Engineering Electromagnetics. Sixth Edition. William H. Hayt, Jr.. John A. Buck. Textbook Table of Contents. The Textbook Table of Contents is your starting. Library of Congress Cataloging-in-Publication Data Hayt, William Hart, – Engineering electromagnetics / William H. Hayt, Jr., John A. Buck. — 8th ed. p. cm. Engineering electromagnetics / William H. Hayt, Jr., John A. Buck. professor and head of the School of Electrical Engineering, and as professor emeritus.

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Solutions of engineering electromagnetics 6th edition william h. hayt, john a., Past Exams for Electromagnetic Engineering. University. Hayt,Buck Engineering Electromagnetics 6e Pdf previous post Harrison Advanced Engineering Dynamics Pdf. next post Heat Exchanger. The ‗Vector approach' provides better insight into the various as ects of Electromagnetic phenomenon. Vector analysis is therefore an essential tool for the study.

B is thus constant everywhere inside the infinitely long solenoid. Faraday's Law Maxwell's integral law !!.. Indnced electric fields and Faraday's law. Any closed line t such as that of a may be superposed anywhere on the example of b , such that. Faraday's law must be trne for it. The relationship of the positive line-integration sense to the positive direction assumed for ds is the same as for Ampere's circuital law. This is tantamount to saying that an E field is generated by a time-varying magnetic flux.

The E field, in general, must also be time-varying if is to be satisfied at every instant. Faraday's law tor strictly time-static fields is I with its right side reduced to zero f. A field obeying is called a conservative field; all static electric fields are conservative. If the electric charges that produce an electric field are fixed in space, that electric field must obey Faraday's law in time-static form, Several examples of the electric fields of charges at rest have been treated in Example The electric field of a point charge Q.

Closed paths constructed about a point charge an. J This resul t 9 is seen to be independent of the choice of the path connecting P land P 2; it is a function only of the radial distances r 1 and r2 to the respective endpoints PI and P 2.

Static charge distributions like those depicted in Figure b are, in general, just collections of differential charge-elements dq Pvdv; whereas their static electric fields are just superpositions vector sums of the conservative differential electric fields dE produced by each of those static charge- elements.

One may thereby agree that Faraday's law for static electric fields is true in generaL Valid field solutions E r, t and B r, t satisfying Faraday's law must also satisfy the remaining Maxwell's integral relations of I through I ; however, if the time variations of the fields arc not too fast, in some cases a static solution for 9The physical interpretation of the result is of interest.

I t that the net work done in moving a unit test charge around a dosed path is zero; such an electric has already been termed conservative. Thus forms the basis of the theory oflhe scalar potential field of static electric charges to be discussed in Chapter 4.

Such a static field on which time variations are imposed is called quasi-static. On inserting the quasi-static field B r, t into Faraday's law , a first-order approximation to the E field can then be obtained; assuming that the field symmetry permits the extraction of the solution for E from An iterative process can sometimes then be employed to improve the accuracy of the quasi-static solution lO although if the time variations of the fields are not excessively rapid, the first-order solution will often suffice.

Determine from Ampere's law the quasi-static magnetic flux density developed inside the coil of radius a, and then use Faraday's law to find the induced electric intensity field both insidc and outside the coil. From Example b , the magnetic flux density inside the long solenoid carrying a static current I was found to be The electric field E induced by this time- varying B held is found by means of Faraday's law i , the line integral of which is first taken around the symmetric path t of radius p inside the coil, as shown in Figure i Faraday's law becomes in which, from the circular symmetry, E,p must be a constant on t.

Observe that E,p varies in direct proportion to p, as shown in Figure J Chl! Electromagnetic Fields, Energy and Forces. Wiley, , Chapter 6. Showing the assumed integration path t' used for finding the induced E field of a solenoid, and the resulting E field. This is symbolized in the time diagram of Figure Both answers arc directly proportional to j because they are governed by the time rate ofehange of the net magnetic flux intercepted by the surface, as noted from I It specifies that the net magnetic flux positive or negative emanating from any closed surface S in space is always zero.

