3. Magnetostatic fields

Transcription

1 3. Magnetostatic fields D. Rakhesh Singh Kshetimayum 1 Electomagnetic Field Theoy by R. S. Kshetimayum

2 3.1 Intoduction to electic cuents Electic cuents Ohm s law Kichoff s law Joule s law Bounday conditions Kichoff s cuent law Kichoff s voltage law Fig. 3.1 Electic cuents 2 Electomagnetic Field Theoy by R. S. Kshetimayum

3 3.1 Intoduction to electic cuents So fa we have discussed electostatic fields associated with stationay chages What happens when these chages stated moving with unifom velocity? It ceates electic cuents and electic cuents ceates magnetic fields In electic cuents, we will study Ohm s law, Kichoff s law Joule s law Behavio of cuent density at a media inteface 3 Electomagnetic Field Theoy by R. S. Kshetimayum

4 3.1 Intoduction to electic cuents Cuent density What is this? Fo a paticula suface S in a conducto, i is the flux of the cuent density vecto j ove that suface o mathematically i = j ds Ohm s law S It states that the cuent passing though a homogeneous conducto is popotional to the potential diffeence applied acoss it and the constant of popotionality is 1/R which is dependent on the mateial paametes of the conducto 4 Electomagnetic Field Theoy by R. S. Kshetimayum

5 3.1 Intoduction to electic cuents Mathematically, i V i = V R Fom the elation between cuent density (j) and cuent (i), electic potential (V) with electic field (E) and esistance (R) with esistivity (ρ) in an isotopic mateial we can obtain the Ohm s law in point fom as 5 Electomagnetic Field Theoy by R. S. Kshetimayum V Edl Edlds E jds = i = j E R = σ dl = ρ ρdl = ρ = ds

6 3.1 Intoduction to electic cuents whee σ is the conductivity and 1 σ ρ = is the esistivity of the isotopic mateial Table 3.1 Conductivities of some common mateials Mateial σ (S/m) Rubbe Wate Gold Aluminum Coppe Electomagnetic Field Theoy by R. S. Kshetimayum

7 3.1 Intoduction to electic cuents Some points on pefect conductos and electic fields: Pefect conductos o metals have infinite conductivity ideally An infinite conductivity means fo any non-zeo electic field one would get an infinite cuent density which is physically impossible Pefect conductos do not have any electic fields inside it Pefect conductos ae always an equipotential suface At the suface of the pefect conducto, the tangential component of the electic field must be zeo 7 Electomagnetic Field Theoy by R. S. Kshetimayum

8 3.2 Equation of continuity and KCL j ds i Fig. 3.3 Equation of continuity dq d d = = = ρ vdv j ds ρvdv dt dt V dt S V 8 Electomagnetic Field Theoy by R. S. Kshetimayum

9 3.2 Equation of continuity and KCL The above equation is integal fom of equation of continuity It states that any change of chage in a egion must be accompanied by a flow of chage acoss the suface bounding the egion It is basically a pinciple of consevation of chage By applying the divegence theoem d ρ d ρ dt dt V V jdv = dv j + dv = V V V 0 9 Electomagnetic Field Theoy by R. S. Kshetimayum

10 3.2 Equation of continuity and KCL Since the volume unde consideation is abitay d ρ dt V j + = 0 Diffeential fom of the equation of the continuity j = d ρ dt V At steady state, thee can be no points of changing chage density j = 0 j ds = 0 S 10 Electomagnetic Field Theoy by R. S. Kshetimayum

11 3.2 Equation of continuity and KCL The net steady cuent though any closed suface is zeo If we shink the closed suface to a point, it becomes Kichoff s cuent law (KCL) I = 0 KCL states that at any node (junction) in an electical cicuit, the sum of cuents flowing into that node is equal to the sum of cuents flowing out of that node 11 Electomagnetic Field Theoy by R. S. Kshetimayum

12 3.3 Electomotive foce and KVL Fig. 3.4 Poof of KVL 12 Electomagnetic Field Theoy by R. S. Kshetimayum

13 3.3 Electomotive foce and KVL When a esisto is connected between teminals 1 and 2 of the battey, The total electic field intensity s (total electic field compise of electostatic electic field as well as the impessed electic field caused by chemical action) elation to the cuent density is given as j = σ E + E ( c n ) whee the supescipt c is fo consevative field and the supescipt n is fo non-consevative field 13 Electomagnetic Field Theoy by R. S. Kshetimayum

