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Maxwell's Equations in Differential Form, and Uniform Plane Waves In RaoCh03v3.qxd 12/18/03 3:32 PM Page 129 CHAPTER 3 Maxwell’s Equations in Differential Form, and Uniform Plane Waves in Free Space In Chapter 2, we introduced Maxwell’s equations in integral form. We learned that the quantities involved in the formulation of these equations are the scalar quantities, electromotive force, magnetomotive force, magnetic flux, dis- placement flux, charge, and current, which are related to the field vectors and source densities through line, surface, and volume integrals. Thus, the integral forms of Maxwell’s equations, while containing all the information pertinent to the interdependence of the field and source quantities over a given region in space, do not permit us to study directly the interaction between the field vectors and their relationships with the source densities at individual points. It is our goal in this chapter to derive the differential forms of Maxwell’s equa- tions that apply directly to the field vectors and source densities at a given point. We shall derive Maxwell’s equations in differential form by applying Maxwell’s equations in integral form to infinitesimal closed paths, surfaces, and volumes, in the limit that they shrink to points. We will find that the dif- ferential equations relate the spatial variations of the field vectors at a given point to their temporal variations and to the charge and current densities at that point. Using Maxwell’s equations in differential form, we introduce the important topic of uniform plane waves and the associated concepts, funda- mental to gaining an understanding of the basic principles of electromagnetic wave propagation. 129 RaoCh03v3.qxd 12/18/03 3:32 PM Page 130 130 Chapter 3 Maxwell’s Equations in Differential Form . 3.1 FARADAY’S LAW AND AMPÈRE’S CIRCUITAL LAW Faraday’s We recall from Chapter 2 that Faraday’s law is given in integral form by law, special case # d # E dl =- B dS (3.1) CC dt LS where S is any surface bounded by the closed path C. In the most general case, the electric and magnetic fields have all three components (x, y, and z) and are dependent on all three coordinates (x, y, and z) in addition to time (t). For sim- plicity, we shall, however, first consider the case in which the electric field has an x component only, which is dependent only on the z coordinate, in addition to time. Thus, E = E z, t a (3.2) x1 2 x In other words, this simple form of time-varying electric field is everywhere di- rected in the x-direction and it is uniform in planes parallel to the xy-plane. Let us now consider a rectangular path C of infinitesimal size lying in a plane parallel to the xz-plane and defined by the points x, z , x, z +¢z , 1 2 1 2 x +¢x, z +¢z , and x +¢x, z , as shown in Fig. 3.1. According to Fara- 1 2 1 2 day’s law, the emf around the closed path C is equal to the negative of the time rate of change of the magnetic flux enclosed by C. The emf is given by the line integral of E around C. Thus, evaluating the line integrals of E along the four sides of the rectangular path, we obtain x, z+¢z 1 2 # = = E dl 0 since Ez 0 (3.3a) Lx, z 1 2 x+¢x, z+¢z 1 2 # = ¢ E dl [Ex]z+¢z x (3.3b) Lx, z+¢z 1 2 y z (x, z)(⌬z x, z ϩ ⌬z) ⌬xSC FIGURE 3.1 (x ϩ ⌬x, z) (x ϩ ⌬x, z ϩ ⌬z) Infinitesimal rectangular path lying in a plane parallel to the xz-plane. x RaoCh03v3.qxd 12/18/03 3:32 PM Page 131 3.1 Faraday’s Law and Ampère’s Circuital Law 131 x+¢x, z 1 2 # = = E dl 0 since Ez 0 (3.3c) Lx+¢x, z+¢z 1 2 x, z 1 2 # =- ¢ E dl [Ex]z x (3.3d) Lx+¢x, z 1 2 Adding up (3.3a)–(3.3d), we obtain # = ¢ - ¢ E dl [Ex]z+¢z x [Ex]z x CC (3.4) = [E ] +¢ - [E ] ¢x 5 x z z x z6 In (3.3a)–(3.3d) and (3.4),[Ex]z and [Ex]z+¢z denote values of Ex evaluated along the sides of the path for which z = z and z = z +¢z, respectively. To find the magnetic flux enclosed by C, let us consider the plane surface S bounded by C. According to the right-hand screw rule, we must use the mag- netic flux crossing S toward the positive y-direction, that