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Unformatted text preview: 172 MAXWELL'S EQUATIONS AND BOUNDARY CONDITIONS (3-103). One concludes that a surface charge is ' con- ' d d on the interface by the normal components ofE if at least on: reg-1;):ng bath m um h hand no free. surface charge exists at the inter ace 1 on the Qt er a the special proportion 61/61 = 01/0; event and oflittlc importance. (3-134») yields the special case in which £1 and £2 are given by ductive. I _ regions are nonconductlve (a1 = 02 H 0) or (t?!) are is true among the region parameters, presuma rt For both regions noneonducttve, putting p, = m o 61113"1 _ 621;": = 03 orjust Dnl _ DnZ : 0 a) = 0-2 = 0 a result agreeing with (3-43) for the nonconductive case. ' - r - ' If: h parat g tW 3 I I [)(‘tt'l [Illllt‘ ll (3 lf‘lr'u llvfi ldW l0! dwth LLlll L'le tit an In llaf I‘ E l! O 1501.! UP“ (Ullduc [HR 16 10115. i; Cl ldllze the 163111‘ l0] 0116 LolldU(UU1tY IllUCll 1.31 C! that]. l g p g . ' ' _ dar [he 0tgimme the J vectors tilted by amounts 91 and 61 as Show" m (a)' rhf boun y Condition (3432) for dc becomes “7.1:an (I) urrent flux 239/ (a) Region 12w” Region 1:071) Region 2:“; >> 0'1) Region 2: {02) (d) Region 21(62 =10 ‘71) m (c) (1 Ref: JCUO 0‘ cut: Iltb b LIE. lt’S Ol urrent flux I' a ‘lltJIl [a = “)0 . ‘ P C CfI' L. I 2 1 l] C . ( ) m (6) current flux for (113310“. 2 conductive. Constraint {0 tangential flOW at Inlet lace lOI region 1 nonconductive. PROBLEMS 173 while the boundary condition involving tangential components is obtained from (3—79), with J = 0E a: a: (2) From the geometry, the tilt angles obey _ Jrl tan 3] _ tan 9; = & nl jnl The latter combines with (l) and (2), whereupon inserting the expression for tan 61 obtains the refractive law [an 91 = 2 tan 61 (3-138) 0'2 The analogy with the refractive laWS (3-76) and (3-80) for B and E might be noted. For an example in which a; = 1001, the refractive effects of direct current streamlines at an interface are shown typically in (b) of the accompanying figure. For :12 » or, the near perpendicularity of the current flux occurs in regions 1, as noted in (r). Ifcri were reduced to zero, then I, = O, constraining the current flow in region 2 to paths tan- gential to the conductor—insulator boundary as in (d), a result evident from the insertion of},l = j” = 0 into the boundary conditions (4-133) and {4-134}. REFERENCES ELLIOTT, R. S. Electromagnetirr. New York: McGrawiflill, 1966. JAVID, M., and P. M. BROWN. Fit-M Analysis and Electromagnelicj. New York: McGraw—Hill, 1963. _IORDAN, E. C., and K. G. BALMAIN. Electromagnetic Waves and Radiating Systems, 2nd ed., Engle- wood Cliffs, NJ; Prentice—Hall, [963: LORRAIN, P., and D. R. Conson. Elcrtramagnetic Fields and Warm. San Francisco: Freeman, 1970. Rsirz, R., and F. J. MILFORD. Foundations qf Electromagnets": Themy. Reading, Mass: Addison~ Wesley, New York: 1960. PROBLEMS SECTION 3-1 3-1. Pure copper, with a free (outer orbit) electron concentration of about 10‘29 electrons/m3, has the conductivity 0 = 5.8 x If.)7 mho/m at room temperature (Table 3—3). (a) Find the mobility of the free electrons in copper. (b) Express the free electron charge density in coulombs per cubic millimeter for this material. (c) Find the drift velocity of the electrons for the unit applied electric field E = a, V/m. What is the corresponding volume current density in this specimen? Sketch the vectors depicting 9‘, J, and E in the sample. (Explain from physical reasoning why u, and E are in opposite directions, although J and E are in the same sense.) {Answerz (c) —3.finx min/sec] 3-2. Find the current density (expressed in A/cmz) in the. following conductors, possessing only negative electronic charge carriers under the given conditions. (a) The average drift velocity is —n,4.5 mm/sec and the charge carriers have the density 2 x If)“ electrons/m3. (b) The volume density of electronic charge carriers is —-3.5 x 108 C/m’ and the carrier average drift velocity is 4.2 mm/see, with E = 101, V/m within the conductor. What is the conductivity of the region in the latter case? [Answer: (a) 3‘1440 A/cm2 (b) 1,147 A/ctn’. 