Quantity: Add to Cart. Electromagnetic Fields and Waves. The latest edition of Electromagnetic Fields and Waves retains an authoritative, balanced approach, in-depth coverage, extensive analysis, and use of computational techniques to provide a complete understanding of electromagnetics—important to all electrical engineering students.
The text also includes sections on computational techniques in electromagnetics and applications in electrostatics, in transmission lines, and in wire antenna designs. The antennas chapter has been substantially broadened in scope; it now can be used as a stand-alone text in an introductory antennas course. Advantageous pedagogical features appear in every chapter: examples that illustrate key topics and ask the reader to render a solution to a question or problem posed; an abundant number of detailed figures and diagrams, enabling a visual interpretation of the developed mathematical equations; and multiple review questions and problems designed to strengthen and accelerate the learning process.
Become a Redditor and subscribe to one of thousands of communities. Nuoy Amoh rated it it was amazing Jul 13, Please try again later. I did not like this book at all. Mark is currently reading it Sep 21, The text describes the properties of the static electric and magnetic fields in terms of their charge and current sources before introducing Maxwell s equations.
Field and Wave Electromagnetics: The content, pedagogical style, and merit of the present text is self-evident in its thoughtful selection of topics, their sequence, and the pertinent applications.
Tna rated it liked it Nov 17, Sarah rated it it was ok Dec 05, Please note that everything on this subreddit is provided under fair use. Books by Magdy F. Amazon Renewed Refurbished products with a warranty. The nonzero div H, on the other hand, is evident from its flux plot because of the discontinuous flux lines, here required to possess an increasing density with x, yielding a net nonzero outgoing flux emerging from the typical dosed S shown.
The radially directed field J, having a constant flux density of magnitude K, on the other hand, e1early must pick up additional flux lines with an increase in p. It is therefore required to possess a divergence.
Find the diverge nee of the E field produced by the uniformly charged cloud of Figure b at any location both inside and exterior to the cloud.
All inverse r2 radial ficlds behave this way. It is shown in Section B that this result is true in general, even for nonuniform charge dis- tributions in free space. Equation implies that the volume integral of div F dv taken throughout any V equals the net flux of F emerging from the dosed surface S bounding V.
A heuristic proof of proceeds as follows. Suppose that V is subdivided into a large number n of volume-elements, any of which is designated AUi with each en- dosed by bounding surfaces ,S; as in Figure a.
Geometry of a. If the limiting process yielding is to be valid, it is necessary that F, together its first derivatives, be continuous in and on V. IfF and its divergence V. Fare not continuous, then the regions in Vor on S possessing such discontinuities or possible must be excluded by constructing closed surfaces about them, as typified b. The following examples illustrate the foregoing remarks concerning the diver-. Because H is x-directed, however, H.
An example occurs in Poynting's theorem of electromagnetic power considered later in Chapter 7. Equivalently, if electric field lines terminate abruptly, their termini must be electric charges. The flux plot of any B field must, therefore, invariably consist of elosed lines; free magnetic charges are thus nonexistent in the physical world.
A divergenceless field is also called a solenoidal field; magnetic fidds are always solenoidal. Suppose that Maxwell's diHcrential equation , instead of its integral form I , had been postulated. Execute the reverse of the process just described, deriv- ing from by the latter over an arbitrary volume Vand applying the divergence theorem. Integrating over an arhitrary volume V yields Assume that E is well-behaved in the region in question.
From a use of , the left: side can be replaced by the equivalent closed-surface integral ts EoE. Many vector functions do not exhibit this conservative property; a physical example is the magnetic B field obeying Ampere's circuital law For example, in the steady current system of Figure , the line integral of'B dt taken about a circular path enclosing all or part of the wire, a nonzero current result is anticipated.
Nonconservative fields such as these are said to possess a circulation about closed paths of integration. Whenever thc elosed-line integral of a field is taken about a small vanishing closed path and the result is expressed as a ratio to the small area enclosed, that circulation per unit area can be expressed as a vector known as the curl of the field in the neighborhood of a point.