14 3.3 Electomotive foce and KVL Consevative electic field exists both inside the battey and along the wie outside the battey, While the impessed non-consevative electic field exists inside the battey only The line integal of the total electic field aound the closed cicuit gives C j σ ( ) c n E + E dl = dl C 14 Electomagnetic Field Theoy by R. S. Kshetimayum

15 3.3 Electomotive foce and KVL Note that the line integal of the consevative field ove a closed loop is zeo The line integal of the non-consevative field in non-zeo and is equal to the emf of the battey souce Since non-consevative field outside the battey is zeo, C 2 j σ ( ) ( ) c n n E + E dl = E dl = ξ = dl 1 C 15 Electomagnetic Field Theoy by R. S. Kshetimayum

16 3.3 Electomotive foce and KVL Note that i = ja o j=i/a Theefoe, the voltage dop acoss the esisto is V=jl/σ=il/σA=iρl/A=iR If thee ae moe than one souce of emf and moe than one esisto in the closed path, we get Kichhoff's Voltage Law (KVL) M N ξ = i R m n n m= 1 n= 1 KVL states that aound a closed path in an electic cicuit, the algebaic sum of the emfs is equal to the algebaic sum of the voltage dops acoss the esistances 16 Electomagnetic Field Theoy by R. S. Kshetimayum

17 3.4 Joule s law and powe dissipation Conside a medium in which chages ae moving with an aveage velocity v unde the influence of an electic field If ρ v is the volume chage density, then the foce expeienced by the chage in the volume dv is df = dqe = ρ dve V If the chage moves a distance dl in a time dt, the wok done by the electic field is dw = df dl = ρ dve vdt = E v dvdt = E jdvdt = j Edvdt V ( ρ ) V 17 Electomagnetic Field Theoy by R. S. Kshetimayum

18 3.4 Joule s law and powe dissipation Then, the elemental wok done pe unit time is dw dp = = j Edv dt If we define the powe density p as the powe pe unit volume, then, point fom of Joule s law is p = j E The powe associated with the volume (integal fom of Joule s law) is given by P = pdv = j Edv V V 18 Electomagnetic Field Theoy by R. S. Kshetimayum

19 3.5 Bounday conditions fo cuent density σ 1 S σ 2 Fig. 3.5 Bounday conditions fo cuent density 19 Electomagnetic Field Theoy by R. S. Kshetimayum

20 3.5 Bounday conditions fo cuent density How does the cuent density vecto changes when passing though an inteface of two media of diffeent conductivities σ 1 and σ 2? Let us constuct a pillbox whose height is so small that the contibution fom the cuved suface of the cylinde to the cuent can be neglected Applying equation of continuity and computing the suface integals, we have, i j ds = ˆ ˆ ˆ = = S ( J ) 1 J1 = J n1 J 2 0 n J1 s n J 2 s = 0 n 0 n 20 Electomagnetic Field Theoy by R. S. Kshetimayum

21 3.5 Bounday conditions fo cuent density It states that the nomal component of electic cuent density is continuous acoss the bounday Since, we have anothe bounday condition that the tangential component of the electic field is continuous acoss the bounday, that is, ) ) J J J ( ) 1 2 t Jt Jt σ n E 1 1 E2 = 0 n = 0 = 0 = σ1 σ 2 σ1 σ 2 Jt 2 σ 2 21 Electomagnetic Field Theoy by R. S. Kshetimayum

22 3.5 Bounday conditions fo cuent density The atio of the tangential components of the cuent densities at the inteface is equal to the atio of the conductivities of the two media We can also calculate the fee chage density fom the bounday condition on the nomal components of the electic flux densities as follows: D D = ρ ρ = ε E ε E J ρ = ε J ε = J ε ε n1 n2 1 2 n1 n2 S S 1 n1 2 n2 S 1 2 n1 σ1 σ 2 σ1 σ 2 22 Electomagnetic Field Theoy by R. S. Kshetimayum

23 3.6 Intoduction to magnetostatics In static magnetic fields, the thee fundamental laws ae Biot Savat s Law, Gauss s law fo magnetic fields and Ampee s cicuital law Biot Savat law gives the magnetic field due to a cuent caying element Fom Gauss s law fo magnetic fields, we can undestand that the magnetic field lines ae always continuous In othe wods, magnetic monopole does not exist in natue Ampee s cicuital law states that a cuent caying loop poduces a magnetic field 23 Electomagnetic Field Theoy by R. S. Kshetimayum

24 3.1 Intoduction to electic cuents Magnetostatics Biot Savat s law Gauss s law fo magnetic fields Magnetic vecto potential Bounday conditions Self and mutual inductance Ampee s law Magnetization Fig. 3.6 Magnetostatics Magnetic vecto potential in mateials 24 Electomagnetic Field Theoy by R. S. Kshetimayum

