Chap1cont Vector Anlysis

8/22/2019 Chap1cont Vector Anlysis

1/19

ELECTROMAGNETICS THEORY(SEE 2523)


2/19


3/19

An orthogonal system is one in which the coordinates are

mutually perpendicular.

Examples of orthogonal coordinate systems include the

Cartesian, cylindrical and spherical coordinates.

There must be three independent variables. e.g: u1, u2

and u3.

, and are unit vectors for each surface and the

direction normal to their surfaces.

1u

2u

3u

1.4 ORTHOGONAL COORDINATE SYSTEM


4/19

The cross product between the unit vector is:

While the dot product is:

2

1

3,

1

3

2,

3

2

1 uuuuuuuuu

13

3

2

2

1

1

01

3

3

2

2

1

uuuuuu

uuuuuu


5/19

Any vector can be represented as

The magnitude for is given by

A

33

22

11 AuAuAuA

2

3

2

2

2

1|| AAAAA

A


6/19

332211 BuBuBuB If and

the vector operations:

33

22

11 AuAuAuA


7/19

1.4.1: CARTESIAN COORDINATE SYSTEM

Defined by three variablesx, yand z. The ranges on the variables are:

A point P(x1, y1,z1) in coordinate system is located at the

intersection of the three surfaces which is determined by

x=x1, y = y

1and z= z

1.

Most of the problems in electromagnetics only can be

solved using line, surface and volume integral.

-< x < , -< y < and

-< z <


8/19

Fig. 1.10 shows the points P and Q whose coordinates are

P(x, y, z) and Q(x+dx, y+dy, z+dz).

The movement from point P to point Q cause the variables

vary fromxtox+dx, yto y+dyand zto z+dz. These changes will cause the differential volume elements

in Cartesian coordinates given by : dv = dxdydz

Differential length, is given by :d

dzzdyydxxd


9/19

Fig.1.10: Differential elemen t in Cartesian coord inate

dz

dy

dx

z

y

x

d


10/19

Three differential surfaces generated, sd

dz

dy

dx

z

y

x

d

)0

)0

)0

dzdxdyzzdszzsd

dydxdzyydsyysd

dxdydzxxdsxxsd

(when

(when

(when


11/19

Defined by three variables : r, and zand the unit vectors

are , and .r z

A variable r, at a point P is directed radially outward,

normal to the z-axis. is measured from the x-axis in the xy-plane to the r.

zis the same as in the Cartesian system.

1.4.2: CYLINDRICAL COORDINATE SYSTEM


12/19

The ranges on the variables are:

0 < r <

0 < < 2

-< z <

Fig. 1.11: A view o f a po int in

cy l indr ica l coordinate system.


13/19

A line, surface and volume will be generated when a single

variable, two variables and three variables, respectively arevaried.

When these changes are differential as shown in Fig.1.12,

we generate the following differential lines, surfaces and

volume.

Fig 1.12: Differential elements of th ecyl indr ical coo rdinate system


14/19

Yield a differential volume when the coordinate increase

from r, and zto r+dr, +ddan z+dz.

When the angle vary from to +d, the changes in thedistance is rd.

r

z


15/19

r

A variable ris defined as a distance from the origin to any point.

is defined as an angle between the +zaxis and the rline.

is an angle and exactly the same as in cylindrical coordinate

system.

1.4.2: SPHERICAL COORDINATE SYSTEM

Defined by three variables : r, and and the unit vectors

are , and .


16/19

The ranges on the variables are: 0 < r < 0 < <

0 < < 2

Fig. 1.13: A view of a poin t in

spherical coordin ate system


17/19

The changes in d and d will cause the

distance change to rdand rsind.

Fig. 1.14: Differential elements in s phericalcoord inate system


18/19

To reduce the length of certain equations found in

electromagnetics : an operator, called del or nabla.

In Cartesian coordinates for example:

The operator itself has no physical meaning unless it is

associated with scalars and vectors.

Should be noted that some operations yields scalars

while others yield vectors.

dz

dz

dy

dy

dx

dx

1.5: DEL OPERATOR


19/19

The following operations involving operator :

Documents

Chap1cont Vector Anlysis