(1.1) V which contains charges. If a charge density ρ, is defined as the limit of the ratio of the charge contained. 0, and if a force density f

Transcription

1 1.0 Review f Electrmagnetic Field Thery Selected aspects f electrmagnetic thery are reviewed in this sectin, with emphasis n cncepts which are useful in understanding magnet design. Detailed, rigrus treatments are presented in standard texts n the subject. [1,,3] 1.1 General Frm f the Equatins The cncepts f electric and magnetic fields are related t the bservatin f frces experienced by an electric charge. A charge f q culmbs mving with velcity v can experience a frce independent f its velcity and perpendicular t it. The ttal is the Lrentz frce F which can be expressed as F = q (1.1) ( E + v B) This serves t define the electric field intensity E and the magnetic flux density B in terms f the charge q, the velcity f the charge relative t the bserver v, and the ttal frce experienced by the charge F. Cnsider a vlume element V which cntains charges. If a charge density ρ, is defined as the limit f the rati f the charge cntained in V t V as V 0, and if a frce density f is defined as the limit f F V as V 0, then (1.1) becmes The quantity f = ρ E + ρ v B (1.) e e ρ e v represents a charge density in mtin which is a current density J. Thrughut these ntes it is implicitly assumed that there is n relative mtin f cmpnents: therefre, cnvectin currents which result frm the mtin f cnductrs are neglected. Equatin (1.) can therefre be written as f E = ρ e + J B (1.3) The first term f this equatin represents a frce n static charges whereas the secnd is a frce n mving charges r currents. In prblems assciated with magnet design, the 1 f 18

2 interactins between currents and magnetic fields are f primary interest. Frces due t the presence f free charge densities are usually negligible by cmparisn and (1.3) reduces t f = J B (1.4) Fr any particular case, the validity f this apprximatin can be checked by evaluating ρ and cmparing it with the result f (1.4). e E If it is pstulated that charge must be cnserved, the cncepts f charge density and current density can be cmbined mathematically t represent a law f cnservatin f charge as fllws d J nda= dt ρ dv e (1.5) In (1.5) n da is an incremental area element f a clsed surface and n is a nrmal, utwardly directed unit vectr at the element. The incremental vlume element within the clsed surface is dv. Equatin (1.5) therefre states that the net flw f charge ut f the clsed surface is equal t the rate f decrease f ttal charge within the enclsed vlume. In the structural and electrical prblems assciated with magnet design, free space charge densities can usually be neglected and thus (1.5) reduces t The equivalent differential frm f (1.6) is J n da=0 (1.6) J = 0 (1.7) Equatins (1.6) and (1.7) nrmally enter magnet design prblems implicitly rather than explicitly. Simply interpreted, bth equatins require that lines f current must frm clsed lps t be physically realizable. f 18

3 The ther cnditins fr physical realizability f the electric and magnetic fields were frmulated by Maxwell as fllws E dl = d dt E B nda d H dl nda J dt = nda (1.8) ε (1.9) The first integral in each equatin is taken arund a clsed cntur having an incremental length dl. The area integrals are taken ver a simply cnnected surface bunded by a cntur. The quantity H is the magnetic field intensity which in free space (that is, when magnetizable material is nt present) is related t the magnetic flux density thrugh B H = µ. The magnitude f the cnstant µ is dependent n the system f units emplyed. This quantity is the permeability f free space r vacuum, and has the value f 4π 10-7 H/m in the SI system. The cnstant ε is permittivity f free space which is a derived quantity. In SI units ε has a value f F/m. The differential frm f these equatins is B E= (1.10) E H ε = J (1.11) The secnd term in (1.9) and its cunterpart in (1.11) represent what is called the displacement current. These terms are usually negligible in magnet design prblems, as discussed in Sectin 1. Equatins (1.9) and (1.11) therefre reduce t H dl = H = J If the divergence f (1.10) is taken, the result is J nda (1.1) (1.13) 3 f 18