25 3.6 Intoduction to magnetostatics It is easie to find magnetic fields fom the cul of magnetic vecto potential whose diection is along the diection of electic cuent density Anothe topic we will study hee is that how do magnetic fields behave in a medium We will also ty to find the self and mutual inductance and magnetic enegy 25 Electomagnetic Field Theoy by R. S. Kshetimayum

26 3.7 Biot Savat s law The magnetic field due to a cuent caying segment is popotional to its length and the cuent it is caying and the sine of the angle between and invesely popotional to the squae of distance of the point of obsevation P fom the souce cuent element Mathematically, dl $ dl $ µ 0 dl $ db I db = ki db = I π Idl 26 Electomagnetic Field Theoy by R. S. Kshetimayum

27 3.8 Gauss s law fo magnetic fields In studying electic fields, we found that electic chages could be sepaated fom each othe such that a positive chage existed independently fom a negative chage Would the same sepaation of magnetic poles exist? A magnetic monopole has not been obseved o found in natue We find that magnetic field lines ae continuous and do not oiginate o teminate at a point Enclosing an abitay point with a closed suface, we can expess this fact mathematically integal fom of 3 d Maxwell s equations Ψ = B ds = 0 27 Electomagnetic Field Theoy by R. S. Kshetimayum S

28 3.8 Gauss s law fo magnetic fields Using the divegence theoem, Ψ = B ds = B dv 0 ( ) = S V In ode this integal to be equal to zeo fo any abitay volume, the integand itself must be identically zeo which gives diffeential fom of 3 d Maxwell s equations u B =0 28 Electomagnetic Field Theoy by R. S. Kshetimayum

29 3.9 Ampee s cicuital law In 1820, Chistian Oested obseved that compass needles wee deflected when an electical cuent flowed though a neaby wie Right hand gip ule: if you thumb points in the diection of cuent flow, then you finges gip points in the diection of magnetic field Ande Ampee fomulated that the line integal of magnetic field aound any closed path equals µ 0 times the cuent enclosed by the suface bounded by the closed path 29 Electomagnetic Field Theoy by R. S. Kshetimayum

30 3.9 Ampee s cicuital law Incomplete integal fom of 4 th Maxwell s equation B dl = µ 0 C I enclosed By application of Stoke s theoem C B dl = S ( B) ds = µ 0 S J ds In ode the integal to be equal on both sides of the above equation fo any abitay suface, the two integands must be equal 30 Electomagnetic Field Theoy by R. S. Kshetimayum

31 3.9 Ampee s cicuital law Incomplete diffeential fom of 4 th Maxwell s equation u B = µ 0 uu J Note that thee is a fundamental flaw in this Ampee s cicuital law Maxwell in fact coected this Ampee s cicuital law by adding displacement cuent in the RHS Loentz foce equation: fo a chage q moving in the unifom field of both electic and magnetic fields, the total foce on the chage is F = F + F = qe + qv B E M 31 Electomagnetic Field Theoy by R. S. Kshetimayum

32 3.10 Magnetic vecto potential Some cases, it is expedient to wok with magnetic vecto potential and then obtain magnetic flux density Since magnetic flux density is solenoidal, its divegence is zeo ( u B) =0 A vecto whose divegence is zeo can be expessed in tem of the cul of anothe vecto quantity u u B = A 32 Electomagnetic Field Theoy by R. S. Kshetimayum

33 3.10 Magnetic vecto potential Fom Biot Savat s law, u B = I dl u µ O ' R 3 4π R u R = (x-x') x$ +(y-y') $ y +(z-z') z$ It is a standad notation to choose pimed coodinates fo the souce and unpimed coodinates fo the field o obsevation point u Q 1 ( ) = - R 3 R R u µ I 1 O B = ( ) d l ' 4π R whee the negative sign has been eliminated by evesing the tems of the vecto poduct 33 Electomagnetic Field Theoy by R. S. Kshetimayum

34 3.10 Magnetic vecto potential Since 1 dl ' 1 ( ) d l ' = ( ) - ( d l ' ) R R R Since the cul in unpimed vaiables is taken w..t. the pimed vaiables of the souce point, we have, d l ' = 0 ' = I ( d u µ l O B ) 4π R 34 Electomagnetic Field Theoy by R. S. Kshetimayum

