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No Slide Title Slide Presentations for ECE 329, Introduction to Electromagnetic Fields, to supplement “Elements of Engineering Electromagnetics, Sixth Edition” by Nannapaneni Narayana Rao Edward C Jor[.]

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

Cartesian Coordinate System

Cartesian Coordinate System

x

y

z

a z

a y

a y

a x

a x

Right-handed system

xyz xy…

ax, ay, az are uniform unit vectors, that is, the

direction of each unit vector is same everywhere in space

ax ay az

ay az ax

az ax ay

1 12 2

 

z

P2

P1 R12

r1 r2

y x

O

Vector from P x y z, , to P x y z, ,

(1)

x2

x1 O

(x2 – x1)a x r1 z1 r2

P1 R12 P2

(z2 – z1)az

(y2 – y1)ay

y1

z2

y2 y

P1.8 A(12, 0, 0), B(0, 15, 0), C(0, 0, –20).

=

=

vector from B to C

= Vector from A to C

• Unit vector along vector from B to C

(0 – 0)ax  (0 – 15)ay  (–20 – 0)az

152  202 25

(c)Perpendicular distance from A to the line through B

and C

=

15 20

400 16

25

  

 



(Vector from A to C) (Vector from B to C)

BC

25

=

180az – 240ay – 300ax

25

12 2

dl dx a x  dy a y dz a z





z

a

x

a

y

a

dx

dy

dz

dl

 , , 

P x y z

Q x dx y dy z dz  

dl = dx a x + dy a y

= dx a x + f (x) dx a y

Unit vector normal to a surface:

an dl2

dl1

Curve 2 Curve 1

an  dl1 dl2

dl1 dl2

dl

dx dy = f (x) dx

z = constant plane

dz = 0

D1.5 Find dl along the line and having the projection dz on

the z-axis.

(a)

(b)

x 3, y –4

dx 0, dy 0

x  y 0, y  z 1

dx  dy 0, dy  dz 0

dy – dz, dx – dy dz

d dz dz dz

dz

(c)Line passing through (0, 2, 0) and (0, 0, 1).

x 0, dy

0 – 2  dz1 – 0

dx 0, dy – 2 dz

2 2

dz

(3) Differential Surface Vector (dS)

Orientation of the surface is defined uniquely by the

normal ± an to the surface.

For example, in Cartesian coordinates, dS in any plane

parallel to the xy plane is

dl1

dl2

an



x

y dS

dx dy

az

  1 2

sin





 l × l

(4) Differential Volume (dv)

In Cartesian coordinates,

dv dl1 • dl2 dl3

dv dx a x • dy a y dz a z

dx dy dz

dx

x

dl2

dl1

dl3 dv