4 B ( ) E = Hwever, since the divergence f the curl f any vectr field is zer, and since the divergence and time peratrs are cmmutative, (1.14) becmes (1.14) ( B) = 0 Equatin (1.15) can be integrated with respect t time t yield (1.15) B = cnstant (1.16) Therefre, the divergence f B is a quantity independent f time. Experimental evidence shws that this cnstant is zer. The last f the equatins fr physical realizability is therefre B = 0 (1.17) Nte that this is a direct cnsequence f (1.10). In integral frm, (1.17) can be expressed as B 0 nda = (1.18) A simple interpretatin f (1.17) and (1.18) is that magnetic flux lines must frm clsed lps t be physically realizable (n magnetic mnples!). Either the integral equatins 91.6), (1.8), (1.1), and (1.18), r the differential equatins (1.7), (1.10), (1.13), and (1.17) frm the set f gverning equatins f interest in magnet design. The reductin f these equatins r f the mre general equatins with displacement current terms t magnetstatics can be dne directly by setting all terms invlving a time derivative t zer. The cnditins under which the displacement current terms can be neglected even thugh the situatin invlves variatins in time is discussed in the fllwing sectin. 4 f 18

5 1. Reductin t Magnet Systems and Magnetstatics In rder t illustrate cnditins under which the displacement current term can be neglected, cnsider a regin f space where B = µ H (1.19) J = σe (1.0) That is, the magnetic flux density is related t H thrugh the cnstant µ and the current density J is related t the electric field intensity E thrugh the cnstant σ, which is the electrical cnductivity. If the curl f (1.11) is taken and (1.19) and (1.0) are used, then ( B) = µ ( J ) + µ ε ( E) (1.1) This can be rewritten using (1.10) and (1.0) as fllws B B ( B) = µ σ µ ε (1.) This can be reduced using a vectr identity and (1.17), as fllws B B B µ σ µ ε = 0 (1.3) This is a wave equatin which gverns the behavir f the magnetic flux density B in materials with hmgeneus istrpic cnductivity and the permeability and permittivity f free space. Similar equatins gvern E and J. Equatin (1.3) frms the basis fr\ study in areas such as waveguides and transmissin lines. Fr the structural and electrical prblems encuntered in magnet design, the last term in (1.3) can be neglected which B = ( B) B 5 f 18

6 reduces (1.3) t a diffusin equatin. If the cnfiguratin being analyzed is als time invariant r varying slwly with time, then the secnd term can als be drpped which reduces the equatin t the vectr frm f Laplace's equatin. It is imprtant t retain the distinctin that it is the vectr frm. The physical cnditins under which sme f the terms f (1.3) be neglected can be illuminated by casting the equatin in dimensinless frm. This can be accmplished by defining the fllwing dimensinless variables. Bˆ = B B x ˆ, yˆ, zˆ = x l, y l, z ˆ = l tˆ = ωt B l = characteristic magnetic flux density l = characteristic length ω = characteristic frequency If the abve are substituted in 91.3) and terms rearranged, the result is that ˆ ˆ ˆ ˆ B B B µ σωl µ ε ω l = 0 (1.4) ˆ ˆ The functinal frm fr the slutin f the dimensinless magnetic flux density is therefre Bˆ ˆ( ˆ, ˆ, ˆ, ˆ, = B x y z t µ ε ω l (1.5) The slutin is thus dependent n the dimensinless space and time variables as well as tw additinal dimensinless parameters. These tw parameters describe the relative strength r imprtance f the secnd and third terms in cmparisn with the first term in the gverning equatin. Fr example, cnsider the parameter assciated with the third term f (1.4). This parameter can be rewritten using the fact that µ ε 1 c where c is the velcity f light = which is equal t the speed f an electrmagnetic wave in free space. [4] 6 f 18

7 ω l c µ ε ω l = (1.6) lf l is a characteristic length f the cmpnent r material under cnsideratin then l c is the time required fr the prpagatin f an electrmagnetic disturbance r wave acrss this length. If the characteristic frequency ω with which the field is changing at a pint is lw, then the time t = π ω assciated with this change is lng cmpared with the time required fr prpagatin f the disturbance acrss the device t anther pint. If the frequency is lw enugh then the disturbance is, in effect, felt everywhere in the device at the same time, and the wave character f the prblem can be neglected. That is, the third term in (1.3) can be ignred if ω l << 1 (1.7) c This is usually the case in magnet design. The third term in (1-4) is therefre neglected in the remainder f this sectin. In additin, the neglect f this term is an implicit assumptin which is made fr the analyses thrughut these ntes. The secnd terms f (1.9) and (1.11) are als neglected since they are the surce f the wave character f the equatin under discussin. Equatin (1.4) therefre becmes ˆ ˆ B B µ σωl = 0 (1.8) ˆ ˆ This is a diffusin equatin in which the strength f the diffusin term is determined by the magnitude f the dimensinless parameter µ σωl. This parameter is frequently called the magnetic Reynlds number. If the cnditins f a particular prblem are such that µ σωl << 1 (1.9) then the fields are essentially static r steady state in nature and (1.8) reduces t ˆ B ˆ = 0 (1.30) This is the frm f the gverning equatin fr magnetstatics. The cncept f the magnetic Reynlds number is a useful tl which is cnsidered in depth in a later sectin. 7 f 18

