M03_RAO3333_1_SE_CHO3.QXD 4/9/08 1:17 PM Page 71 CHAPTER Maxwell’s Equations in Differential Form 3 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, elec- tromotive force, magnetomotive force, magnetic flux, displacement 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 equations 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 differential 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. In this process we shall also learn two important operations in vector calculus, known as curl and divergence, and two related theorems, known as Stokes’ and divergence theorems. 3.1 FARADAY’S LAW We recall from the previous chapter that Faraday’s law is given in integral form by 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 elec- tric 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 simplicity, we shall, how- ever, 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 = Ex(z, t)ax (3.2) 71 M03_RAO3333_1_SE_CHO3.QXD 4/9/08 1:17 PM Page 72 72 Chapter 3 Maxwell’s Equations in Differential Form In other words, this simple form of time-varying electric field is everywhere directed 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 par- allel to the xz-plane and defined by the points x, z , x, z +¢z , x +¢x, z +¢z , and x +¢x, z , as shown in Figure 3.1. According to Faraday’s law, the emf around the closed path C is equal to the negative of the1 time2 1 rate of change2 1 of the magnetic2 flux 1enclosed by2 C.The emf is given by the line integral of E around C.Thus,evaluat- ing the line integrals of E along the four sides of the rectangular path, we obtain x, z+¢z 1 2E # dl = 0 since Ez = 0 (3.3a) Lx, z 1 x+¢2 x, z+¢z 1 2E # dl = [Ex]z+¢z ¢x (3.3b) Lx, z+¢z 1 (x+¢x,2 z) E # dl = 0 since Ez = 0 (3.3c) L(x+¢x, z+¢z) 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 = [Ex]z+¢z - [Ex]z ¢x (3.4) 5 6 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. 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 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 magnetic flux crossing S toward the positive y-direction, that is, into the page, since the path C is tra- versed 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 M03_RAO3333_1_SE_CHO3.QXD 4/9/08 1:17 PM Page 73 3.1 Faraday’s Law 73 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 rectangular path C under consideration, we get d [Ex]z z - [Ex]z ¢x =- [By] x, z ¢x ¢z +¢ dt 5 6 5 1 2 6 or [Ex]z+¢z - [Ex]z 0[By] x, z =- (3.6) ¢z 0t1 2 If we now let the rectangular path shrink to the point (x, z) by letting ¢x and ¢z tend to zero, we obtain [Ex]z+¢z - [Ex]z 0[By] x, z Lim =-Lim ¢x:0 ¢z ¢x:0 0t1 2 ¢z:0 ¢z:0 or 0Ex 0By =- (3.7) 0z 0t 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 the above derivation can be carried out for any arbi- trary point (x, y, z), it is valid for all points. It tells us in particular that a time-varying By at a point results in an Ex at that point having 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 Given B = B0 cos vt ay and it is known that E has an x-component only, let us find Ex. From (3.6), we have B 0Ex 0 y 0 =- =- (B0 cos vt) = vB0 sin vt 0z 0t 0t Ex = vB0z sin vt We note that the uniform magnetic field gives rise to an electric field varying linearly with z. M03_RAO3333_1_SE_CHO3.QXD 4/9/08 1:17 PM Page 74 74 Chapter 3 Maxwell’s Equations in Differential Form Proceeding further, we can verify this result by evaluating A E # dl around the rectangular path of Example 2.8.This rectangular path is reproduced in Figure 3.2.The required line integral is given by b a E # dl = [Ez]x=0 dz + [Ex]z=b dx C C L z=0 L x=0 0 0 + [Ez]x=a dz + [Ex]z=0 dx L z=b L x=a = 0 + [vB0b sin vt]a + 0 + 0 = abB0v sin vt which agrees with the result of Example 2.8. x = 0 y z z = b z = 0 FIGURE 3.2 x = a Rectangular path of Example 2.8. x We shall now proceed to generalize (3.7) for the arbitrary case of the electric field having all three components (x, y,andz), each of them depending on all three coordinates (x, y,and z), in addition to time (t), that is, E = Ex x, y, z, t ax + Ey x, y, z, t ay + Ez x, y, z, t az (3.8) To do this, let us consider1 the three2 infinitesimal1 rectangular2 1 paths in2 planes parallel to the three mutually orthogonal planes of the Cartesian coordinate system, as shown in Figure 3.3.Evaluating AE # dl around the closed paths abcda, adefa,and afgba,we get E # dl = [Ey] x, z ¢y + [Ez] x, y+¢y ¢z Cabcda 1 2 1 2 - [Ey] x, z+¢z ¢y - [Ez] x, y ¢z (3.9a) 1 2 1 2 E # dl = [Ez] x, y ¢z + [Ex] y, z+¢z ¢x Cadefa 1 2 1 2 - [Ez] x+¢x, y ¢z - [Ex] y, z ¢x (3.9b) 1 2 1 2 M03_RAO3333_1_SE_CHO3.QXD 4/9/08 1:17 PM Page 75 3.1 Faraday’s Law 75 E # dl = [Ex] y, z ¢x + [Ey] x+¢x, z ¢y Cafgba 1 2 1 2 - [Ex] y+¢y, z ¢x - [Ey] x, z ¢y (3.9c) 1 2 1 2 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 f(x ϩ ⌬x, y, z) g(x ϩ ⌬x, y ϩ ⌬y, z) FIGURE 3.3 Infinitesimal rectangular paths in three mutually orthogonal planes. In (3.9a)–(3.9c) the subscripts associated with the field components in the various terms on the right sides of the equations denote the value of the coordinates 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,keeping 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 [Ez] x, y+¢y - [Ez] x, y [Ey] x, z+¢z - [Ey] x, z 0[Bx] x, y, z - =- (3.11a) 1 ¢2 y 1 2 1 ¢2 z 1 2 01t 2 [Ex] y, z+¢z - [Ex] y, z [Ez] x+¢x, y - [Ez] x, y 0[By] x, y, z - =- (3.11b) 1 ¢2 z 1 2 1 ¢2 x 1 2 01t 2 M03_RAO3333_1_SE_CHO3.QXD 4/9/08 1:17 PM Page 76 76 Chapter 3 Maxwell’s Equations in Differential Form [Ey] x+¢x, z - [Ey] x, z [Ex] y+¢y, z - [Ex] y, z 0[Bz] x, y, z - =- (3.11c) 1 ¢2 x 1 2 1 ¢2 y 1 2 01t 2 If we now let all three paths shrink to the point a by letting ¢x, ¢y, and ¢z tend to zero, (3.11a)–(3.11c) reduce to 0Ez 0Ey 0Bx - =- (3.12a) 0y 0z 0t 0Ex 0Ez 0By - =- (3.12b) 0z 0x 0t 0Ey 0Ex 0Bz - =- (3.12c) 0x 0y 0t Equations (3.12a)–(3.12c) are the differential equations governing the relationships be- tween the space variations of the electric field components and the time variations of the magnetic field components at a point.