F Q E v B MAGNETOSTATICS. Creation of magnetic field B. Effect of B on a moving charge. On moving charges only. Stationary and moving charges

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1 MAGNETOSTATICS Ceation of magnetic field. Effect of on a moving chage. Take the second case: F Q v mag On moving chages only F QE v Stationay and moving chages dw F dl Analysis on F mag : mag mag Qv. vdt Magnetic foces do no wok!

2 Cuent-caying wie v v t I v F mag vdq vdl I dl K di Kˆ K v dl Fmag v da K da Chage-flow ove a suface with a suface cuent density K dl K V l t d it Volume cuent density di da da v Flow F mag v d mag d vˆ

3 The amount of chage passing though aea A in time Δt is nq(avt). Amount flowing pe unit aea pe unit time is nqv, giving you the cuent density. j = nqv. dq I jds (ate of flow of chage) dt s Conside a closed suface enclosing volume V. If be the chage density fo an infinitesimal volume dv, then dv epesents the total chage inside the volume V. Accoding to the law of consevation o of chage, the ate of flow of chage though the enclosed suface is equal to the ate of decease of chage in it. s j ds - t V dv V dv t

4 Accoding to divegence theoem s --> j dv dv t j ds jdv V V V j t This is called the equation of continuity it and epesents the physical facts of consevation of chages.

5 IOT SAVAT LAW Ceation of magnetic field with the movement of chages. Unde steady state movement of chages (steady cuent), themagnetic field poduced is given by the iot-savat Law: (i) d I (ii) d dl 1 (iii) d (iv) d s in with k as constant of popotionality, p in SI units, k NA Idl sin d 4 dl Idl p

6 Idl ( l ) d 4 3 ( l ) I 1 dl ( l) 3 4 ( l ) P l dl

7 EXAMPLE - 1 I dl (yeˆ ˆ z le x) 3/ 4 y l P y 1 Using the tan = l /y /y, we get the final esult as I sin sin eˆ 4y 1 z I sin eˆ 4y If - 1 = : z If = / Infinite line: y O l dl I e ˆ y z x I

8 EXAMPLE - I adeˆ zeˆ ˆ z ae 4 3/ z a azd e a d e I z 4 3/ z a ˆ ˆ I P O dl deˆ I a I a eˆ ˆ e 3/ z 3/ 4 z a z a z

9 If thee ae n tuns, then ni a z a 3/ eˆ z At the cente of the loop: I e ˆ a z

10 EXAMPLE - 3 P z 1 dz Use the evaluation of the magnetic field fo a loop and follow supeposition pinciple to evaluate fo a solenoid. The vaiable is now z. What about cuent? If thee ae n no. of tuns pe unit length and each tun having cuent I, then I a ni a dz d (ndz)e ˆ ˆ z e 3/ 3/ z z a z a

11 z a cot dz a cosec d z a a cot a a cosec ni ni sindeˆ cos cos eˆ 1 z 1 z Case 1 If we take a long solenoid (adius of the solenoid vey small compaed to its length) and obsevation point p is well with in the solenoid, then 1 =and = nieˆ z

12 Case When the obsevation point P is taken one end of the solenoid 1 = and = / ni Hence in case of semi-infinitely long solenoid, the magnetic field at y g, g a point at the end of the solenoid is half the magnetic field at a point well inside the solenoid.

13 Fo suface and volume cuents, iot-savat law becomes da K ˆ ) ( 4 ) ( d ˆ ) ( 4 ) ( 4 4

14 Infinite plane of unifom cuent sheet Constant cuent density K / y K / Cuent sheet lies in the xy plane and K Keˆ y xeˆ x yeˆ y x eˆ yeˆ x e y zeˆ xx eˆ y y eˆ zeˆ da z x y z x x y y z dxdy K Kzeˆ O da / / x x eˆ x z x P (x,y,z) z

15 Use iot-savat law fo a suface cuent density and integate ˆ ˆ d d 3 ˆ ˆ ' 4 x z ze x x e dx dy K x x y y x x y y z ˆ ˆ K ' ze X e dx dy 3 4 x z ze X e dx dy K X Y z x x X y y Y z-component will vanish because the integand is an odd function of X / x e x z z K K e ˆ 1 ˆ 1

