3 3 Curl and Divergence Slide Presentations for ECE 329, Introduction to Electromagnetic Fields, to supplement “Elements of Engineering Electromagnetics, Sixth Edition” by Nannapaneni Narayana Rao Edw[.]
3.3-1 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 3.3 Curl and Divergence 3.3-3 Maxwell’s Equations in Differential Form B × E = t D × H = J t Curl ax ∂ × Α ∂x Ax D B ay ∂ ∂y Ay az ∂ ∂z Az A y A z A x Divergence A = x y z 3.3-4 Basic definition of curl Lim C A d l × A = an S S max × A is the maximum value of circulation of A per unit area in the limit that the area shrinks to the point Direction of × A 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 3.3-5 Curl Meter is a device to probe the field for studying the curl of the field It responds to the circulation of the field 3.3-6 3.3-7 a 2x for x v0 a az v 2x a v0 az for x a a ax ay az × v x y z vz × vy vz ay x a negative for x positive for a x a 2v0 a a y 2v0 a y a 3.3-8 Basic definition of divergence Lim A= v A d S v is the outward flux of A per unit volume in the limit that the volume shrinks to the point Divergence meter 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 3.3-9 Example: At the point (1, 1, 0) (a) x 1 y 1 a y Divergence positive (c) x a y y ax Divergence zero (b) x z x y 1 z x y 1 Divergence negative z y 3.3-10 Two Useful Theorems: Stokes’ theorem A d l = × A d S C S Divergence theorem A d S = A dv S V A useful identity × A 3.3-11 ax ay az × Α x Ax y Ay z Az × A = × A x × A y × A z x y z x x Ax y y Ay z 0 z Az