0.147 MU/m] 174 MAXWELL's EQUATIONS AND BOUNDARY CONDITIONS SECTION 3-2 3-3. At some particular temperature, helium gas has 1025 atomsj'm’ and is measured to have the dielectric susceptibility of 1.5 x 10“. What is its electric polarizatiOn field P for the applied field E = 103 V/m? What is the charge density p“ and the average displacement cl of the nucleus relative to the electron cloud for the given E? What is 6,? [Answer: p, -—- 3.2 x 10‘ Cfm’] 3-4. At low frequencies, the measured relative permittivity of water is 81 (Table 3-3). What is then its electric susceptibility? What electric field E must be applied to produce, at the sinusoidal frequency a), the polarization field P = 1,10 sin to! ,uC/mz in a water sample? (Ex ress E in kV/m.) Firfd the corresponding electric displacement density D, expressed in [AC/m . Without using field values, form appropriate ratios to determine by what factor the magnitude of D is larger than that of P; similarly, compare P with EOE. 3-5. (a) To make the electric polarization density P and the applied field EOE exactly the same in a material, what must its relative permittivity be? (b) What is the relative permittivity of a material ifP has 10 times the value of 50E therein? (c) H!) has 10 times the strength of £013 in a material, what is its relative permittivity? (d) If D is 10P in some region, what is its 6,? [Answers (b) 11 (d) 1.111] 3-6. The same electric field, E = 103az V/m, is applied to the following regions having the dielectric susceptibilities: (a) zero (what sort ofregion is this?); (b) 10-3; (c) l; (d) 103. Deter- mine the relative permittivity, the applied field 60E, the electric polarization field P, and the electric displacement field D for each region. SECTION 3-2A. 5-7. The following E fields are given to exist in some block of polyethylene. for which E, = 2.26 (from Table 3-3): (a) 1,103.13 sin to! V/m; (b) 31,1031: sin tot V/m; (c) n,(103/r1) sin cot V/m. Find the fields EOE, P, D, the polarization (bound) charge density p,, and the volume polarization (bound) current density J, for each applied E field. [Answerz (c) p,=0, J, = 1,.(1 1.14/r2)cu cos cu! nAfmz] 3-8. Corresponding to the electric polarizatitm field P = 11,10 sin to! pG/m2 of Problem 3-4, find the polarization (bound) current density J, at the frequencies: (a) 1 kHz; (b) 1 MHz. SECTION 3-23 _ 3-9. Apply the Gauss—Maxwell integral law (3-36) to a vanishing volume element Ar: in a dielectric region, to rederive its differential form (3-24). [Hint Divide (3-36) by An and consider the meaning of each ratio as As —a 0.] 3-10. Making use of the divergence theorem, show how the differential expression (3-21) can be manipulated to yield the integral form (3-38). Explain the physical meaning of this result. SECTION 3-20 3-11. The coaxial, circular cylindrical conductor pair (coaxial line) of great length and with the dimensions shown contains a homogeneous dielectric sleeve with the permittivity £.