This statement is illustrated in Figure ; in a of that figure is an arbitrary closed surface S constructed in the region containing a generalized magnetic flux configura- tion having a density B r, t in space. Gaussian closed surCrcc relative to magnetic fields. This means that magnetic flux lines always form closed lines.

Equivalently, it states that magnetic fields cannot terminate on magnetic charge sources for the reason that free magnetic charges do not exist physically. This is in contrast to the conclusion drawn from Gauss's law I f f electric fields; the presence of a nonzero right side involving the electric charge density function Pi' in that relation attests to the physical existence of free electric charges.

It is easy to find physical examples that illustrate the closed nature of magnetic flux lines. The magnetic field of a long, straight current-carrying wire of Figure is shown once more in Figure b ; observe how the uninterrupted flux lines account for' precisely as many magnetic flux lines entering the typical closed surface S as are emerging from it.

Such is the case for all closed surfaces that might be constructed in space for that field.

Table of contents

To accomplish such transformations, two steps are required: The details of transforming a vector field A from the rectangular to the circular cylindrical co- ordinate system considered in the following. Thus, in Figure a is shown superposed the rectangular and systems, with the unit vector a p displayed for the purpose of projections onto the ullit vectors ax, a y , and a z.

From the it that Similarly geometry, With the unit Now with the xl a' j, Geomelli", circular cylindrical cmmli",,'," a vector aq, at P on Figure would reveal, from the a onto ax, a y , a z as follows.

Step 2 concerns the transformation of the coordinate variables x,y, z , ap- in A, to the variables p, cP, z. Conversely, if A were given in circular cylindrical coordinate fc rm and its trans- formation to rectangular coordinates were desired, the reverse of the foregoing pro- CE?

A compilation of the transformations is f lUnd in Table A similar geometrical procedure can be used to transfc rm some vector field A between the rectangular and spherical coordinate systems.

I t is left. Transl rm the given vector lield to circular cylindrical coordinates. In this system, length is expressed in meters, mass in kilograms, and time in seconds. A fi urth unit, that of either eleetric charge coulomb or electric current coulomb per second, or ampere , is needed in the dimensional description of electromagnetic phenomena. The rationalized mks system, which elimi- nates a Ll,ctor 4n from the Maxwell equations, has been almost universally adopted, and it is used in this text.

The Giorgi mks system is especially noteworthy in that it deals with the primary electromagnetic quantities directly in the practical units in which they arc measured: The choice of the dimension of the fourth unit charge adopted for the mks system is seen to depend on the values chosen for the constants Eo and Ito that appear in the Maxwell equations, I and In Table are listed units of the mks system by name, unit, and symbol.

Similarly, 3 X 12 F is abbreviated 3 pF, with p pica denoting the factor Other literal prefixes to be used in this way are listed in Table The Classical Theory of Electricity and Magnetism. Blackie, Electrical Engineering Science. McGraw-Hill, FANO, R. CHU, and R. Wiley, HAYT, W. Engineerin,g Electromagnetin, 4th ed. McGraw-Hili, !. Electromagnetic Fields and Waves. San Francisco: Freeman, Vector Ana ysis. Foundations of Electromagnetic Theory.

Reading, Mass.: Addison- Wesley, Given are the vector constants: Sketch them at the origin in the rectangular coordinate system and evaluate the following. If the same E were given to exist at another point, say P 2 3, 0, I in this region, explain why these E vectors are considered equal at these different points. Also shown arc the coordinate surfaces p 5, 30, z 6, the intersection ofwhieh defines Pl' a Find the magnitude ofB.

Deter- mine the expression for the unit vector a B directed along B in circular eylindrical coordinates. Label aB on your reproduction of this sketch.

Sketch and label the additional coordinate surface needed to identify the P z location. Explain why these B vectors at the different points PI and P z arc in fact not equal vectors, despite the identical expressions for B and its unit vector in this coor- dinate system. Identify the components ofB that are perpendicular to the three coordinai: Label the coordinates of the points P 3 and P 4 shown. What is the vector field expression for G x, 3, 2 applieable over this t?