It follows that a conservative field has a zero value of curl everywhere; it is also called an irrotational field.
Historically, the concept of curl comes from a mathematical model of effects in hydrodynamics. The early work of Helmholtz in the vortex motion of fluid fields led ultimately to the mathematical postulates by Maxwell of Faraday'S con- ceptiollS of the electric fields induced by magnetic fields. A connection between curl and fluid phenomena can be established by supposing a small paddle wheel to be immersed in a stream of water, its velocity field being represented by the flux map shown in :Figure Let the paddle wheel be oriented as at A in the figure.
In this example, the velocity field l' is said to have a vector curl directed into the paper along the axis of the paddle wheel, a s 'nse determined by the thumb of he right hand if the fingers point in the direction of the rotation; the vector curl of v has a negative z direction at A.
Similarly, physically rotating the paddle wheel axis at right angles as at B in the figure provides a way 10 determine the x component of the vector curl of v, symbolized [curl v]x. In rectangular coordinates, the total vector curl of v is the vector sum Generally, the curl 2 of a vector field F ub U2, U3, t , denoted curl F, is expressed as the vector sum of three orthogonal components, as follows Each component is defined as a line integral ofF, dt about a shrinking closed line on a per-unit-area basis with the al component defined The vanishing suriace bounded by the closed line t shown in Figure is As l , with the direction of integration around t assumed to be governed by the right-hand rule.
A closed line l bouuding the vanishing area As lo used in defining the a l component of curl Fat P. Similar definitions apply to the other two components, so the total value of curl F at a point is expressed curlF A difierential expression for curl F in generalized coordinates is found from by a procedure resembling that used in finding the differential expression for div F in Section 2.
The shape of each closed line l used in the limits of is of no con- sequence, as long as the dimensions of Lls inside l tend toward zero together. Thus, in finding the a l component of curl F, t is deformed into the curvilinear rectangle of Figure b with edges Lll z and Lll 3.
Along the top edge, F2 changes an incremental amount, but in general so does the length increment, Llt z , because of the curvilinear coordinate system.
The line-integral contribution along the top edge is found from a Taylor's expansion of W about P. Relative to curl F in generalized orthogonal coordinates. It is seen that I also leads to the following pvt. So if G were a fluid velocity field with a paddle wheel immersed in it as in Figure , a clockwise rotation looking along the negative z direction would result, agreeing with the direction of curl G. Find the curl of the B fields both inside and outside the long, straight wire carrying the steady current J shown in Figure This special case demollstrates the validity of a Maxwell's diflerential relation to be developed in Section 2-SB.
You may fi. This is ealled the theorem Jf Stokes. Suppose the arbitrary S is subdivided into a large number n of surface-elements, typical ofwhieh is bounded by til as in Figure a.
Relative to Stokes's theorem. If the left side of is surnmed over all closed contours t; on the surface S of Figure a , the common edges of adjacent elements are traversed twice and in opposite directions to cause the integrations about t; to cancel everywhere on S except on its outer boundary t. As with the divergence theorem, It IS necessary in that F together with its first derivatives be continuous.
The positive sense of ds should as usual agree with the integration sense around t according to the right-hand rule. Given the vector field x 1 illustrate the validity of Stokes's theorem by evaluating over the open surface S defined by the five sides of a cube measuring 1 m on a side and about the closed line t bounding S as shown.
From To accomplish this, a small circle t:3 at. Ifds is assumed positive outward on S, then the sense of the line integration is as noted, the integrals cancelling along z and 4 oftlw connective strip a jr Integration sense b EXAMPLE You might consider how the results would have compared had one ignored the singularity. Maxwell's Curl Relations for Electric and Magnetic Fields in Free Space In Section B, the divergence of a vector fUllction was put to use in deriving the differential Maxwell equations and from their integral versions and The definition of the curl may similarly be used to obtain the differential forms of the remaining equations I and i , Because the latter are correct for closed lines of arbitrary shapes and sizes, one may choose t in the form of any small closed path bounding a j Lls 1 in the vicinity of any point, as in Figure Taking the ratio of I to Lls 1 yidds, with the assignment of the vector sense a l to each side, d r Bods dt Jl1s, Lls 1 , the left side, as AS l , becomes a1rcurlEl 1.