8 1.3 Bundary Cnditins The differential equatins given in Sectins 1.1 and 1. tgether with the cnstituent relatins presented in Sectin 1.4 gvern the relatinship between the field variables in any regin f space. If several regins having different prperties are invlved, bundary cnditins are required t determine hw the fields crss the surface which separates ne regin frm anther. These bundary cnditins can be derived using the integral frm f the equatins given earlier. Since the primary interest is in magnet design, nly thse cnstituent relatins and bundary cnditins which are necessary fr use with (1.7), (1.10), (1.13), and (1.17) are cnsidered. Tw bundary cnditins n the magnetic field must be cnsidered, ne t specify the relatinship between the cmpnents f field nrmal t a bundary, and the ther t specify the relatinship between cmpnents tangent t a bundary. These can be fund frm (1.18) and (1.1) respectively. First, (1.18) is applied t a small, clsed cylindrical surface placed such that its faces are in tw regins parallel t the bundary between regins, as shwn in Figure The dimensins f the cylinder are reduced abut a pint P lcated n the bundary which is within the cylindrical surface. The result is a cnditin which requires that the cmpnent f B nrmal t the bundary be cntinuus. This can be expressed mathematically as n ( B B1 ) = 0 (1.31) where B 1 and B are the magnetic flux densities at P in Regins 1 and. respectively, and n is a unit vectr at P nrmal t the bundary and directed frm Regin 1 t Regin. The secnd bundary cnditin can be fund by applying (1.1) t a cntur which surrunds a small plane perpendicular t the bundary between Regins 1 and. as shwn in Figure If n is again a unit vectr at P which is nrmal t the bundary and directed frm Regin 1 t Regin, then shrinking the dimensins abut P results in 8 f 18

9 n (1.3) ( H H ) = K 1 where K is a surface current density r current sheet which flws in the bundary. This is ften absent in practical prblems but is useful in certain idealized mdels. The bundary cnditin f (1.3) requires that the cmpnent f H tangential t the surface be discntinuus at the bundary if a current sheet exists, and that the discntinuity in H be equal in magnitude t the surface current density and at right angles t it. 1.4 Cnstituent Relatins In a typical magnetic field system, the cnductin prcess accunts fr the free current density in materials. The mst cmmn cnstituent relatinship is Ohm's law, J = σe (1.33) where σ is the electrical cnductivity. The cnductivity is typically assumed t be cnstant within a regin, which requires that the cnductivity in that regin be hmgeneus and istrpic. Thrughut these ntes it is implicitly assumed that there is n relative mtin f the magnet cmpnents being analyzed. Equatin (1.33) is used in Sectin 1. tgether with the fllwing cnstituent relatinship which relates the magnetic flux density tt the magnetic field intensity in free space r in materials having the permeability f free space. B H = µ (1.34) The simplest frm f cnstituent relatin fr a magnetic material is t assume that the material is hmgeneus and istrpic and that the field vectrs are related by the permeability µ. which is cnstant, as fllws B = µ H (1.35) 9 f 18

10 Often magnetic materials are used in which B is nt directly prprtinal H. Fr these the cnstitutive relatinship takes the frm B = µ ( H )H (1.36) where H = H. Sme materials exhibit hysteresis which means that (1.36) is nt single valued. In sme applicatins this effect can be imprtant; hwever, fr many magnet designs a single-valued functinal dependence is usually adequate. Data frm which (1.36) can be derived is given in ne f several frms. Fr example, the permeability may be pltted as a functin f H, that is, µ versus H. Frequently, either B r the B H µ will be pltted versus H. quantity ( ) The cncept f magnetizatin arises in electrmagnetic thery when the magnetic field is cnsidered as being generated by tw surces, ne assciated with an applied current density J which can be cntrlled directly, and the ther assciated with magnetizable material. The magnetizable material can be cnsidered t cnsist f a surce f current density J which is nt cntrllable in that these magnetic currents cannt be circulated m thrugh an external circuit. With these tw surces, (1.13) becmes B = J + J µ m (1.37) If a magnetizatin density M is defined such that M = (1.38) J m then (1.37) can be rewritten as B µ M = J (1.39) 10 f 18