16 The Divegence and cul of Nonzeo cul?? dl I dl s I I s dl If we use cylindical coodinates (s,, z) with cuent along z axis dl dseˆ I ˆ e s sdeˆ dzˆ s e z I 1 sd I d s I

17 undle of staight wies I 5 I 1 I Each wie that passes though the loop contibutes I dl I enc If the flow of chage is epesented by a volume chage density I enc da Applying stokes theoem Integal taken ove the suface bounded by the loop da da Above deivation is esticted by the condition that we need infinitely staight line cuents I 4 I 3

18 Divegence and cul of iot-savat law fo a volume cuent distibution is ( ) ˆ ( ) dv ' 4 is a function of is a function of x, y, z x, y, z x xeˆ x y y eˆ y z z eˆ z dv ' dxdydz d x, y, z x, y, z Integation is done ove the pimed coodinates Divegence and cul is done ove the unpimed coodinates

19 Applying divegence to the magnetic field due to a volume chage distibution d ˆ 4 4 A A A A A A ˆ ˆ ˆ because does not depend on unpimed coodinates ˆ and

20 Applying cul to the magnetic field due to a volume chage distibution d ˆ 4 ) ( ) ( A A A A A ˆ ˆ ˆ ˆ ˆ ˆ (a) The tems involving deivatives of is dopped since does not depend on (x,y,z) ) ( 4 ˆ 3 The second tem in (a) ) ( 4 ( ) integates to zeo 3 ( )4 ( )d ( ) 3 ( )4 ( )d () 4

21 How does the othe tem vanish? th d i ti t l t it h t ecause the deivative acts only on tem, we can switch to ˆ ˆ Conside the x-component x x x x x x ) ( ) ( ) ( ) ( We ae dealing with steady cuents, hence second tems in zeo x x 3 ) ( ˆ Contibution to the integal fom this tem is da x x d x x 3 3 ) ( ) ( a d d S V 3 3 We ae integating ove the souce egion that include all the cuent. On the bounday the cuent is zeo and hence the suface integal vanishes

22 FUTHE EXAMPLES FO IOT SAVAT LAW b P a b I P a

23 AMPEE S LAW in diffeential fom Using Stokes theoem da da dl da is the total cuent passing though the suface-- I enc dl I enc in integal fom

24 EXAMPLES Example 1 s Ampeian loop I y symmety, the magnitude of is constant aound an ampeian loop of adius s dl dl s I enc I eˆ s I

25 Example Magnetic field of an infinite unifom suface cuent flowing ove the xy plane K Keˆ x can only have a y-component It points towads the left above the plane and towads the ight in the plane below K dl l I enc Kl Ampeian loop z K K z l

26 Example 3 Magnetic field of a vey long solenoid consisting of n closely wound tuns pe unit length on a cylinde of adius and caying a steady cuent I Loop1 dl Loop ( a) ( b) L I (a) (b) enc dl L I enc nil nieˆ Keˆ Inside the solenoid z z Outside the solenoid K ni a b 1 Ampeian loops L

27 EXAMPLE 4 A long coppe pipe with thick walls has an inne adius and an oute adius. What is the cuent density fo all? Use Ampee's Law to find the magnetic fields as a function of adial distance fom the cente of the pipe. <<: < <: () I 3 eˆ z I > :

28 EXAMPLE 5 Along coppe wie of coss-sectional adius caies a k cuent density ( ) e eˆ. Use Ampee's Law to z detemine as a function of the distance fom the cente of the wie. inside the wie: 1 ke e e k k k e e k k k outside the wie: k k k e e k k 1ke e e k k k

29 () A( ) POISSON S EQN. IN MAGNETOSTATICS da S Flux though any closed suface is always zeo. No monopole fo the magnetic chage A() is the vecto potential Divegence of a cul is always zeo A A A We can add to magnetic potential any function whose cul vanishes, with no effect on

30 We use this feedom to eliminate the divegence of A A If the oiginal potential A is not divegenceless, then add to it gadient of function such that, A becomes divegenceless. Gauge Tansfomation: A A A A 1 A d 4 A μ P i In the cuent fee egion, Poisson s eqn. Coulomb gauge A A Laplace Laplace eqn.