Equation states that the curl of the field E at any position is precisely the time rate of decrease of the field B there. This implies that the presence of a time-varying magnetic field B in a region is respon- sible for an induccd time-varying E in that region, such that is cverywhere satisfied. If the electric and magnetic fields in free space are static, the operator Ojat appearing in and should be set to zero.
The difTerential Maxwell equations usually oH;:,r a much hroader class of solutions; obtaining a number of these solutions will constitute the task of rnueh of the remaining text. Also of importance are the sinusoidal steady state, or time-harmonic solutions of Maxwell's equations. Time-barmonie fields E and B are generated whenever their charge and current sources have densities varying sinusoidally in time.
Assuming the sinusoidal sourees to have been active long enough that the transient field components have decayed to negligible levels permits the further assumption that E and B have reached a sinusoidal steady state.
The alternative and equivalent t wmulation is achieved if the fields are assumed to vary according to the complex exponential factor. This assumption leads to a reduction of the field fimetions of space and time to fimctions of space only, as ohserved in the following. They represent a simplification of the real-time fOIIns in that the tipe variable t has been eliminated.
One can show that a similar procedure using the replacements leads to a complex, time-harmonic set of the integral forms of Maxwell's equations in free space.
A comparison wi th their time-dependent versions is provided in Table Applications of the complex time-harmonic forms through to ele- mentary wave solutions in free space are considered in Section A preliminary discussion or the Laplacian operator and a development of the so-called wave equations are desirable prerequisites to finding such solutions.
These are discussed next. Moreover, the divergence of the vector function grad], denoted symbolically by V - V] , is by the definition a scalar measure of the flux source-per-unit-volurne condition of V] at every point in a region. The expansions and for V] and its di- vergence can be combined to obtain V - Vf in a desired coordinate system, a result to be found useful for obtaining both time-varying and time-static field solutions.
No corresponding simplicity occurs in other coordinate systems because of the spatial dependence of the unit vectors already noted. Then if V. It is worth wile to observe that one can more easily expand V 2 F by use of the vector identity a than by ddinition Several vector identities involving the difterential operators grad, div, and curl are listed in Table along with vector algebraic and integral identities.
The integral identities 7 and 8 are recognized as those of diver'gence and Stokes's theorem, respectively. Extensions of the divergence theorem lead 'to Green's integral identities 9 and 10 , proved in the next section. The integrand in the volume integral may be expanded by use of 15 in Table , whence becomes Green's first integral identity fVg.
CVllJ dv Subtracting the latter from obtains Green's second integral identity also knowll as Green's symmetric theorem.
Green's theorems and are important in applications to theorems ofbound,lIy-value problems of field theory, as well as to special theorems concerning integral properties of scalar and vector functions. One such. This theorem, not proved here,5 shows that the specification of both the divergence and the curl of a vector function F in a region V, plus a particular boundary condition on the surface S that bounds V, are sufficient to make F unique.
Maxwell's equations , 1 , , and specify the divergence and the curl of both the E and the B fields in a region in terms of charge and current densities as well as the B or E field , so that these relationships, together with appropriately specified boundary conditions, can similarly be expected to provide unique field solutions.
Finding solu- tions of Maxwell's differential equations is facilitated for some problems hy first mani- pulating them simultaneously to obtain differential equations in terms of only B or E, as is discussed next. In a time-varying electromagnetic field problem, one is generally interested in obtaining E and B field solutions of the tour Maxwell relations, a process that can often be facilitated by combining Maxwell's equations such that one of the fields B or E is eliminated, yielding a partial differential equation known as the wave equation.
This is accomplished as follows. Ramo, J. Whinnery, and T. Van Duzer. Fields and Waves in Communication Electronics, 2nd cd. New York: Wiley, , p. A wave equation similar to can be obtained in terms of B.