11 Cmparisn f this equatin with (1.13) indicates that the magnetic field intensity H can be written in terms f the magnetizatin density as H B = M µ (1.40) versus H is the same as a plt f µ M versus H. This is a cnvenient frmulatin because M is frequently saturated when magnetic Thus a plt f the quantity ( B µ H ) materials are used in large, high field magnets. A magnetic susceptibility χ can als be defined such that M = χh (1.41) The magnetic susceptibility is related t the permeability thrugh ( χ ) µ = µ 1+ (1.4) 1.5 Ptential Functins Ptential functins can be used t reduce the number f variables r therwise simplify the prcess f finding a slutin t the gverning equatins. The requirement f (1.17) that the divergence f B be zer naturally leads t the definitin f a vectr ptential A f the frm A = B (1.43) This frmulatin autmatically satisfies (1.17) since the divergence f the curl f any vectr field is zer. Fr this analysis it is assumed that the cnstituent relatins are given by (1.33) and (1.36). These require that (1) any electrically cnducting material be hmgeneus and istrpic with a cnstant electrical cnductivity and () any magnetically permeable material be hmgeneus and istrpic but nt necessarily linear, since the permeability can be a functin f H. It is assumed that this functin is 11 f 18

12 single valued, which requires that hysteresis effects be negligible. Equatin (1.43) alne is nt sufficient t define A : hwever, if A = 0 (1.44) is impsed a cnstraint, A is fully defined. Substitutin f (1.43) int (1.10) leads t A E + = 0 (1.45) and t the definitin f a scalar ptential functin φ. This scalar ptential autmatically satisfies (1.45) since the curl f the gradient f a sclar functin is zer. Therefre A E = φ = 0 (1.46) Equatins (1.46) and (1.33) can be used with (1.7) t yield A σ φ + = 0 (1.47) In additin, substitutin f (1.33), (1.360, and (1.46) int (1.13) yields 1 A A + σ φ + = 0 (1.48) µ Equatins (1.47) and (1.48) are the tw gverning equatins in terms f the tw unknwn ptential functins A and φ which can be used in place f the field equatins fr E and B. Equatins (1.47) and (1.48) are the gverning equatins which shuld be used in the fllwing manner fr regins in which there are n current surces. If, as is ften the case, the current density distributin is knwn in a regin and is the driver in the particular cnfiguratin then the gverning equatin fr that regin becmes 1 A = µ J c (1.49) 1 f 18

13 subject t (1.44). Nte als that J c must satisfy J c = 0 t be physically realizable. Equatin (1.49) drives the slutin in the ther regins, which are gverned by (1.47) and (1.48). If the prblem is tw dimensinal with a driving current density directed in the third dimensin, the vectr ptential reduces t a single cmpnent in the directin f the driving current density. Frm (1.49) the gverning equatin in the current-carrying regin becmes 1 A = µ Frm (1.48) the gverning equatin in the ther regins is J c (1.50) 1 A A = σ µ (1.51) The advantage in this situatin is that the vectr ptential A cnsists f a single cmpnent since jc has nly ne cmpnent. If the tw-dimensinal prblem is further simplified t cnsist f a steady-state situatin, (1.50) remains unchanged but the right side f (1.51) becmes zer. In a regin f cnstant permeability µ, (1.13) and (1.43) require that A = µ J In rectangular crdinates the cmpnents f this vectr equatin reduce t a scalar Laplacian, as fllws: A x = µ J x A y = µ J y A z = µ J z (1.53) Nte that each f these represents cmpnents f the vectr equatin (1.5) and that the vectr frm is equivalent t the scalar frm f the differential equatin nly in 13 f 18

14 rectangular crdinates. In ther crdinate systems, (1.5) des nt reduce t the simple frm f (1.53). Each f the vectr cmpnents must satisfy Pissn's equatin which has a slutin f the frm [3] A i = µ 4π J dv i r qp (1.54) where i = x, y, r z and r = distance frm pint p where A i is measured, t pint qp q where J is measured. This can then be written in vectr frm as A = µ 4π JdV r qp (1.55) where the integral is taken ver the vlume f the entire regin. Equatin (1.55) leads t a physical interpretatin f A. Cnsider a clsed circuit f small-crss-sectin wire which carries a current density J. Outside the wire there is n cntributin t the vlume integral f (1.55) because J = 0. Inside the wire, JdV = Ids where I is the current and ds is the length f the vlume element in the directin f J. Equatin (1.55) then becmes A = µ 4π Ids r qp (1.56) Therefre, the vectr cntributin t A at a pint by a current-carrying element in a clsed circuit is parallel t that element. Equatins (1.43) and (1.55) can be manipulated t shw that [4] 14 f 18

