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

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

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:
There is no unique solution to this.

4. 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.

5. 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.

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

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

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

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

11. 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

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

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