The complex time-harmonic forms of the wave equations may be obtained by re- placing Band E with their complex exponential forms, The simplest solutions of these scalar wave equations are uniform plane waves, involving as few as two fIeld components. They are considered in the next section. Simplifying fea- tures are that the solutions are amenable to the rectangular coordinate systeln, and the number offIeld components reduces to as few as two. These simplifications provide a background for the more complex wave structures discussed in later chapters.
Uniform plane waves have the property that, at any fixed instant, the E and B fields are uniform over plane surfaces. It will be shown that waves propagating in the z direction result from this restriction. If the waves propagate in empty space, one requires an additional assumption. The complex time-harmonic forms of the Maxwell differential equations deter- mining the wave solutions are through Plane wave concepts are included here becanse of their universal relevance to all dynamic field phenomena, and because they are essential to a more complete understanding of conduction and polarization eflects in materials under other than purely static comtitions.
Before atlempting to extract solutions from the wave equations, one may note that the curl relations, 08 and , furnish some interesting properties of the solutions, restricted by assumptions 1 and 2. No z component of either E or :B is obtained, thus making the field directions entirely transverse to the axis.
This is seen to be the case on setting Ex 0 in Ob , for example, forcing By to vanish while yet leaving the field pair Ey, Bx intact, the lall! When field pairs are of each other, they are said to be uncoupled. Alter- llatively, one can make use of either vect.?
It is to be shown that the exponential solutions and DzeifJ oz arc representations of constant amplitude waves travelytg in tl;e posi- tive z and negative z directions, respectively. The complex coefficients C't and C 2 must have the units of volts per meter, denoting arbitrary complex amplitudes of t;.
Complex amplitudes rep- resented in the complex plane. Once a solution of one wave equation has been obtained, the remaining can be l und by use of Maxwell's equations. Thus, the solution for Ex The real-time f rm of By of is similarly found to be By. The following symbols are chosen to denote them.
Electric field sketches ofa positive traveling unil rm plane wave. To display the field throughout a cross section any x-z plane, the flux plot of Figure b is more suitable. Vector plot ofthc fields ofa uniform plane wave along the z axis. Note the typical equiphase surface, depicting fluxes of E; and 13;. Its amplitude? Its vector direction in space? Thus in Figure , the z traveling uniform plane wave shown with the field components Ex Hy is said to be polarized in the x direction or simply x-polarized.
Similarly, the plane wave with the components E y , Hx described in Problem is polarized in the y direction. Both these waves are linearly polarized, because the electric field vector in any fixed z plane describes a straight-line path as time passes. Because Maxwell's equations are linear equations, a vector superposition, or summing, of the two linearly polarized uniform plane waves just introduced will also provide a valid field solution.
The resultant vector sum will not necessarily be linearly polarized, however, depending on the phase condition between the x- and the y- polarized electric field components. For example, with Ex Related H field components arc omitted for clarity. Thus, the tip of the total E vector describes an elliptical locus in any fixed Z plane as the wave moves by, indicating the elliptical polarization of the wave. Wave polarization is of practical importance in radio communication transmit- receive links because the power extracted by a receiving antenna from the arriving wave is usually dependent on the orientation of the antenna relative to the polarization of that wave.
The common half-wave, thin wire dipole antenna, for example, picks up the maximum power from a linearly polarized oncoming wave when the electric field of the arriving wave is aligned with the antenna wire, while accepting zero power from the wave if the electric field and the wire are at right angles. If the arriving wave were circularly or elliptically polarized, a component of the arriving E-field vector is made available to the receiving dipole regardless of its tilt in the plane ofE, so that the orientation of the receiving antenna, in any fixed z-plane, would have little or no effect on the amount of signal received.
This could be of considerable importance in satellite communications, in which the receiving antenna on the satellite is tumbling in space and therefore changing its attitude relative to the oncoming wave. Antennas capable of transmitting circularly polarized waves, such as helical antennas or phased crossed dipoles, are readily constructed to accommodate this need. CIlD, and R. Electromagnetics, 2nd ed. Vector Analysis. RAMO, S. Held" and Waves in Communication Electronics.
