Vector Notation in Electromagnetic Theory: Rectangular, Cylindrical, Spherical, Lecture notes of Electromagnetic Engineering

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EEE2 0 1:

Electromagnetic Theory

Lecture 2

VECTOR NOTATION

VECTOR NOTATION:

A Axa x Ayay Azaz

ˆ ˆ ˆ   



Rectangular or

Cartesian

Coordinate

System

x

z

y

A  B  Ax Bx  AyBy  AzBz

 

Dot Product

x y z

x y z

x y z

B B B

A A A

a a a

A B

ˆ ˆ ˆ

 

 

Cross Product

 

2

1

2 2 2

x y z

A  A  A  A



Magnitude of vector

(SCALAR)

(VECTOR)

x

y

z

A

x

A

y

A

z

A



B



Dot product:

x x y y z z

A  B  A B  A B  A B





Cross product:

x y z

x y z

B B B

A A A

x y z

A B

Back

Cartesian Coordinates

Page 108

VECTOR REPRESENTATION: CYLINDRICAL COORDINATES

Cylindrical representation uses: r , f , z

A Ar ar A a Azaz

ˆ ˆ ˆ   f f 



A  B  Ar Br  AfB f  AzBz

UNIT VECTORS:

r z

a a a

f

Dot Product

(SCALAR)

r

f

z

P

x

z

y

x

z

y

VECTOR REPRESENTATION: UNIT VECTORS

y

a

ˆ x

a

ˆ

z

a

ˆ

Unit Vector

Representation

for Rectangular

Coordinate

System

x

a

ˆ

The Unit Vectors imply :

y

a

ˆ

z

a

ˆ

Points in the direction of increasing x

Points in the direction of increasing y

Points in the direction of increasing z

Rectangular Coordinate System

r

f

z

P

x

z

y

VECTOR REPRESENTATION: UNIT VECTORS

Cylindrical Coordinate System

z

a

ˆ

f

a

ˆ

r

a

ˆ

The Unit Vectors imply :

z

a

ˆ

Points in the direction of increasing r

Points in the direction of increasing j

Points in the direction of increasing z

r

a

ˆ

f

a

ˆ

r z

a a a

f

q f

a a a

r

x y z

a a a

RECTANGULAR

Coordinate

Systems

CYLINDRICAL

Coordinate

Systems

SPHERICAL

Coordinate

Systems

NOTE THE ORDER!

r,f, z r,q ,f

Note: We do not emphasize transformations between coordinate systems

VECTOR REPRESENTATION: UNIT VECTORS

Summary

METRIC COEFFICIENTS

1. Rectangular Coordinates:

When you move a small amount in x -direction, the distance is dx

In a similar fashion, you generate dy and dz

Unit is in “meters”

METRIC COEFFICIENTS

2. Cylindrical Coordinates:

Distance = r df

x

y

d f

r

Differential Distances:

( dr, rd f , dz )

3. Spherical Coordinates:

Distance = r sinq df

x

y

d f

r sin q

Differential Distances:

( dr, rd q , r sin q d f )

r

f

P

x

z

y

q

METRIC COEFFICIENTS

AREA INTEGRALS

• integration over 2 “delta” distances

dx

dy

Example:

x

y

2

6

3 7

AREA =



7

3

6

2

dy dx = 16

Note that: z = constant

In this course, area & surface integrals will be

on similar types of surfaces e.g. r =constant

or f = constant or q = constant et c….

Representation of differential surface element:

z

ds dx dy a

ˆ

  



Vector is NORMAL

to surface

SURFACE NORMAL

Base

Vectors

A

1

r radial distance in x-y plane

Φ azimuth angle measured from the positive

x-axis

Z

0  r

0  2 

  z

Cylindrical Coordinates

 

 

 

ˆ

ˆ ˆ

,

ˆ ˆ

ˆ

,

ˆ

ˆ

ˆ

z r

z r

r z

r z

A a A rA A zA

ˆ

ˆ

ˆ ˆ

   



 

Back Pages 109-

( r, Φ, z)

Vector representation







    

2 2 2

r z

A A A A A A

  

Magnitude of A

Position vector A

Base vector properties

1 1

ˆ

ˆ

rr  zz

Dot product:

r r z z

A B  A B  A B  A B

f f





Cross product:

r z

r z

B B B

A A A

r z

A B

f

f

f

B

A

Back

Cylindrical Coordinates

Pages 109-