Sketch curves showing the variation of G x and of IGI versus x over this range. Shown is the "distance vector," R, a vector directed from the point PdXl,Yl' zd to P Z x 2 , 'z, zz in space, the position vectors of the latter being r 1 and rz. Sketch two vectors A and B in the same plane, showing from the definition and the geometry that A.

Use the ddinitionof the dot product to find the projection F cos 0 ofF onto G. Sketch this projection. Find the smaller angle between F and G.

Find their magnitudes, and find the smaller angle between them in their common plane by use of 1 their dot product, and 2 their cross product. Label these angles on your diagram. Employ the concept of "projection" from Problem ; e. Xc the direction cosine. B, is zero. What does it mean? Use a graphical construction to reinforce your remarks. C [Answer: Applying the definition of the cross produ.

Avoid employing the determinant in your arguments. A parallelepiped has edges given by ax, 2a p and a z Sketch this "box. Label it on the sketch. Label it also noting that as a free vector, B can be translated parallel to itself without altering its magnitude and direction. Relative to the pivot point given, find the vector moment torque associated with the owing vector force and distance in meters. Sketch the applicable vector diagrams, indicating from the right-hand rule the rotation associ- ated with the moment M.

Find the value of the line integralofH. Is H a conservative field? Explain [Answer: Find the line integral ofE.

Using standard scalar volume-integration methods, make use of the triple integral of p"dv given by to find the charge q inside the following volume regions. Sketch each con- figuration with dimensions, labeling a volume element at the typical point P inside.. Employing standard scalar surface-integration methods, make use of the double integral of Ps ds to find the total charge on the following surfaces.

Sketch the dimensional layout, labeling a scalar surface element at the typical point P Ul, U2, u3 on S, appropriate to the coordinate system required. Given is the E-field solution 7b for the point charge Qlocated at the origin. Add to the sketch the details of the vector surface element ds suggested by Figure b. A spherical shell of charge possesses the constant volume charge density Pv between its inner and outer radii a and b.

Show an appropriately labeled sketch along with the details of your proof. Determine the k that will make the total charge in the sphere zero. For this k, why is the E field external to the sphere zero? Find E. An infinitely long, cylindrical clond of radius p a in free space contains the static, uniform volume charge density p". With a suitably labeled sketch, make use of the symmetry and Gauss's law to obtain the following.

Label the Gaussian surfaces used. Show from your solution that the field oLltside the cloud is the same as that expected if the total clurge were concentrated along the z-axis. Do not usc Gauss's law. A hollow, circular cylindrical conductor in free space, assumed infinitely long to avoid end efiects, and having the inner and outer radii band c, respectively, carries the direct current 1. A coaxial pair of circular cylindrical conductors, infinitely long in frec space, have the dimensions shown and carry the equal and opposite total currents f.

Use Ampere's law, together with an appropriately labeled diagram showing the closed s sphere. Show l,riven by fill make zero? Show that the static B fields of the coaxiallinc of Problem arc the superposition of the fields of the hollow conductor of Problem ' and those of the isolated conductor of Example The currents are assumed charge-compensated, making electric fields absent in this problem.

Choose a rectangular dosed path t with one side parallel to the known field of a , and its other side aligned with the unknown field. Two parallel, round conductors, infinitely long and carrying the currents 1, 1, are 2d m apart.

Sketch a top view of the conductors in the x: Show its vector field contribution Bl at the normal distance P 1 from 1 to the typical location P x,] , making use of Showing Bl decomposed into its Bxl and Byl components, use the geometry to develop the expression lor Bl solely in terms ofx and]. Find also the vector B at the following x,. Introducing the unit vectoraql at Pon Figure a , from the geometry verify b. Similarly verify c.

Transform this to its spherical coordinate form, r 2a sin 0 cosljJ. Transform the following veetor fields to the circular cylindrical coordinate system. Transform the given vector fields to the spherical coordinate system.