From the substitution of the appropriate coordinate variables and metric coefficients into the gradient expression , show that a,b,c follow in the three common coordinate systems. Also convert the magnitude expression to correct forms in those systems. In Problem is depicted the "distance-vector," R, defined as the difference r 2 r 1 of the position vectors to the endpoints ofR.
Carry out a direct proof resembling that leading to the expression fi r div F, but carried out in the rectangular coordinate system. Begin with , expressed in rectangular coordinates with reference to a diagram like Figure but adapted to the rectangular system. By the substitution of the appropriate coordinate variables and metric coefficients into , show that the expressions a,b,c follow, in the three common coordinate systems.
Determine for each of the following vector fields whether or not it has Hux sources; that is, find its divergence. Prove, by expansion in rectangular coordinates, that V. G, the identity 12 in Table Show that the following fields are, source-free. By comparison with results found in Example , with what kinds of static-charge sources are these field-types identified? Comment on this conclusion relative to , applicable to the uniform line charge.
Which choice of the parameter n provides a divergenceless field? Comment on this conclusion with respect to 7b , the electric field of the point charge. Illustrate the validity of the divergence theorem by evaluating its volume and surface integrals in and on the given parallelepiped. Sketch it.
Given that the field F r, e, 4 arlO - a. Show that inserting this field into the Maxwell divergence relation yields the charge density originally assumed. What is the physical interpretation of the zero value of the divergence expected of each and every B field?
With reference to a diagram resembling Figure but adapted to the rectangular co- ordinate system, give the details of a proof of the curl expression carried out in rectangular coordinate form. Find the curl of each of the vector fields given in Problem Which of those fields are irrotational conservative? Find in detail the curl of the vcctors Vg, VG, and Vh generated in Problem , [These results exemplify the validity of the vector identity 20 in Table Illustrate the validity of Stokes's theorem using the same closed line t and vector ficld of Example , but this time employ the surface S, consisting simply of the square located at y I.
What is the required expression for ds on S, ifit is to contorm to the line integration sense chosen about t? Is E conservative? One such surface S is shown. Show that this B field satisfies the time-static Maxwell curl relation Show that this magnetic field satisfies Substituting the correct coordinate variables and metric coefficients, show that the definition of the Laplacian operator of a vector field becomes in rectangular coordinates. Repeat Problem , except this time show that the definition , in the cylindrical coordinate system, yields Show that the use of the vector identity expanded in the cylindrical coordinate system yields the result In a manner similar to that employed in obtaining the wave equation in terms ofE, derive the vector wave eqnation in terms ofB.
Show how it may be reduced to for empty space. Using the replacements for real-time with complex time-harmonic fields, convert the vector, inhomogenous wave equations and to their corresponding complex time-harmonic forms. With the proper assumptions, show how these reduce to I and , appropriate to source-free empty space. Suppose that you are told that the complex wave fLlIlction Ex ;;:.
By direct substitution, prove that this is so. Its direction of travel? Its vector direction in space polarization? What is the A? Express the H field in its real-time form, H; z, t. The unit. Answer the questions asked in Problem concerning this given traveling wave. Begin with the other indepepdellt pair of Maxwell dillcrential equations a and b , involving the field-pair E y , Bx. Compare them with the ratios applicable to the x-polarized case.
Compare the results with the x-polarized case depicted in Figure , looking for similarities. Show that it becomes circular polarization when the component amplitudes are equal. Use a polarization diagram in the z 0 plane as suggested by Figure b to prove which or these two polarization cases has the electric field vector rotating cloekwise in time, and which counterclockwise looking in the positivc-z direction.
Comment on the analogy between this diagram and thc "Lissajou figures" observable with an oscilloscope on exciting its vertical and horizontal amplifiers with sinusoidal signals differing in phase. As an added option, modify the three-dimt'llsional diagram in Figure to illustrate the details of this polarization problem.
What kind of polarization exists here? An electric or magnetic field impressed on a material exerts Lorentz forces on the particles, which undergo displacements or rearrangements to modify the impressed fields accordingly.
