Fundamentals of Electromagnetics: A Two-Week, 8-Day, Intensive Course for Training Faculty in Electrical-, Electronics-, Communication-, and Computer- Related Engineering Departments by Nannapaneni Narayana Rao Edward C. Jordan Professor Emeritus of Electrical and Computer Engineering University of Illinois at Urbana-Champaign, USA Distinguished Amrita Professor of Engineering Amrita Vishwa Vidyapeetham, India Amrita Viswa Vidya Peetham, Coimbatore August 11, 12, 13, 14, 18, 19, 20, and 21, 2008

3-1 Module 3 Maxwell’s Equations In Differential Form Faraday’s law and Ampere’s Circuital Law Gauss’ Laws and the Continuity Equation Curl and Divergence

3-2 Instructional Objectives 8.Determine if a given time-varying electric/magnetic field satisfies Maxwell’s curl equations, and if so find the corresponding magnetic/electric field, and any required condition, if the field is incompletely specified 9.Find the electric/magnetic field due to one- dimensional static charge/current distribution using Maxwell’s divergence/curl equation for the electric/magnetic field 10. Establish the physical realizability of a static electric field by using Maxwell’s curl equation for the static case, and of a magnetic field by using the Maxwell’s divergence equation for the magnetic field

3-3 Faraday’s Law and Ampère’s Circuital Law (FEME, Secs. 3.1, 3.2; EEE6E, Sec. 3.1)

3-4 Maxwell’s Equations in Differential Form Why differential form? Because for integral forms to be useful, an a priori knowledge of the behavior of the field to be computed is necessary. The problem is similar to the following: There is no unique solution to this.

3-5 However, if, e.g., y(x) = Cx, then we can find y(x), since then On the other hand, suppose we have the following problem: Then y(x) = 2x + C. Thus the solution is unique to within a constant.

3-6 FARADAY’S LAW First consider the special case and apply the integral form to the rectangular path shown, in the limit that the rectangle shrinks to a point.

3-7

3-8 General Case Lateral space derivatives of the components of E Time derivatives of the components of B

3-9 Combining into a single differential equation, Differential form of Faraday’s Law

3-10 AMPÈRE’S CIRCUITAL LAW Consider the general case first. Then noting that we obtain from analogy,

3-11 Thus Special case: Differential form of Ampère’s circuital law

3-12 Ex.For in free space find the value(s) of k such that E satisfies both of Maxwell’s curl equations. Noting that

3-13

3-14 Thus, Then, noting that we have from

3-15

3-16

3-17 Comparing with the original given E, we have Sinusoidal traveling waves in free space, propagating in the z directions with velocity,

3-18 Gauss’ Laws and the Continuity Equation (FEME, Secs. 3.4, 3.5, 3.6; EEE6E, Sec. 3.2)

3-19 GAUSS’ LAW FOR THE ELECTRIC FIELD

3-20

3-21 Divergence of D =  Ex. Given that Find D everywhere.

3-22 Noting that  =  (x) and hence D = D(x), we set

3-23 Thus, which also means that D has only an x- component. Proceeding further, we have where C is the constant of integration. Evaluating the integral graphically, we have the following:

3-24  00 From symmetry considerations, the fields on the two sides of the charge distribution must be equal in magnitude and opposite in direction. Hence, C = –  0 a

3-25

3-26 GAUSS’ LAW FOR THE MAGNETIC FIELD From analogy Solenoidal property of magnetic field lines. Provides test for physical realizability of a given vector field as a magnetic field.  

3-27 LAW OF CONSERVATION OF CHARGE   Continuity Equation

3-28 SUMMARY (4) is, however, not independent of (1), and (3) can be derived from (2) with the aid of (5). (1) (2) (3) (4) (5)

3-29 Curl and Divergence (FEME, Secs. 3.3, 3.6; EEE6E, Sec. 3.3)

3-30 Maxwell’s Equations in Differential Form Curl Divergence

3-31 Basic definition of curl is the maximum value of circulation of A per unit area in the limit that the area shrinks to the point. Direction of is the direction of the normal vector to the area in the limit that the area shrinks to the point, and in the right-hand sense.

3-32 Curl Meter is a device to probe the field for studying the curl of the field. It responds to the circulation of the field.

3-33

3-34

3-35 Basic definition of divergence Divergence meter is the outward flux of A per unit volume in the limit that the volume shrinks to the point. is a device to probe the field for studying the divergence of the field. It responds to the closed surface integral of the vector field.

3-36 Example: At the point (1, 1, 0) Divergence zero Divergence positive Divergence negative (a) (b) (c)

3-37 Two Useful Theorems: Stokes’ theorem Divergence theorem A useful identity

3-38

The End