35 3.10 Magnetic vecto potential The integation and cul ae w..t. to two diffeent sets of vaiables, so we can intechange the ode and wite the peceding equation as u µ 0I dl ' u µ 0I dl ' B = [ ] A = 4 π R 4 π R Genealizing line cuent density in tems of the volume cuent density, u A µ 0 J = V dv ' 4π R 35 Electomagnetic Field Theoy by R. S. Kshetimayum

36 3.10 Magnetic vecto potential v v Fig. 3.8 (a) Electon obit aound nucleus ceating magnetic dipole moment; Magnetization in (b) nonmagnetic and (c) magnetic mateials 36 Electomagnetic Field Theoy by R. S. Kshetimayum

37 3.10 Magnetic vecto potential Magnetization The magnetic moment of an electon is defined as u $ 2 m = n I π d = n$ I S whee I is the bound cuent (bound to the atom and it is caused by obiting electons aound the nucleus of the atom) $n is the diection nomal to the plane in which the electon obits and d is the adius of obit (see Fig. 3.8 (a)) 37 Electomagnetic Field Theoy by R. S. Kshetimayum

38 3.10 Magnetic vecto potential Magnetization is magnetic moment pe unit volume The magnetization fo N atoms in a volume v in which the i th atom has the magnetic moment is defined as m uu i uu M N 1 uu A = lim mi [ ] v 0 v m i= 1 Mateials like fee space, ai ae nonmagnetic (µ is appoximately 1) Fo non-magnetic mateials: (see fo example Fig. 3.8 (b), in a volume, the vecto sum of all the magnetic moments is zeo) 38 Electomagnetic Field Theoy by R. S. Kshetimayum

39 3.10 Magnetic vecto potential Fo magnetic mateials: (see fo instance Fig. 3.8 (c), in a volume, the vecto sum of all the magnetic moments is nonzeo) Given a magnetization which is non-zeo fo a magnetic mateial in a volume, the magnetic dipole moment due to an element of volume dv can be witten as u dm uu = M dv The contibution of due to is d A u M d m u 39 Electomagnetic Field Theoy by R. S. Kshetimayum

40 3.10 Magnetic vecto potential µ dm ˆ µ d A = 4π 4π ( ) ' M dv 0 0 = ˆ 2 2 The magnetic vecto potential and magnetic flux density could be calculated as uu u µ o M A = dv' 3 4π ' V u u B = A 40 Electomagnetic Field Theoy by R. S. Kshetimayum

41 3.10 Magnetic vecto potential Magnetic vecto potential in mateials Let us ty to expess this magnetic vecto potential in tems of bound suface and volume cuent density 1 ' $ u µ uu 0 1 Q ' ( ) = 2 A = M ' ( ) dv' 4π We also have, uu M 1 uu 1 uu Q ' ( ) = ' ( ) M + ' M uu uu 1 1 uu M M ' ( ) = ' M - ' ( ) 41 Electomagnetic Field Theoy by R. S. Kshetimayum

42 3.10 Magnetic vecto potential uu µ 0 1 uu M A= ( ' M - ' ) dv' 4π ' v ' M ' M ' Q dv = ds ' ' V S The poof fo the above equality, we will solve in example 3.5 = A = µ 0 4π V ' µ 0 4π 1 V ' 1 ( ) ( ) ' ' 0 ' M dv + M ds µ 4π µ 4π 1 1 ( ) ( ) ' ' 0 ' M dv + M nˆ ds S ' S ' 42 Electomagnetic Field Theoy by R. S. Kshetimayum

43 3.10 Magnetic vecto potential The above equation can be witten in the fom below Whee A = µ 0 4π J vb ' dv ' + ' V S J bound volume cuent density is given by uuu J vb = uu M bound suface cuent density is expessed as uuu uu J = M nˆ sb sb ds ' 43 Electomagnetic Field Theoy by R. S. Kshetimayum

44 3.10 Magnetic vecto potential Magnetized mateial can always be modeled in tems of bound suface and volume cuent density But they ae fictitious elements and can not be measued Only the magnetization is consideed to be eal and measuable 44 Electomagnetic Field Theoy by R. S. Kshetimayum

45 3.11 Magnetostatic bounday conditions S h J S h h S Fig. 3.9 Magnetostatic bounday conditions 45 Electomagnetic Field Theoy by R. S. Kshetimayum

46 3.11 Magnetostatic bounday conditions Nomal components of the magnetic flux density Conside a Gaussian pill-box at the inteface between two diffeent media, aanged as in the figue above The integal fom of Gauss s law tells us that B ds = 0 pillbox As the height of the pill-box h tends to zeo at the inteface, thee will be no contibution fom the cuved sufaces in the total magnetic flux, hence, we have 46 Electomagnetic Field Theoy by R. S. Kshetimayum

