Chapter One
1.1
Vector analysis
Vector Representation
This introductory chapter is a review of mathematical concepts required
for the electromagnetism. It is assumed that the reader is already familiar with
elementary vector analysis.
Physical quantities that we deal with in electromagnetism can be scalars or
vectors.
A scalar is an entity which only has a magnitude. Examples of scalars are
mass, time, distance, electric charge, electric potential, energy, temperature etc.
A vector is characterized by both magnitude and direction. Examples of vectors
in physics are displacement, velocity, acceleration, force, electric field, magnetic
field etc.
A vector is represented by a straight line of specific length having an
arrow head which shows its direction. This straight line is called the
representative line of that vector. Vector variables are sometimes notated with
arrow above them (particularly in handwritten physics formulae), and sometimes
notated by using a bold-face font (particularly in print).
A field is a quantity which can be specified everywhere in space as a
function of position. The quantity that is specified may be a scalar or a vector.
For instance, we can specify the temperature at every point in a room. The room
may, therefore, be said to be a region of “temperature field” which is a scalar
field because the temperature T(x, y, z) is a scalar function of the position. An
example of a scalar field in electromagnetism is the electric potential.
In a similar manner, a vector quantity which can be specified at every point
in a region of space is a vector field. For instance, every point on the earth may
be considered to be in the gravitational force field of the earth. We may specify
the field by the magnitude and the direction of acceleration due to gravity (i.e.
force per unit mass) g(x, y, z) at every point in space. As another example
consider flow of water in a pipe. At each point in the pipe, the water molecule
has a velocity ⃗ (x, y, z). The water in the pipe may be said to be in a velocity
field. There are several examples of vector field in electromagnetism, e.g., the
electric field ⃗⃗ , the magnetic flux density ⃗⃗ etc.
1
Chapter One
Vector analysis
Orthogonal coordinate systems
Rectangular (Cartesian) Coordinates P (x, y, z)
In Cartesian (rectangular) coordinates a point P is
specified by x, y, and z, where these values are all measured
from the origin (see figure at right).
Cylindrical Coordinates
P (r, Φ, z)
In cylindrical coordinates a point P is specified by r, Φ,
z, where Φ is measured from the x axis (see figure at right).
Spherical Coordinates
P (r, θ, Φ)
In spherical coordinates a point P is specified by r, θ,
Φ, where r is measured from the origin, θ is measured from
the z axis, and Φ is measured from the x axis (see figure at
right).
1.2
Cartesian Coordinates
A vector at the point P is specified in terms of three mutually perpendicular
components with unit vectors 𝑥̂, 𝑦̂, 𝑎𝑛𝑑 𝑧̂ , (also called 𝑖̂, 𝑗̂ and 𝑘̂, or aˆ x, aˆ y , and aˆ z ).
Vector representation

A  Axaˆ x  Ayaˆ y  Azaˆ z

or A  xˆ A x  yˆ A y  zˆ A z
Magnitude of A
   
A  A  A   A 2x  A 2y  A 2z
Position vector A
xˆ x1  yˆ y1  zˆ z1
2
Chapter One
Vector analysis
Base vector properties
xˆ  xˆ  yˆ  yˆ  zˆ  zˆ  1
xˆ  yˆ  yˆ  zˆ  zˆ  xˆ  0
xˆ  yˆ  zˆ
yˆ  zˆ  xˆ
zˆ  xˆ  yˆ
Dot product:
 
A  B  AxBx  Ay By  Az Bz
Cross product:
xˆ
 
A  B  Ax
yˆ
zˆ
Ay
Az
Bx
By
Bz
1.3
Cylindrical Coordinates
Cylindrical representation uses: r, Φ, z

A  Araˆ r  Aaˆ   Azaˆ z
 
A  B  ArBr  AB  AzBz
Dot Product (Scalar)
aˆ
Unit Vectors
1.4
r
aˆ 
aˆ z 
Spherical coordinates
Spherical representation uses: r, θ, Ф

