5 1 Gradient, Laplacian, and the Potential Functions Slide Presentations for ECE 329, Introduction to Electromagnetic Fields, to supplement “Elements of Engineering Electromagnetics, Sixth Edition” by[.]
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 5.1 Gradient, Laplacian, and the Potential Functions 5.1-3 Gradient and the Potential Functions ax × A x Ax ay y Ay az z Az × A × A x × A y × A z x y z x x Ax 0 y y Ay z z Az 5.1-4 Since B 0, B can be expressed as the curl of a vector Thus B = A A is known as the magnetic vector potential Then × E = × A t A × t 5.1-5 A × E + 0 t A E+ t A E = t is known as the electric scalar potential ax ay + az y z x ax ay az x y z is the gradient of 5.1-6 ax × x x ax x x 0 ay y y ay y y az z z az z z 5.1-7 Basic definition of Q x dx, y dy, z dz d dl d d l P x, y , z From this, we get d an dn Maximum rate of increase of an direction of the maximum rate of increase, which occurs normal to the constant surface 5.1-8 B × E = t D × H = J + t D = B = B = × A A E = t and using A = t 2 t Potential function A equations A J t 5.1-9 Laplacian of scalar 2 A = A × × A Laplacian of vector In Cartesian coordinates, 2 x y z 2 A = 2 Ax ax 2 Ay a y 2 Az az