- Assume the static surface charges totaling 1 Qon every axial length if of the inner and outer conductors respectively. (a) Making use of the symmetry and Gauss's law (3-37), determine for each region between the conductors the D and the E fields. (b) Determine P in the dielectric region. By use of the criterion (3-21), determine whether there is any volume density of excess polarization (hound) charge, of density pp, within the dielectric. (c) Making use of the appropriate boundary conditions, find the free charge densities on the conductor surfaces at p = a and d, as well as the surface polarization (bound) densities at p = b and c. (d) Letting a = 2 mm, b = 4- mm, c = 8 mm, d = 1 cm, Q]! = 10'2 ,uC/m, and e, = 2.26 (polyethylene), find the values oil. and p, at the conductor surfaces at p = a and d. Find also D, P, and E at the surface ,0 = b-l- (just within the dielectric), comparing their values with those at p = b— (just outside the dielectric). [Answen (d) 13(6) = 90 kV/ln, p,(a) = 0.796 ,uC/mz, E(b+) = 19.9 kV/in] PROBLEMS 175 sin PROBLEM 3-1 1 3-12. IAssume that the region a < p < d between the coaxial conductors of Problem 3-11 is filled With a Single, inhomogeneous dielectric material for which the permittivity is €(p) a func- tion ofonly p. (a) Make use of the symmetry and Gauss's law (3—37) to establish the fuiictional dependence of e on p required to make E between the conductors independent of ,0. Express the answer such that 6(p) has the value c, at the outer radius ,0 = d. What is then E? (b) Find both the polarization density field P and the volume density p of polarization (bound) charge for this choice ofe(p). (c) Note that this nonuniform design of the dielectric region provides a way to avmd high electric fields in a coaxial configuration, thus reducing the possibility of di- electric breakdown. Suggest how the nonuniform permittivity conditions of this problem might be met approxtmately, using, say, three or four diflerent but homogeneous dielectric materials. [Answer: (a) €(p) = Elsi/p (b) p, = QED/21:6,dtp] The concentric, spherical conductor pair is separated by two dielectric shells of permit- I tivmes 61 and 62 as shown, the interface between them appearing at r = 5. Assuming the static surface charges totaling i Qon the inner (r = a) and outer (r = r) conductor surfaces respec- tively, answer the following. (a) Use Gauss’s law (3-37) and the symmetry to deduce D and E Withinthe two regions. Both these fields are normal to the interface at r = [1. Which boundary condition in Table 3-2 is applicable at this interface? (b) Find the expression for P in each region From (3-21),_deduce whether there is a polarization (bound) charge density ,0 within either dielectric region. (c) Employ the proper boundary conditions to find the free liiurface charge density p, on the conductor surfaces, as well as the surface polarization charge density p at r = b. (d) With a = 1m, 11: 1.02 m, r I: 1.05 m, 6,, = 2.26, 6,: =1 (air), and Q= 0.1 ,uép find the values of E at the radii b' and 15*. as well as on thehconductor surfaces 1* = a and c. .‘iketch E versus r from a to c. [Answer (a) E” = Q/47telr’ (d) iE',,(a) = 398 kV/m] r PROBLEM 3-13 176 MAXWELL‘S EQUATIONS AND BOUNDARY CONDITIONS C" N 3-3 _ v v :54 gased on a pillbox construction suggested by Figure 3-4, prove the boundary condition ' ' f an interface. (3 50} concerning the continuity of the normal components of B to either side O 3-4 ‘ ‘ - :Elfnglzinning with the force (3-51) acting on each edge of the current loop of Figure .5 7(a), Fill in the remaining details to prove (3-54). . . . tar 3-16 Prove that the net magnetic force 1",, acting on an-arbitrary, closed, thiiyfifaflii‘igmy‘): circtiit carrying the uniform current I and immersed in a uruform B field, is zero. [ iii . ' ' l (3-52) about the circuit path {, noting that f B x «if can be written 3 x See also Examp e 1-6.] ‘ n ' ‘ - located in the z = 0 plane as in - h d mensions of the square current loop, ‘ 3717. 