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

1.1 Vector Algebra

(1) Vectors (A) vs. Scalars (A) Magnitude and direction Magnitude only Ex: Velocity, Force Ex: Mass, Charge (2) Unit Vectors have magnitude unity denoted by symbol a with subscript Useful for expressing vectors in terms of their components.

(3) Dot Product is a scalar A • B = AB cos a B Useful for finding angle between two vectors.

(4) Cross Product is a vector A B = AB sin a B is perpendicular to both A and B. Useful for finding unit vector perpendicular to two vectors. an

where (5) Triple Cross Product in general.

(6) Scalar Triple Product is a scalar.

Volume of the parallelepiped

D1.2 A = 3a1 + 2a2 + a3 B = a1 + a2 – a3 C = a1 + 2a2 + 3a3 (a) A + B – 4C = (3 + 1 – 4)a1 + (2 + 1 – 8)a2 + (1 – 1 – 12)a3 = – 5a2 – 12a3

(b) A + 2B – C = (3 + 2 – 1)a1 + (2 + 2 – 2)a2 + (1 – 2 + 3)a3 = 4a1 + 2a2 – 4a3 Unit Vector =

(c) A • C = 3 1 + 2 2 + 1 3 = 10 (d) = = 5a1 – 4a2 + a3

A • (B C) = (3a1 + 2a2 + a3) • (5a1 – 4a2 + a3) = 3 5 + 2 (–4) + 1 1 (e) = 15 – 8 + 1 = 8 Same as A • (B C) = (3a1 + 2a2 + a3) • (5a1 – 4a2 + a3) = 3 5 + 2 (–4) + 1 1 = 15 – 8 + 1 = 8

P1.5 D = B – A ( A + D = B) E = C – B ( B + E = C) D and E lie along a straight line.

What is the geometric interpretation of this result?

Another Example Given Find A.

To find C, use (1) or (2).