No Slide Title Slide Presentations for ECE 329, Introduction to Electromagnetic Fields, to supplement “Elements of Engineering Electromagnetics, Sixth Edition” by Nannapaneni Narayana Rao Edward C Jor[.]
Slide Presentations for ECE 329, Introduction to Electromagnetic Fields, to supplement “Elements of Engineering Electromagnetics, Sixth Edition” by Nannapaneni Narayana Rao Edward C Jordan Professor of Electrical and Computer Engineering University of Illinois at Urbana-Champaign, Urbana, Illinois, USA Distinguished Amrita Professor of Engineering Amrita Vishwa Vidyapeetham, Coimbatore, Tamil Nadu, India 3.1 Faraday’s Law and Ampère’s Circuital Law 3.1-3 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: If 0 y(x) dx 2, what is y(x)? There is no unique solution to this 3.1-4 However, if, e.g., y(x) = Cx, then we can find y(x), since then x 1 0 Cx dx 2 or C 2 0 2 or C 4 y(x) 4x On the other hand, suppose we have the following problem: dy If 2, what is y? dx Then y(x) = 2x + C Thus the solution is unique to within a constant 3.1-5 FARADAY’S LAW First consider the special case E E x (z,t) a x and H H y (z, t) a y and apply the integral form to the rectangular path shown, in the limit that the rectangle shrinks to a point y z (x, z) x z (x, z + z) S C (x + x, z) (x + x, z + z) x 3.1-6 d C E d l dt S B dS Ex z z d By x z x Ex z x x,z dt E Lim x z z x z Ex z x x z d Lim x z By Ex z t B dt y x, z x z x z 3.1-7 General Case E E x (x, y, z,t)a x E y (x, y, z,t)a y Ez (x, y, z, t)a z H H x (x, y, z,t)a x H y (x, y, z,t)a y Hz (x, y, z, t)a z Ez E y Bx – – y z t By E x E z – – z x t E y E x Bz – – x y t Lateral space derivatives of the components of E Time derivatives of the components of B 3.1-8 Combining into a single differential equation, ax ay az x Ex y Ey B – z t Ez B E – t Differential form of Faraday’s Law a x ay az x y z B Del Cross E or Curl of E = – t 3.1-9 AMPÈRE’S CIRCUITAL LAW Consider the general case first Then noting that d C E • dl – dt S B • dS E – (B) we obtain from analogy, t d C H • dl S J • dS dt S D • dS H J (D) t 3.1-10 D H J t Thus Special case: E E x (z,t)a x , H H y (z,t)a y ax a y az D 0 J z t Hy H y Dx – Jx z t Differential form of Ampère’s circuital law 3.1-11 H y Dx – Jx – z t Ex For E E0 cos 6 ×10 t kz a y in free space 0 , , J = , find the value(s) of k such that E satisfies both of Maxwell’s curl equations Noting that E E y (z,t)a y , we have from B E – , t 3.1-12 ax ay az z B – E – 0 t Ey Bx Ey t z E0 cos 6 10 t kz z kE0 sin 6 108 t kz kE0 Bx cos 10 t kz 6 10 3.1-13 Thus, kE0 B cos 10 t kz ax 6 10 B B H 0 4 10 kE0 cos 10 t kz ax 240 Then, noting that H H x (z,t)a x , we have from D H , t 3.1-14 ax D × H t Hx ay 0 az z Dy H x t z k E0 sin 10 t kz 240 3.1-15 k E0 Dy cos 10 t kz 1440 10 k E0 D cos 10 t kz a y 1440 10 D D E 9 10 36 k E0 cos 6 10 t kz a y 4 3.1-16 Comparing with the original given E, we have k E0 E0 4 k 2 E E0 cos 6 108 t 2 z a y Sinusoidal traveling waves in free space, propagating in the z directions with velocity, 10 (c) m s