15 where i qp B µ ( J i ) qp = 4π rqp dv (1.57) is a unit vectr directed frm q t p. This equatin is cmmnly called the Bit-Savart law. In magnetstatic prblems, cnditins are steady in time. In current-free regins (1.13) becmes H = 0 (1.58) which is autmatically satisfied by a scalar ptential functin defined by H = φ (1.59) Equatins (1.59) and (1.17) tgether with (1.35) imply that the field can be fund by slving ψ = 0 (1.60) where ψ is the magnetstatic ptential. 1.6 Inductance and the Vectr Ptential The mutual inductance between tw circuits b and c can be thught f as the flux linked by c per unit current in b with zer current in c. Equatin (1.43) defines the vectr ptential as the integral frm f this equatin is A = B (1.61) l A dl = B nda (1.6) 15 f 18

16 where l is a clsed cntur and the right-hand side is the flux thrugh a simplycnnected area bunded by l. If the cntur cincides with circuit c which has zer current and if the field B and thus A are generated by a current I b in circuit b, then the right side f (1.6) is the ttal flux generated by b and linked by c, as fllws Hwever, in linear systems B nda = Φ bc bc b bc (1.63) Φ = M I (1.64) where M bc is the mutual inductance between circuits b and c. Therefre M bc 1 = I b c A dc b (1.65) The subscript is added t A in (1.65) t indicate that the vectr ptential is generated by the current I b in circuit b althugh it is measured and integrated arund circuit c which has zer current. This relatinship can be used t btain accurate inductance calculatins thrugh numerical integratin. 1.7 Electrmagnetic Frces The frce f electrmagnetic rigin is mentined in Sectin 1.1 in cnnectin with the definitins f E and f B. In a magnet system, the electrmagnetic lads are calculated by starting with (1.4) which gives the lcal frce density f n an element which carries a current density J while immersed in a magnetic field B. If the element and fields are such that J and B are unifrm ver the vlume f the element as sketched in Figure 1.7.1, then the net frce n the element is F = (1.66) ( J B) dh dw dl If this incremental element is part f a larger vlume f material carrying current in the magnetic field then the net frce n any current-carrying vlume r sectin can be fund by integratin as fllws 16 f 18

17 F = J B dh dw dl (1.67) This equatin can be used t find the net frce n a current-carrying bdy by perfrming a vlume integratin. It can be useful t find the net frce by perfrming an integral ver a clsed surface which surrunds the bdy. This invlves the use f the Maxwell Stress Tensr, which is given by [4] Where T δ mn µ = µ H nh m δ mnh k H k (1.68) = 1when m mn = n δ mn = 0 when m n and subscripts dente crdinate directins. Repeated subscripts such as k imply a summatin. The m th cmpnent f the lcal frce density is given by f m T mn = (1.69) x n and the net frce n a bdy can be fund frm F m = T n da (1.70) mn n where n n, is the n th cmpnent f the utward-directed unit vectr n nrmal t the clsed surface surrunding the bdy. The use f (1.67) requires that the fields be knwn thrughut the vlume f integratin whereas (1.70) requires that they be knwn nly n the clsed surface. Furthermre, the clsed surface can be cnsiderably larger than the bdy n which the net frce is desired. 17 f 18

18 This cncept is particularly useful when the gemetry f the prblem is such that sme f the cmpnents f the tensr r f the integrand in (1.70) are zer. It can be shwn that the frm f (1.68) thrugh (1.70) are the same whether the fields are generated by free current densities J, by magnetized materials, r by a cmbinatin f these surces prvided that the ttal field is used. The permeability µ must be istrpic, but can be a functin f psitin. Furthermre, the permeability cannt be a functin f the material density althugh this magnetstrictin effect can be included in (1.68) thrugh an additinal term. These cnstraints are usually unimprtant in fusin magnet systems. REFERENCES - Sectin I 1. R.M. Fan, L.J. Chu, R.B. Adler, Electrmagnetic Fields, Energy, and Frces, Wiley, N. Y., J. D. Jacksn, Classical Electrdynamics. Wiley, N. Y., W. R. Smythe, Static and Dynamic Electricity, 3rd ed., McGraw-Hill, N. Y., H. H. Wdsn and J. R. Melcher, Electrmechanical Dynamics, Parts I, II, and III, Wiley, N. Y., f 18