The Maxwell equations that describe the electric and magnetic field behavior in a material are thus expected to require modifications from their free-space versions to account for whatever additional fields the material particles produce. It is the task in this chapter to diseuss these extensions of the free-space Maxwell equations. The topic of conduction is discussed from the viewpoint of a collision model.
The chapter continues with a consideration orthe added effects of electric polarization within a material, providing a Maxwell divergence relation valid for materials as well as free space. Next is treated the added effect of magnetic polarization, yielding a suit- ably altered Maxwell curl expression lor the magnetic field. The field vectors D and H are thereby defined. Boundary conditions prevailing at interfaces separating difter- ently polarized regions are developed fi'om the integral forms of the Maxwell equations, to compare the normal components ofD and the tangential components ofH at adja- cent points in the regions.
The discussion continues with related treatments of the Maxwell div B and curl E equations for material regions, their integral forms, and corresponding boundary conditions. The chapter concludes with a discussion of uni- form plane waves in a material possessing the parameters J, E, and j1, exemplifYing the use of the Maxwell equations for a linear, homogeneous, and isotropic material. A representation of the production of a drift component of the velocity of free electrons in a metal.
Electric charge conduction 2. Electric polarization 3. In terms of their charge-conduction property, materials may for some purposes be classified as insulators dielectrics , which possess essentially no free electrons to pro- vide currents under an impressed electric field; and conductors, in which free, outer orbit electrons are readily available to produce a conduction current when an electric field is impressed.
An electrically conductive solid, commonly known as a conductor, is vi- sualized in the submicroscopic world as a latticework of positive ions in which outer- orbit electrons are free to wander as free electrons1-negative charges not attached to any particular atoms.
On this structure are superposed thermal agitations associated with the temperature of the conductor --the light, agile conduction electrons moving about the more massive ion lattice, imparting some of their momentum to that lattice in exchange for new random directions of flight until more interactioIls occur.
This cir- cumstance is depicted in Figure a for a typical conduction electroIl. That a very large number of particles are present in a small volume increment is appreciated on noting that a typical conductor, sodium, possesses about 2.
When free electrons collide interact with the ion lattice, they give up, on the average, a momentum rrtvd in the mean free time Tc between collisions, ifm is the electron mass.
On equating this to the Lorentz electric field force applied within the conductor, one obtains and solving tor Vd yields the steady drift velocity The expression , linearly relating the drift velocity to the applied E field, is of the form in which the proportionality constant Pe' taken to be a positive number, is termed the electron mobility, which from is evidently eTc 2 m IV-sec m '.
Eq uation is an expression exhibiting a linear dependence ofJ on the applied E field in the conductor. Experiments show that this is an exceedingly accurate model for a wide selection of physical con- ductors. Find the mean free time and the electron mobility for sodium, having the measured dc conductivity 2. Sodium has an atomic density of 2.
Thus from , the mean free time becomes rna 9. Its electron mobility is found from either of the relations 2. The current density is proportional to the drift velocity Vd, im- plying from that current decays with time at the same rate on removing the E field. The differential equation can be simplified ifE is assumed sinusoidal.
The model of electrical conductivity just described is essentially that proposed by Karl Drude in The so-called band theory of solids, an outgrowth of quantum mechanics, is useful for describing the intrinsic differences among the con- ductors, semiconductors, and insulators.
New York: Wiley, , for details. The mechanism of the dielectric polarization effects resulting from applied electric fields may be explained in terms of the microscopic displacements of the bound positive and negative charge constituents from their average equilibrium positions, produced by the Lorentz electric field forces on the charges.
Such displacements are usually only a fraction of a molecular diametcr in the material, but the sheer numbers of particles involved may cause a significant change in the electric field from its value in the absence of the dielectric substance. Dielectric polarization may arise from the following causes.
Electronic poLarization, in which the bound, negative electron cloud, subject to an impressed E field, is displaced from the equilibrium position relative to the positive nucleus. Ionic polarization, in which the positive and negative ions of a molecule are dis- placed in the presence of an applied E field. Orientational polarization, occurring. The tendency for the so-called polar molecules of such a material to align parallel with the applied field is opposed by the thermal agita- tion effects and the mutual interaction forces among the particles.
