Section 1 2TRƯỜNG ĐIỆN TỪ

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


1.2
Cartesian
Coordinate System


1.2-3

Cartesian Coordinate System

z
az
O
ay
ax
y
x

az
ax

ay
z
x

y


1.2-4

Right-handed system
a x a y a z
a y a z a x

xyz xy…

a z a x a y
ax, ay, az are uniform unit vectors, that is, the
direction of each unit vector is same everywhere in
space.


1.2-5

(1) Vector from P1  x1 , y1 , z1  to P2  x2 , y2 , z2 
z

r1  R12 r2

P1

R12 r2  r1

P2
r2

r1
O

 x2ax  y2a y  z2az 

  x1ax  y1a y  z1az 

R12

x

 x2  x1 ax   y2  y1 a y   z2  z1 az

y


1.2-6

z

R12

P1
(x 2 – x1)ax

x1

x

x2

r1
O

z1

r2
(y2 – y1)ay
y1

P2

(z2 – z1)az

z2

y2

y


1.2-7

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

Distance from B to C

=(0 – 0)a x  (0 – 15)a y  (–20 – 0)a z
= 15 2  20 2 25
(b) Component of vector from A to C along
vector from B to C
= Vector from A to C
• Unit vector along vector from B to C


1.2-8

 12ax  20az 

 15a



y

 20az 

 15a y  20az

400


16
25
(c)Perpendicular distance from A to the line through B
and C

(Vector
from A to C) (Vector from B to C)
=
BC


 12ax  20az  15a y 
25

20az 


1.2-9

=
(2)

180a z – 240a y – 300a x
25

= 12 2
Differential Length Vector (dl)
az
dl

P  x, y , z 


Q  x  dx, y  dy, z  dz 

dx

dz



ax



dy

ay

dl dx a x  dy a y dz a z


1.2-10

dl
dx

y = f(x)

z = constant plane

dy = f (x) dx
dz = 0

dl = dx ax + dy ay
= dx ax + f (x) dx ay
Unit vector normal to a surface:
dl1 dl 2
an 
dl1 dl 2

an

dl2
dl1

Curve 2
Curve 1


1.2-11

D1.5

Find dl along the line and having the projection dz on
the z-axis.
(a)
x 3, y –4

dx 0, dy 0
 dl dz a z

x  y 0, y  z 1
(b)
dx  dy 0, dy  dz 0
dy – dz, dx – dy dz

d l dz ax  dz a y  dz az
ax  a y  az  dz


1.2-12

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

dy
dz
x 0,

0 – 2 1 –0
dx 0, dy – 2 dz
d l  2 dz a y  dz az
 2a y  az  dz


1.2-13

(3)

Differential Surface Vector (dS)

dS dl1 dl2  sin 

 d l1 × d l2

an





dl2
dS
dl1

Orientation of the surface is defined uniquely by the
normal ± an to the surface.

dS  dS a n dl1 dl 2 a n dl1 dl 2
For example, in Cartesian coordinates, dS in any plane
parallel to the xy plane is
az

dx dy a z dx a x dy a y

dy
dx dS

x

y

1.2-14

(4)

Differential Volume (dv)

dv dl1 • dl 2 dl3

dl2

dl3

dv

dl1

In Cartesian coordinates,
dv dx a x • dy a y dz a z
dx dy dz

dz

z

dy
dx

y
x