31 dl ˆ ( ) I 4 dl ˆ 1 dl dl dl I dl I dl 4 C 4 C Idl K( )da A( ) C S dl A ( )d 4 V

32 DIVEGENCE OF A Idl I A( ) 4 C 4 C dl 1 1 dl dl A( I 1 ) dl 4 C oi 1 da 4 S dl 1 = dl

33 MAGNETIC SCALA POTENTIAL In the cuent fee egion, Theefoe, can be expessed as: m We call m as magnetic scala potential. We see that m satisfies the Laplace s equation. m m

34 ELECTOSTATICS AND MAGNETOSTATICS E E q enc Eds dl I V A V A I enc

35 MULTIPOLE EXPANSION FO MAGNETIC VECTO POTENTIAL Simila to the electic scala potential, one can use multipole expansion to find out the magnetic vecto potentialatafa away place due to a cuent distibution. p d d = dl

36 cos 1 cos d This expession can be ewitten as d cos 1 d Using bionomial expansion x x x x cos 8 3 cos d 8 d Vecto potential due to the cuent loop is I dl A 4 d I A d l cos d l ( ) cos d l Vecto potential due to the cuent loop is 4 d 3 A dl cos dl ( ) cos dl 4

37 As in multipole expession of V, The fist tem, goes like 1/, is monopole tem. The second tem, which goes like 1/,isdipole tem. The thid tem, goes like 1/ 3,isquadupole tem. The magnetic monopole tem is always zeo as the total vecto displacement aound close loop is dl= Hence no magnetic monopole exists in natue. In the absence of the monopole tem, the dominant tem is the dipole-except p in the ae case whee it, too vanishes. A dipole I 4 I 4 cos dl ˆ The integal in the above expession afte some manipulations can be witten as 1 ˆ dl ˆ dl μi 1 A = - ˆ dl dipole 4π dl

38 μ ˆ m A dip= 4π 1 m= I d l Whee m is the magnetic dipole moment of the loop, defined as If the cuent loop is a plane loop (cuent located on the suface of a plane, then dl Is the aea of the shaded tiangle as shown in figue. So the integal is the aea of whole loop line 1 dl Aea,a

39 In this case, the dipole moment of the cuent loop is equal to m I a Whee the diection of a must be consistent with the diection of the cuent loop (ight hand ule) Since the magnetic monopole tem is always zeo, the magnetic dipole moment always independent d of oigin. i is Assuming that the magnetic dipole is located at the oigin of ou coodinate system and that m is pointing along the positive z axis, μ m ˆ μ msinθ A dipole= = e ˆ 4π 4π Coesponding magnetic field is equal to dipole A dipole 1 m sin 1 m sin sin ˆ ˆ sin 4 4 m 4 dipole cos e 3 sin e ˆ ˆ 1 ˆ ˆ 4 3 me 3 e m

40 Poblem 5.38: A phonogaph ecod of adius, caying a unifom suface chage, is otating with a constant angula velocity. Find the magnetic dipole moment. Poblem 5.39: Find the magnetic dipole moment of a spinning spheical shell of adius, caies a unifom suface chage. Show that fo > the potential is that of pefect dipole. Poblem 5.41: Show that magnetic dipole moment of an abitay localized cuent loop is independent of the location of efeence point.

41 Magnetic dipole and magnetic dipole moment: A magnetic dipole consists of pai of magnetic dipole of equal and opposite stength sepaated by small distance. Examples of magnet dipoles ae Magnetic needle, a magnet, Cuent caying solenoid, A cuent loop etc. Atom is also consideed to behave like a dipole - so the fundamental magnetic dipole in natue is associated with the electons.

42 The poduct of pole stength of eithe poles and distance between them is called as magnetic dipole moment. The distance between two poles is called magnetic length. o m M l m M ( l ) m N l S The vecto l is diected fom south to noth. Thus the diection of magnetic dipole moment is fom south to noth S.I. units of M is ampee-mete (Am) and that of m is ampee-mete (Am ) o joule tesla -1

43 Field at a point due to magnetic dipole in the end on-position (on the axis) Let NS be a magnetic dipole of pole stength M and length l. Let P be the point on its axis at a distance d fom the cente of the dipole. l S N S p N d The magnetic field at point P due to noth pole is M 4 l d And will be diected away fom the magnet. The magnetic field at point P due to South pole pole is

44 M 4 l d And will be diected towads fom the magnet. Theefoe esultant field is = N - S 4Mld 4 d l md 4 l d Hee m = Ml is dipole moment of the magnet If d>>l mm 4 d 3