47 3.11 Magnetostatic bounday conditions S u u B d s+ B d s=0 S S S 1 2 B ds - B ds =0 n n 2 S (B - B )ds=0 n B =B n n 1 2 n The nomal components of the magnetic flux density ae continuous at the bounday 47 Electomagnetic Field Theoy by R. S. Kshetimayum

48 3.11 Magnetostatic bounday conditions Tangential components of the magnetic field intensity Applying Ampee s law to the closed path PQRSP H dl = H dl + H dl + H dl + H dl = PQ QR whee I is the total cuent enclosed by the closed path PQRS which lies in the xy plane Assume that x is along the diection of PQ in Fig. 3.9 At the inteface, h 0, the line integal along paths QR and SP ae negligible, hence, RS SP I 48 Electomagnetic Field Theoy by R. S. Kshetimayum

49 3.11 Magnetostatic bounday conditions PQ uu uu H d l + H d l =I RS uu uuu uu ( H - H ) x$ d l = J $ y dl h PQ 1 2 uu uu Q Lim J V h = J S h 0 is the definition of suface cuent density uu uuu uu ( H $ $ 1 - H 2 ) ( y z$ ) d l = J S y d l PQ Fom the popety of vecto scala tiple poduct, we have, V 49 Electomagnetic Field Theoy by R. S. Kshetimayum

50 3.11 Magnetostatic bounday conditions uu uuu uu uuu $ { ( - )} d l = { ( - )} $ uu y z$ H H z$ H H y d l = J $ y d l PQ PQ uu uuu uu z$ H H J ( 1-2 ) = S S The tangential component of the magnetic field intensity at the inteface is continuous unless thee is a suface cuent density pesent at the inteface 50 Electomagnetic Field Theoy by R. S. Kshetimayum

51 3.12 Self and mutual inductance A cicuit caying cuent I poduces a magnetic field which causes a flux to pass though each tun of the cicuit If the cicuit has N tuns, we define the magnetic flux linkage as Λ = Nψ. ψ = B ds Also, the magnetic flux linkage enclosed by the cuent caying conducto is popotional to the cuent caied by the conductos L= Λ/I= Λ I Λ = LI whee L is the constant of popotionality called the inductance of the cicuit (unit: Heny) Nψ I 51 Electomagnetic Field Theoy by R. S. Kshetimayum

52 3.12 Self and mutual inductance The magnetic enegy stoed in an inducto is expessed fom cicuit theoy as: Wm = 1 LI 2 2 L = 2W m 2 I If instead of having a single cicuit, we have two cicuits caying cuents I 1 and I 2, a magnetic induction exists between two cicuits Fou components of fluxes ae poduced The flux ψ, 12 fo example, is the flux passing though the cicuit 1 due to cuent in cicuit 2 52 Electomagnetic Field Theoy by R. S. Kshetimayum

53 3.12 Self and mutual inductance ψ 12 = B2 S Define M 12 = Similaly, 1 u uu ds M Λ = I 21 Nψ I Λ = = I N ψ I The total enegy in the magnetic field is due to the sum of enegies 1 1 Wm = W + W + W = L I + L I ± M I I Electomagnetic Field Theoy by R. S. Kshetimayum

54 3.13 Summay Electic cuents Fig (a) Electic cuents in a nutshell Ohm s law Kichoff s law Joule s law Bounday conditions j = σeσ p = j E J n = = 1 J n2 Kichoff s cuent law I = 0 Kichoff s voltage law M N ξ = i R m n n m= 1 n= 1 ρ s J J t1 t 2 = J σ1 = σ n1 2 ε1 σ1 ε 2 σ 2 54 Electomagnetic Field Theoy by R. S. Kshetimayum

55 3.13 Summay Fig (b) Magnetostatics in a nutshell Biot Savat s law µ 0 dl db = I 4π R R Gauss s law fo magnetic fields 55 Ψ = B ds = 0 S Ampee s law B dl = µ 0 C 2 I enclosed uu M Magnetostatics Magnetic vecto potential Magnetization i= 1 u A N 1 uu A = lim mi [ ] v 0 v m Electomagnetic Field Theoy by R. S. Kshetimayum µ 0 J = V dv ' 4π R Self and mutual inductance L=Λ/I=NΨ/I M 21 Λ = = I N ψ I Bounday conditions Magnetic vecto potential in mateials A = µ 4π B n1 =B n2 zˆ ( H1 H 2 ) = J S J 0 vb ' dv ' + ' V S J sb ds '