n Dn n D n ) ds σ f whee σ f is the suface chage density on the inteface between the dielectics and n the unit vecto along the outwad dawn nomal to the suface D n and D n ae the nomal components of the displacement vecto in media and espectively. When σ, we get n D n D n n Now D εe and D ε E n n n n ds D ε E ε ε n E n n E n o n E n E n n ε ε Thus we find that the nomal component of E is discontinuous. 9

120 PHY4 ELECTROMAGNETISM UNIT SOLUTION & ANSWERS SELF ASSESSMENT EXERCISE'S () The potential diffeence (V) between the plates is not changed. But the electic field between the plates is V /( d / ) ( V / d) twice the value of the electic field E. The doubling of the electic field doubles the chage on each plate. Theefoe, C Q / V also doubles. Thus if we halve the distance of sepaation between the plates, the capacitance doubles. () We know that C Q / V C µ F and V. F 4 V Q CV. 4 C.4 C (3) The enegy stoed in a capacito is W C φ It can be witten W Cφ φ (i) We know that Q C φ (ii) Using Eq. (ii) in Eq. (i), we get W Qφ Hence pove, the esult. (4) C A / d ε Hee, ε 8.85 F/m, A 4 m, d 3 m

121 PHY4 MODULE Theefoe, C Hee C is the chage that aises the potential by unity o the chage holding capacity. (5) We have ε Capacit. with dielectic between plates Capacit. with fee space between plates Hee ε 3. Thus the capacitance of the capacito will get tebled when the dielectic ( ε 3) is filled up in all the ai space. Now a dielectic mateial is intoduced. Let its thickness be t. The capacity of the capacitance is F ε A, d C ai C dielectic ε A ( d t + t / ε ) Cdielectic [ ε A/( d t ε )] d Cai ε A d d t + t / ε Hee Theefoe, 3 t d and ε d d d t + t / ε d d Theefoe C C dielectic ai d d / That is, the capacitance will get doubled. (6) C πε ε / ln( / 8) and

122 PHY4 ELECTROMAGNETISM C C C πε ε ln(5 / 4) ln( / 8) / ln(5/ 4) o C C (7) When the capacito ae connected in seies, the equivalent is given by C C T. µ F Q V C T.. 6 C 6 Q. C.5 V 44 V Q. C. V V Q. C3. V 3 V (8) The aangement is shown in Fig... Let C 4 be the effective capacitance of C and C. Using seies law of capacitos C 4 C + C o CC C4 C + C

123 PHY4 MODULE This capacitance C 4 then adds to C 3 to give the total capacitance C of the combination i.e., o C C 4 + C 3 C C 3 CC + C + C Fig.. 3