The divergence theorem and the theo- rem of Stokes are used to derive the differential forms of Maxwell's divergence and curl equations in free space fi'om their integral versions postulated in Chapter L The appropriate manipulations of Maxwell's time-varying differential equations are seen to lead to the wavc equations in terms of the Band E fields, and the wavelike nature their solutions is exemplified by considering in detail the field solutions of uniform waves in free space.

A pursuit of these ideas requires some background in the differentiation of vector fields, to be discussed in the following section. This notion has already been introduced in Section in connection with the position vector r. I t is now considered in general for any differentiable vector field.

The derivatives of the sum or product comhinations of scalar and vector I tions are often of interest. For example, iff and F are respectively scalar and VI functions of the variable u, the derivative of their product is, from dUF.

If F has continuous partial derivatives of at least the second order, i permissible to differentiate it in either order; thus 2 The partial derivative of the sum or product combinations of scalar 'and vect functions sometimes is useful. In particular, one can use to prove tbat t following expansions are valid Ns in the in- tor incre- ion of the in Figure liffercncc erivative on. For example, in the scalar temperature field T ull Uz, U3, t depicted in Figure I-I a , one can surmise from graphical considerations that the maximum space rates of tem- perature change OCCllr in di rections normal to the constan t temperatu re surfaces shown.

Generally, the maximum space rate of change of a scalar function, induding the vector direction in which the rate of change takes place, can be characterized by means of a vector di! It is developed here. A physical example of such a surface is any of the constant temperature surfaces of Figure l-l a.

Balanis - Advanced Engineering Electromagnetics - Solutions (Balanis) - menor.pdf

Then the amount by which I changes in going from P to P' is zero, but from , gradf. Grad I is therefore a vector everywhere perpendicular to any surface on which I constant. Thus, any path connecting Po and P will provide the same result, Carrying out over some path A from Po to the point P and then back to Po once more over a different path B, the contributions of the two integrals would cancel exactly, making the result.

This is not done here because of its lengthy form and because it serves no particular need in connection with the objectives of this text. You may wish to consult other sources relative to extending to other coordinate systems.

Sketch a few isotherms constant temperature surfaces of this static thermal field and determine the gradient of T. The isotherms arc obtained by setting T equal to specific constant temperature values. Brown, Field AnalYsis and Elfictromagnetics. McGraw-Hill, , p. The x andy components of the temperature gradient denote space rate of change of temperature along these coordinate axes.

From , the magnitude is denoting the maximal space rate of change of temperature at any pomt. One may observe that heat will flow in the direction of maximal temperature decrease; that is, along lines perpendicular to the isotherms and thus in a dir'cction opposite to that of the vector grad T at any poin t.

If a vector field r is representable by a continuous system of unbroken flux lines in a volume region as for example, in Figure a , the region is said to be sourcefree; or equivalently, field F is said to be divergenceless. The divergence ofF is zero. On the other hand, l1li;;;; , fjl! Concerning the divergence of flux fields.

The characterization of the divergence of a vector field on a mathematical basis is described here. The divergence of a vectorfield F, abbreviated div F, is defined as the limit of the net outward flux ofF, fs F ds, per unit volume, as the volume! The shape of S is immaterial in this limit, as long as the dimensions of! The definition leads to partial diHerential expressions for div F in the various coordinate systems.

For example, in generalized orthogonal coordinates, div F is shown to become The derivation of the differential expression for div F in generalized ortho- gonal coordinates proceeds from the definition The contribution! A volume-clement L'iu in the generalized orthogonal coordinate system nsed in the development of the partial diflcrcntial expression for div F.

The remaining four sides are similarly treated. On taking the dot product of V with F in the rectangular system of coordinates, one finds or precisely a. Sketch nux plots [or each o[the following vector fields, and find the divergence of each: Tnspection reveals a zero value or divergence is obtained for the fields F and G; a tcst closed surface placed any- where in the region will have zero net flux emanating from it. 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. Given the p-dependent field: 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: 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.

The right side of applied to t: From To accomplish this, a small circle t: 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.

