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

Ampère’s Circuital Law 3.1 Faraday’s Law and Ampère’s Circuital Law

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.

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.

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.

General Case Lateral space derivatives of the components of E Time derivatives of the components of B

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

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

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

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

Then, noting that we have from Thus, Then, noting that we have from

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