is, into the page, since the path C is traversed in the clockwise sense. The only component of B normal to the area S is the y-component. Also since the area is infinitesimal in size, we can assume By to be uniform over the area and equal to its value at (x, z). The required magnetic flux is then given by # = ¢ ¢ B dS [By] x, z x z (3.5) LS 1 2 Substituting (3.4) and (3.5) into (3.1) to apply Faraday’s law to the rectan- gular path C under consideration, we get - ¢ =-d ¢ ¢ [Ex]z+¢z [Ex]z x [By] x, z x z 5 6 dt 5 1 2 6 or 0 [E ] +¢ - [E ] [By] x, z x z z x z 1 2 =- (3.6) ¢z 0t If we now let the rectangular path shrink to the point (x, z) by letting ¢x and ¢z tend to zero, we obtain 0 [E ] +¢ - [E ] [By] x, z x z z x z 1 2 lim =-lim ¢x:0 ¢ ¢x:0 0 ¢z:0 z ¢z:0 t or 0 0 Ex By =- (3.7) 0z 0t RaoCh03v3.qxd 12/18/03 3:32 PM Page 132 132 Chapter 3 Maxwell’s Equations in Differential Form . Equation (3.7) is Faraday’s law in differential form for the simple case of E given by (3.2). It relates the variation of Ex with z (space) at a point to the variation of By with t (time) at that point. Since this derivation can be carried out for any arbitrary point (x, y, z), it is valid for all points. It tells us in par- ticular that an Ex associated with a time-varying By has a differential# in the z- direction. This is to be expected since if this is not the case,AE dl around the infinitesimal rectangular path would be zero. Example 3.1 Finding B for a given E Given E = 10 cos 6p * 108t - 2pz a V/m, let us find B that satisfies (3.7). 1 2 x From (3.7), we have 0 0 By Ex =- 0t 0z 0 8 =- [10 cos 6p * 10 t - 2pz ] 0z 1 2 =-20p sin 6p * 108t - 2pz 1 2 -7 10 8 B = cos 6p * 10 t - 2pz y 3 1 2 -7 10 8 B = cos 6p * 10 t - 2pz a 3 1 2 y Faraday’s We shall now proceed to derive the differential form of (3.1) for the gen- law, general eral case of the electric field having all three components (x, y, z), each of them case depending on all three coordinates (x, y, and z), in addition to time (t); that is, E = E x, y, z, t a + E x, y, z, t a + E x, y, z, t a (3.8) x1 2 x y1 2 y z1 2 z To do this, let us consider the three infinitesimal rectangular paths in planes par- allel to the three mutually orthogonal planes# of the Cartesian coordinate sys- tem, as shown in Fig. 3.2. Evaluating AE dl around the closed paths abcda, adefa, and afgba, we get # = ¢ + ¢ E dl [Ey] x, z y [Ez] x,y+¢y z 1 2 1 2 Cabcda (3.9a) - ¢ - ¢ [Ey] x, z+¢z y [Ez] x, y z 1 2 1 2 # = ¢ + ¢ E dl [Ez] x, y z [Ex] y, z+¢z x C 1 2 1 2 adefa (3.9b) - ¢ - ¢ [Ez] x+¢x, y z [Ex] y, z x 1 2 1 2 RaoCh03v3.qxd 12/18/03 3:32 PM Page 133 3.1 Faraday’s Law and Ampère’s Circuital Law 133 d(x, y, z ϩ ⌬z) c(x, y ϩ ⌬y, z ϩ ⌬z) ⌬z z e(x ϩ ⌬x, y, z ϩ ⌬z) a(x, y, z) b(x, y ϩ ⌬y, z) y ⌬y ⌬x x FIGURE 3.2 Infinitesimal rectangular paths in f(x ϩ ⌬x, y, z) g(x ϩ ⌬x, y ϩ ⌬y, z) three mutually orthogonal planes. # = ¢ + ¢ E dl [Ex] y, z x [Ey] x+¢x, z y C 1 2 1 2 afgba (3.9c) - ¢ - ¢ [Ex] y+¢y, z x [Ey] x, z y 1 2 1 2 In (3.9a)–(3.9c), the subscripts associated with the field components in the vari- ous terms on the right sides of the equations denote the values of the coordi- nates that remain constant along# the sides of the closed paths corresponding to the terms. Now, evaluating 1B dS over the surfaces abcd, adef, and afgb, keep- ing in mind the right-hand screw rule, we have # = ¢ ¢ B dS [Bx] x, y, z y z (3.10a) Labcd 1 2 # = ¢ ¢ B dS [By] x, y, z z x (3.10b) Ladef 1 2 # = ¢ ¢ B dS [Bz] x, y, z x y (3.10c) Lafgb 1 2 Applying Faraday’s law to each of the three paths by making use of (3.9a)–(3.9c) and (3.10a)–(3.10c) and simplifying, we obtain - - 0 [Ez] x, y+¢y [Ez] x, y [Ey] x, z+¢z [Ey] x, z [Bx] x, y, z 1 2 1 2 1 2 1 2 1 2 - =- (3.11a) ¢y ¢z 0t - - 0 [Ex] y, z+¢z [Ex] y, z [Ez] x+¢x, y [Ez] x, y [By] x, y, z 1 2 1 2 1 2 1 2 1 2 - =- (3.11b) ¢z ¢x 0t - - 0 [Ey] x+¢x, z [Ey] x, z [Ex] y+¢y, z [Ex] y, z [Bz] x, y, z 1 2 1 2 1 2 1 2 1 2 - =- (3.11c) ¢x ¢y 0t RaoCh03v3.qxd 12/18/03 3:32 PM Page 134 134 Chapter 3 Maxwell’s Equations in Differential Form .
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