Yt Is B' ds J,B. The difTerential Maxwell equations usually oH;: 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: 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. 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!

engineering electromagnetic fields and waves 2nd edition.pdf

When field pairs are of each other, they are said to be uncoupled. Alter- llatively, one can make use of either vect.? Either approach obtains the following wave equation in terms of Ex: 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;. Employing amplitude symbols E: 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. Consider only the first terms of each: 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.

The first octant of a sphere centered at the origin is bounded by the four coordinate surfaces: 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 Make use of in accounting elr the space derivatives of the unit vectors a p and a",. 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. A particular plane wave in empty space has the electric field given, in time- harmonic form, by E: What is its amplitude?

Its direction of travel? Its vector direction in space polarization?

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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.

A uniform plane wave has the time-harmouic electric field: What kind of polarization exists here? Is it clock,: 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. Magnetic polarization For large classes of materials, these effects are often adequately described through use of three parameters: 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.

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. Determine E at P 0 , y, 0: The field will be.

This field will be:. Now, since the charge is at the origin, we expect to obtain only a radial component of E M. This will be:. Calculate the total charge present: A uniform volume charge density of 0. If the integral over r in part a is taken to r 1, we would obtain[. With the limits thus changed, the integral for the charge becomes:. What is the average volume charge density throughout this large region? Each cube will contain the equivalent of one little sphere. Neglecting the little sphere volume, the average density becomes.

Find the charge within the region: The integral that gives the charge will be. Uniform line charges of 0. This field will in general be:. Find E in cartesian coordinates at P 1 , 2 , 3 if the charge extends from. With the infinite line, we know that the field will have only a radial component in cylindrical coordinates or x and y components in cartesian. Therefore, at point P:. So the integral becomes. Since all line charges are infinitely-long, we can write:.

Substituting these into the expression for E P gives. What force per unit length does each line charge exert on the other? The charges are parallel to the z axis and are separated by 0. Thus the force per unit length acting on the line at postive y arising from the charge at negative y is.

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The integral becomes:. Since the integration limits are symmetric about the origin, and since the y and z components of the integrand exhibit odd parity change sign when crossing the origin, but otherwise symmetric , these will integrate to zero, leaving only the x component. This is evident just from the symmetry of the problem. Performing the z integration first on the x component, we obtain using tables:. The integral becomes. First, we recognize from symmetry that only a z component of E will be present.

The superposition integral for the z component of E will be:. Surface charge density is positioned in free space as follows: For this reason, the net field magnitude will be the same everywhere, whereas the field direction will depend on which side of a given sheet one is positioned.

We take the first point, for example, and find. The magnitude of E A is thus 3. This will be the magnitude at the other two points as well. Find E at the origin if the following charge distributions are present in free space: The sum of the fields at the origin from each charge in order is:.

A uniform surface charge density of 0. Find E at the origin: Since each pair consists of equal and opposite charges, the effect at the origin is to double the field produce by one of each type. Taking the sum of the fields at the origin from the surface and line charges, respectively, we find:. We write.

Since this vector is to have no z compo-. If you don't receive any email, please check your Junk Mail box. If it is not there too, then contact us to info docsity.

If even this does not goes as it should, we need to start praying! This is only a preview. Load more. Search in the document preview. Plots are shown below. So the projection will be: The projection is: The angle is found through the dot product of the associated unit vectors, or: Describe the surfaces defined by the equations: This is the equation of a cylinder, centered on the x axis, and of radius 2.

Express in cylindrical components: Sketch F: The force will be: This force in general will be: With this restriction, the force becomes: Now the x component of E at the new P 3 will be: This expression simplifies to the following quadratic: The field will take the general form: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.

From , the B field must be -directed if the line integration counter- clockwise looking ii'om above is to yicld the positive current 1 emerging from St. Now the x component of E at the new P 3 will be: The net density includes the eHect of both species of surface polarization charge positive and negative accumulated just to either side of the interface. Then is written more generally in which the dependence of Xe on E is noted. Antennas capable of transmitting circularly polarized waves, such as helical antennas or phased crossed dipoles, are readily constructed to accommodate this need.

Log in. We take the first point, for example, and find.