A  Ar aˆ r  A aˆ   A aˆ 
 
A  B  ArBr  AB  AB
aˆ
r
aˆ 
aˆ  
Dot Product (Scalar)
Unit Vectors
3
Chapter One
Vector analysis
Vector Representation: Unit Vectors
Rectangular Coordinate System
Unit Vector Representation for
Rectangular Coordinate System
The Unit Vectors imply:
â x
Points in the direction of increasing x
â y
Points in the direction of increasing y
â z
Points in the direction of increasing z
Cylindrical Coordinate System
The Unit vectors imply:
â r
Points in the direction of increasing r
â 
Points in the direction of increasing Ф
â z
Points in the direction of increasing z
Spherical Coordinate System
The Unit Vectors imply:
â r
Points in the direction of increasing r
â 
Points in the direction of increasing θ
â 
Points in the direction of increasing Ф
4
Chapter One
1.5
Vector analysis
Differential length, area, and volume
1- Cartesian Coordinates
When you move a small amount in x-direction, the distance is dx. In a similar
fashion, you generate dy and dz.
Differential quantities:
Length (displacement):

d l  xˆ dx  yˆ dy  zˆ dz
Area (normal area):

d sx  xˆ dydz

d sy  yˆ dxdz

d sz  zˆ dxdy
Volume:
dv  dxdydz
2- Cylindrical Coordinates ( r, Φ, z):
0r
r
radial distance in x-y plane
Φ
azimuth angle measured from the positive x-axis
Z
  z  
Vector representation


ˆ A   zˆ A z
A  aˆ A  rˆA r  
Magnitude of A

 
A   A  A   A r2  A 2  A 2z
Position vector A
rˆr1  zˆ z1
5
0    2
Chapter One
Vector analysis
Base vector properties
ˆ  zˆ ,
rˆ  
ˆ  zˆ  rˆ ,

ˆ
zˆ  rˆ  
Dot product:
 
A  B  Ar Br  A B   A z B z
Cross product:
rˆ
 
A  B  Ar
ˆ
A
zˆ
Az
Br
B
Bz
Length (displacement):

ˆ rd  zˆ dz
d l  rˆdr  
Area (normal area):

d sr  rˆrd dz

ˆ drdz
d s  

d sz  zˆ rdrd 
Volume:
dv  rdrd dz
3- Spherical Coordinates (R, θ, Φ)
Vector representation

ˆ A R  ˆ A   ˆ A 
AR
Magnitude of A
   
A  A  A   A 2R  A 2  A 2
6
Chapter One
Vector analysis
Position vector A
ˆ R1
R
Base vector properties
ˆ 
ˆ 
ˆ,
R
ˆ 
ˆ,
ˆ R

ˆ 
ˆ
ˆ R

Dot product:
 
A  B  A R B R  A B  A B
Cross product:
ˆ
R
 
A  B  AR
ˆ
A
ˆ
A
BR
B
B
Differential quantities:
Length (displacement):

ˆ dl R  
ˆ dl   
ˆ dl 
dl  R
ˆ dR  
ˆ Rd  
ˆ R sin d
R
Area (normal area):

ˆ dl dl   R
ˆ R 2 sin dd
d sR  R

ˆ dl R dl   
ˆ R sin dRd
d s  

ˆ dl R dl   
ˆ RdRd
d s  
Volume:
dv  R 2 sin dRdd
dl R  dR
dl   Rd
dl   R sin d
7
Chapter One
Vector analysis
Cartesian to Cylindrical Transformation
A r  A x cos   A y sin 
A    A x sin   A y cos 
Az  Az
r   x2  y 2
  tan 1 (y / x)
[
zz
rˆ  xˆ cos   yˆ sin 
ˆ  xˆ sin   yˆ cos 
zˆ  zˆ
The fundamental parameters of the rectangular, cylindrical, and spherical coordinate
systems are summarized in the following table:
8