3E;(l:‘infotatiarlge scale by assuming each side to be 2 meters‘long (side (1 lgcated :I. Figure. ihe 1*! = 0 plane etc}. The current flows clockwise when looking in the +1: intent;g . j _ gsiriime the current lbop to be immersed in a magnetic field having only the CDmpOflTEJ‘cgé £fitJetch the system. Determine the force JF, due to the magnetic field actizfgflonBansy :1 men—y . 1- of the side !.. as well as the total force on (1. showing that F, a: 3.! y y “tin , ; a"ital-hat the net magnetic force on the loop is zero. Show also that the differenittia torqhue ri lug 0: current element of t,. relative to the moment arm R -= a; + o,o measure roir: t 1;: tgrim; iindT = 3,133 23:11:: and that the total torque on (I is zero. (b)dR{epeat (a). afucfll‘lrhgl the (ma: I' . Sh wh the forces on the sides {I an ‘ are zero, n that only lihiislopfipscdiie tool; , isylyi-RZIBI. Find the total torque, due to B, pnly,Il)3yfiat-ialoglyé :l‘luihree components of]! present, what is the total torquez on the loop. (all e 111:: u mzig-netic moment ofthis finite sized current loop as In = In: :31) tzfiashgvyhtiiziagfigg‘tzamomtlm -' -‘ ed 'n {b is e uivalem to (3-54), = m x . I _ I :finrthifin clirreni 100‘; of sides 2n = 10 cm and carrying 10 A, immersed 1n the field B = 0.3.; + 0.4., + 0.5., bem2? ' ' ’ ‘ “I 3 - d with the steady " tizable material has 5.3 x 10 atomsj'ni , an ‘ 3-18. ‘Afigfduliluraifyi‘n'i applied to its interior, there results the average-dipole mgjiiient magnltifa: Iii—2“ A-mi. (a) Find the density of magnetiZation M. the magnetic Siliscepti 1121', title—relative permeability, and the permeability of this material. (b) Find 3 in t is mater: . [Answerz M = 39,600 Aim, pi = 12.4 nil-Hm] . . . . d ' ' ‘ ' " ' ‘ lar s ecimen of magnetic material is measure . Th ma netic SUSCCpUblllty of a particu p _ A I ‘ B tsdlge 59. Viliatgis the magnetic polarization M and the magnetic intensity H, if the field in the material is 0.0lax bemz? 3-20. Given the following volume magnetization fields M within certair;I regions of magnetic materials, find the volume densities J. produced by the bound Currents t eretn. (a) 150”, (b) 1,200,351I (c) n.520 (cylindrical) (d) aglotlr cos 9 (e) 1,160,“:- [Answen (b) 0 (d) “‘200 cm 6] h l l t' hip (3 56) relating the ‘ ’ - l to show how t e cur re 3 ions - , :3;etifrtiiiioriziiiiik:ii:iiiilifiraltbztlifinagnetization current density J“, is transformed to the integral relation (3-6? )1. SECTION 341‘ _ z A I 3-22 Show that the magnetization current denstty j”, = —a,l[} Afrn as:fociat‘edt writ; $1: bourid currents in the sample of Example 3-3 yields, from an appropriatie :fiiatastfmtglgrogmn d rrent flow of [06’ A through any fixed it cross section 0 I . ihiaiaifiglhnnsiv: by use of the line integral of (3-67}. Sketch the system, appropriately labeled. PROBLEMS 177 SECTION 3-43 3-23. Employ a suitable sketch, showing how the quantity n I: I'll, boundary condition (3-72}, specifies the tangmtia! component of the in both magnitude and direction. used in the magnetic-field surface current density L 3-24.. Apply the appropriate boundary condition in answering the following. (a) An air-to- perfect-conductor interface is at z = 0. the region z > El being air. With H = 150:, Afm in the air region, what is the surface current density on the perfect conductor? How much total