Water is a common example of a substance exhibiting orientational polarization effects, In each type of dielectric polarization, particle displacements are inhibited by powerful restoring forces between the positive and negative charge centers.
In Figure is illustrated the polarization mechanism in a material involving two species of charge. One should imagine thermal agitations superimposed on the average positions of the particles shown.
Displacement equilibrium is attained when the applied forces are balanced by the internal attractive Coulomb forces of the couplets. Positive cloud I--,:!
Electric polarization eflects in simple models of nonpolar and polar dielectric materials. IfE were applied in the x direction as shown, a net component ofP would be induced. In other words, div P becomes a negative, polarization-created, effective charge density, given the symbol pp. Equating to Pp liZ! Thc divergence of EoE, inIree space, has been given by to express the density Pv of fi'ee charge.
For such materials 4 PocE in which the parameter Xc is called the electric Eo is retained in to make Xe dimensionless. XC' Pv, D, and E,. If E is made strong enough, a material may experience polarization displacements that result in permanent dislocations of the molecular structure, or a dielectric breakdown, for which case does not hold. In a nonlinear material the magnitude of D is not proportional to the applied E field, though the E and the P vectors may have the same directions.
Then is written more generally in which the dependence of Xe on E is noted. Dielectric Polarization Current Density If the electric field giving rise to dielectric polarization effects is time-varying, the resulting polarization field is also time-varying.
Then the displacements of the positive charge constituents in one direction, together with the negative charges moving oppositely, give rise to charge displacements through cross sections of the material identifiable as currents through those cross sections. The field jp, along with the polariza- tion charge density field Pp described by , acts as an additional source of electric and rnagnetic fields.
In particular, the special role played by jp in relation to magnetic fields in a material region is discussed later in Section Integral Form of Gauss's Law of Materials The dielectric polarization effects attributed to material regions have been seen to lead to the divergence expressions and , relating the field quantities p and D to the polarization charge and free charge sources.
The divergence theorem can be used to transform these differential equations into corresponding integral forms. The most important of these is for the D field; that is, V.
Equation is the integral form of Maxwell's equation for a material region, sometimes called Gauss's law for material regions. It states that the net outward flux of D over any closed surface is a measure of the total free charge contained by the volume V bounded by S, at any instant of time. Equation states that the net outward flux ofP emanating from the sur- face of V is a measure of the net polarization charge summed throughout V.
Spatial Boundary Conditions for Normal D and P In many electromagnetic field problems ofphysical interest, it becomes necessary to discuss how the fields behave as one traverses the boundary surfaces, or interfaces, separating the various material regions that comprise the system. In such problems, a matching or fitting of the field solutions is required so that the boundary conditions at the interfaces may be satisfied.
The proper boundary conditions for the fields are determined, as will be shown, from the integral forms of Maxwell's equations for material regions. If, however, a surface charge density denoted by Ps and defined by the limit Region 2: 2 D2 a P.
Gaussian pillbox surface constrncted for deriviug the boundary condition on the normal component ofD. Two special cases of of physical interest are mentioned in the following, while a more general result is left for discussion in Section CASE A. Both regions perfect dielectrics.
Then reduces to The normal component oj D is continuous at an interface separating two perfect dielectrics, as illustrated in Figure a. Two cases of the boundalY condition lor normal components of D. One region is a perfect dielectric; the other is a perfect conductor. Electric currents are limited to finite densities in the physical world. Thus reduces to ,5. A boundary condition similar to can be derived comparing the normal components of the dielectric polarization vector P.
The net density includes the eHect of both species of surface polarization charge positive and negative accumulated just to either side of the interface. Two parallel conducting plates of great extent and d m apart are statically charged with q C on every area A of the lower and upper plates, respectively, as noted in a , The conductors are separated by air except for a homogeneous dielectric slab of thickness c and permittivity E, spaced a distance b from the lower plate.
Sketch the flux ofD. A Gaussian closed-surface S in the form of a reetangular box is placed as in Figure d , to contain the free charge q.
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