45 Field due to a magnetic dipole in the boad side on position Let NS be a magnetic dipole of pole stength M and length l. Let P be a point on the boad side on position of the dipole at the distance d fom its cente. N P S The magnetic field at point P due to noth pole is d N M PN 3 4 PN S l N The magnetic field at point P due to noth pole is

46 S M PS 3 4 PS Now NP = PS = (d +l ) 1/ Theefoe esultant field at P M d l 3 / M d l 3 / M d l 3 / m 4 l 3 / d ( NP NS l PS)

47 If d >> l Thus m 3 4 d axial = equatoial The same conclusion we had dawn fo electic field

48 When a pemanent magnet is placed in a field, Noth pole will expeience a foce in the diection of field and south pole has a foce opposite to the pole. If the field is unifom the net foce is zeo, but thee is a toque. Fo electic dipole, the toque is given by elation N p E p: electic dipole moment and E: unifom electic field The coesponding expession fo toque of magnetic dipole in magnetic field is N m In addition to the pemanent magnets being dipole, we see that cuent loops ae also magnetic dipole.

49 Toque on a dipole (ba magnet) in a magnetic field: If a magnetic dipole is placed in a unifom magnetic field as shown in figue, the Noth and South poles of the magnet will expeience equal and opposite foces. N l m M M S Z Let M be the pole stength of each pole and moment m and magnetic field, then be the angle between magnetic dipole Foce on Noth pole = M along Foce on South pole = M opposite to These foces will constitute a couple which tends to otate the magnet in the diection of. Thus a magnet expeience a toque

50 N = Foce distance between the foce = M ZN = M (SN sin) = M (l sin) because In tiangle SZN, sin = ZN/SN o ZN = SN sin o N = (M l ) sin = msin In vecto fom N m When = 1 unit and = 9, then N = m Thus the magnetic dipole moment can be defined as the toque acting on a magnetic dipole placed nomal to the unifom magnetic field of unit stength

51 Let us calculate the toque on a ectangula cuent loop in a unifom field. Conside a ectangula loop of sides a and b as shown in figue placed in a unifom magnetic field and let the diection of the field is along z-axis. The magnetic dipole moment is pependicula to the cuent loop and makes an angle with magnetic field. F 1 F 4 I 1 4 I 3 a F F 3 b

52 Since the cuents ae opposite on opposite sides of the loop, the foces ae also opposite, so thee is no net foce on the loop (when the field is unifom). The foces on the loop sides 3 and 4 tend to stetch the loop, but do not otate the loop. ecause of the foces on the two sides maked 1 and, tend to otate the loop about y-axis and geneates the toque. The magnitude of foces F 1 and F is F 1 F Ib And thei moment o leve am is asin So the toque N is equal to N Iabsin msin

53 Whee m = Iab is the magnetic dipole moment of the loop O N m The toque given by above equation is a special case, the esult is ight fo small cuent loop of any shape in the unifom magnetic field. Wok Done on a Magnetic Dipole. Since a magnetic dipole placed in an extenal magnetic field expeiences a toque, wok (positive o negative) must be done by an extenal agent in ode to change the oientation of the dipole. Let us calculate how much wok is done by the field when otating the dipole fom angle A to. W A Nd

54 W m sin d A m cos cos A If the dipole is initially at ight angle to the field i.e. A = 9 and finally makes an angle with field i.e. =, then W m cos cos9 -mcos This wok done is equal to the potential of the dipole U p -m cos - m

55 In case of non-unifom field the above discussion is exact only fo a pefect dipole of infinitesimal size. Now we will calculate foce of a infinitesimal loop of dipole moment m in the field. We have seen that the potential enegy of a magnetic dipole m in a magnetic field is U p -m cos - m We know foce is elated to potential enegy by the elation F U Theefoe F m Using poduct ule (ule 4)

56 A A A A A Theefoe F m m m m m Since m is not function of space co-odinate Theefoe m and m And Theefoe

57 F m Povided thee is no extenal cuent at the actual location of the dipole We must be vey caeful about the analogies between electic and magnetic dipoles. Fo example foce on a magnetic dipole in non unifom field is F m Whee as fo electic field F p E So one should be vey alet when solving the poblems.