current I flows in a 20-cm-wide x-directed strip of this conductor surface? Sketch this system showing H, J,, and a few current flux lines. (b) Find the current density on the conductor surface of (a), this time assuming H = 301,, + 40.3, Afrn. Sketch this system. (c) Suppose in the geometry of Figure l-19(a} that the long, straight wire shown is a perfect conductor, and that surface currents totaling 1 flow on the conductor surface ,0 = a. The B field for p > a is still given correctly by (1-64}. Use this field to deduce the surface current density J, on the wire. Formulate a vector integral relationship between I and L, showing a related sketch. 3-25. What two simultaneous boundary conditions are being satisfied by the magnetic field refraction expression (3-76}? Establish that, if region 1 is air and region 2 is iron with pr, = if)4 (a case of high contrast in permeabilities), the tilt angle 3, Of Bl from the normal in region is very small for most values of 92. For example, find 91 if 91 = 0, 45°, 89°, and then 39.9”. How far from the normal must 6'; be if 91 is to become as large as 10“? Sketch this example. 3-26. The toroidal iron core of rectangular cross section partly fills the closely Wound toroidal coil of it turns and carrying the direct current I as shown. (it) Use the right-hand rule (thumb in the sense of I) to establish the direction of H inside the winding. (b) Use the static form of Ampere's law (3-66) to deduce H at any radius p within the winding, and determine B for the two regions. Which boundary condition for magnetic fields (Table 3-2) is being satisfied at the air-iron interface? (c) From H deduce expressions for the magnetization density field M in the [W0 regions. Sketch flux plots showing (in side views) the relative densities of H, Ema, and M in the two regions. assuming it, >> 1 for the iron. (d) Find J," within the iron as well as j”, on the four sides of the iron core. Sketch representatEVe vectors or fluxes depicting these quantities. {e} If a = 1 cm, 6 = 1.5 cm, (‘2 2 cm, :1: 1 cm, it, =1000, it = 100 turns, .f= 100 mA, find the values ofH and B at p = a+ and {9— (just within the iron], at p = b-I» and ,0: c. 3-27. As a simple exercise in applying boundary conditions, an air space (region 1) defined for all e > U and a magnetic substrate with p, = 4 (region 2) occurring for all e < 0 are separated by the infinite plane interface at e = 0. The constant, static magnetic field in region 1 is given to be .3, = 0.3a, + 0.4-, + 0.50Jr Wit/nil. Sketch 31 (shown for convenience at the origin) and the normal unit vector In at the interface (its direction taken as going from region 2 to region 1). (a) Make use of the boundary conditions (Table 3-2), concerning the continuity of appropriate tangential or normal field components at the interface, to deduce the vector fields H1, 3,, and H, in the two regions, as well as the field magnitudes. {Leave H expressions in terms of PROBLEM 3-26 5 ye. mans-ram o ownsiuno mu munmur UUNDI'I'IUNS 2: M 3: W PROBLEM 3-28 the symbolic 11..) (b) By use of the definition of n' 3. find the angles 9. and 9, between ss andI(orfl)inthetworegiom.(LabelO,onthesketch.)Oieckyourames-slryuseof (3-76). [Am-m: (a) a, - 1.2., + 1.6:, + 0.5-, bem' (b) a, - 76°] 3-”. Averylong, nonmagneticconductorm- l) afradiuscearriesthestadccurtentlas shown. Thecontluctorissunounded byaeylindrical sleeveoi'nomconducting magnetic material withathicknasumdingfinmp—amp-banddtepmnuhifityn.1hemundingregim isair. (a) Makeuseof and Ampére‘a law (3-66) tofindllandlinthe three regions. (Iabelthecloaedlincscmployedin thepmofidepictingflinthepmperfinsecneachline.) (b) Find thehlfieldinthemagneticregion. [fl-628A,¢- l cth- l.5ust,p,-6forthe magnetic sleeve, sketch h", B‘. and M‘ versus ,9 for this system. Comment on the continuity (or otherwise) of these tangential fields at the interfices. (c) By use of (3-56) and (3-73b), find the volume magnetization current density-l“I and the bound suffice current densities J_ within andcn themagneticaleeve. SECTION 3-5 . 3-29. Twom-infininresimair (region l)fiirz>fl,andadideen-ic (regions,th e - til-en) ibrz < 0, arcseparated by the interfisce ate - 0. In the airregicn, the constant electric fieldEl - —l5n,+2m,+30a,Wmisgiven.Sketchl,for-eonvenienoeattheorigin.(a)F'md D and E for both regions, making use of boundary conditions (Table 3-2). (Leave to explicitly intheDettpmcslions.) (b) Fuidtherefiacdonanglufl, and9,&omthemrmalinbolhregions, makingusecfthedefinitionofn°lifnisdirectcdfiomngion2toregion LUsetherefi'action law (3-80) at a check. [Answen E: It —l§n,+ 20;,4- 7.5.. Villa, 92 I 73.30"] CTION 3-7 3!”. Provedqu (3-90a) and (3-90b) brtheattcnuation oonstantaand thephase constant 3 associated with uniform plane waves in an unbounded, [any region. 3-81. Asurneuniformplanewavestobetravelingatthefiequencyf- lWMHzinalo-y region having the constitutive parameters is u 110, e - 6:0, 0 - 10'3 mbofm. (a) By direct sub- stitution into (3-38), determine the value of the complex pmpagation onnstant associated with the waves, expressing 7 in its complex rectangular form denoted by (3-89). From this result infer the values of the wave attenuation constant and phase constant. (b) Find the attenuation constant and the constant by use of (3-903) and (3-90b). [Answen a -0.76l prm, )3 I 5.187 radfm] 342. ReputhblemS-Sl,thistisneasunfingthepammetersofthelmsyrcgiontoben-p0, e:- l.8£o. er- lOmhofm, and in which uniformplane f- 10 61-11. [A'nswen y: 597.? +fi60.5 m"] 3-83. Making use ofthe free-space parametersp-po, (-6., and 6-0. show that the «missions (S-Na}. (3-90b) and (3-998) reducetothefi'eehspaoetuultstl - OJ I 500112-118), and q = no of(2-130b). 34¢. Provedlatthcpenmfionoflhreeskhtdepthsbyapbnewaveinmamndncfiw region produces an amplitude reduction to 5% of the reference value. Show that six skin depths yields 0.25%. mounts 179 I345. Giventheelecu-ic-ficldplanewaveaolution 3-91!) ' ' prepay.“ comm 1s defined by (Retail:wa substitution into the Mixwulcrnwelfiut (3-33) 3.1.9 cone: magnetic utionbecomes{3-Nc,iftho’mnmc' ‘ 'peflance 'tlefined by (3-99a). [Hulk Show that the coefficient mini}: reduce:II to 8' 1:" W m q u as; Showthatthcuprusinnforinuinsicwaveimpcdancefi,dcfinedby(3-97)as wave penetrate before teaching 5% of its surface value’ Perfiarm this cal ula ' ‘ lotvradiofi'equendesHOItI-la intheVLFrange l. mummgc ‘umnmm tltecosutantse,-8landa-(40lm atmmtl‘ezmuefiflzmul’ mum-Ins undersea radio communication, basedouyour results. SECTION 3-8 349. Uaethemlm' (390 ‘ . conduct-(forum 0/095) 1), ' 90h 99" m we'll”, a[alilllttlliletoa good mgionattlusfiequencyisaboummm.Useasketchcl‘thereal-tinseelecuiclield ' .mns' ' ' I tithe z-ongIthocapiainthenseaningofHepth ofpeneo-stion.” (1:) find theta-aunt: this cavealabelsngtton the sketchofpart (b). Comparethiswavelength ‘ Intimregionassunsingnowthatitismmpictelyloale- ...
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This note was uploaded on 11/22/2010 for the course ECSE ecse 351 taught by Professor Dennis during the Winter '07 term at McGill.

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asmt4,5book - 172 MAXWELL'S EQUATIONS AND BOUNDARY...

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